1. technology and computing
2. mp3 and midi

THE RELATIONSHIP BETWEEN THE ANGLE OF RELEASE AND
THE VELOCITY OF RELEASE IN THE SHOT-PUT, AND THE
APPLICATION OF A THEORETICAL MODEL TO
ESTIMATE THE OPTIMUM ANGLE OF RELEASE
By
Andreas V. Maheras
B.A., University of Athens, 1986
M.A., Western Michigan University, 1990
Submitted to the Department of Health, Physical Education
and Recreation and the Faculty of the Graduate School of
the University of Kansas in partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
Dissertation Committee:
Carole J. Zebas_____________
Chairperson
James D. LaPoint____________
Matthew Adeyanju____________
Thomas E. Mulinazzi_________
D. B. Tracy__________________
Dissertation defended: June 13, 1995
ABSTRACT
Given the relative importance of the angle of
release in the shot-put event, the estimation of the
optimum angle of release in a way that the distance
thrown is maximized, is sought.
Theory has shown that
the optimum angle of release fluctuates between 41 and 43
degrees.
Biomechanical analyses have indicated that
shot-putters release their shots at angles between 32 and
38 degrees, with few cases reported in the above 40
degree category.
As a result, a discrepancy between
theory and practice has been observed.
The purpose of
the study was to examine whether a thrower specific
dependency of the angle of release on the velocity of
release exists in the shot-put event.
Such dependency
would probably make the theoretical model inapplicable to
real life shot-putting.
The agreement between the real
life models and the theoretical model was also
investigated.
Five male, collegiate, shot-putters from four
universities in the state of Kansas were employed.
Each
shot-putter threw under five different angles of release,
from a very low to a very high.
Each thrower attempted
10 throws in each of these five angles for a total of 50
throws.
High speed videography was used to record the
throwers’ attempts.
A Peak Performance system analysis
along with a FORTRAN computer program were used to obtain
ii
the velocity, the angle, and the height of release for
each of the 250 throws filmed.
Correlational analyses were used to obtain the
relationship between the obtained parameters.
Multiple
regression techniques were used to estimate the real life
angle of release.
For all five subjects there was a significant
relationship between the angle of release and the
velocity of release thus, showing dependency of the angle
of release on the velocity of release.
For all five
subjects there was no agreement between the real life
angles and those estimated by the theoretical model.
It was concluded that at present, the theoretical
model is not applicable in the shot-put event.
The
optimum release conditions other than velocity depend
crucially on how the maximum achievable release velocity
is functionally related to the other release conditions.
iii
ACKNOWLEDGMENTS
I would like to thank my advisor Dr. Carole Zebas
for her help and encouragement in doing this project, and
also for her leadership and guidance during the course of
my studies.
Thanks to my committee members Dr. Matthew Adeyanju
and Dr. James LaPoint, of the H.P.E.R department, Dr.
Dick Tracy of the E.P.R department, and Dr. Tom Mulinazzi
of the Engineering department, for their leadership.
Special thanks go to Dr. Jesus Dapena of Indiana
University, and Dr. Elvin Eltze of Fort Hays State
University, for sharing their valuable insights
pertaining to the present study.
I also thank Dr.
Raymond McCoy of the college of William and Mary, Dr.
Francis Mirabelle of the United States Air Force
Ballistics department and, Dr. Charles Votaw of Fort Hays
State University, for their cooperation with this study.
To coaches Jim Krob of Fort Hays State University,
Laurie Trapp of the University of Kansas, Will Wabaunsee
of Emporia State University, and Tom Hays of Wichita
State University, thanks for their cooperation.
I would like to thank my brothers Panagiotis and
Manthos Maheras for sharing their knowledge with me over
the years.
I would also like to acknowledge Mr. Petros
Papageorgiou, coach, professor and mentor who first
introduced me to the “secrets” of the human movement.
iv
I must also acknowledge the authorities of the
Ministry of Culture of Greece for the opportunity they
gave me to study abroad.
Last, but not least, a big thanks goes to those who
participated in the study and made it possible.
Andreas Vassilios Maheras
v
DEDICATION
I would like, here, to especially acknowledge Mrs.
Donna Fleischacker/Maheras, and also my parents Zoe and
Vassilios Maheras, for the abundance of support they have
provided me over the years.
This study is dedicated to
them.
vi
TABLE OF CONTENTS
ABSTRACT ................................ ii
ACKNOWLEDGMENTS ......................... iv
DEDICATION .............................. vi
LIST OF TABLES .......................... xi
LIST OF FIGURES ....................... xiii
Chapter 1.
THE PROBLEM .............................. 1
Introduction ........................... 1
The Relationship Between the Angle of
Release and Velocity of Release and
the Application of the Model ......... 3
The Theoretical Model .................. 6
Statement of the Purpose .............. 10
Scope of the Study .................... 11
Assumptions ........................... 14
Significance of the Study ............. 15
Hypotheses ............................ 19
Definition of Terms ................... 19
Chapter 2.
REVIEW OF RELATED LITERATURE ............
Introduction ..........................
Review of Literature Related to the
Shot-Put event with Emphasis on the
Basic Mechanics and also the Angle,
Velocity and Height of Release .....
Basic Mechanical and Other
Principles .......................
Methods of Estimating the Optimum
Angle of Release .................
The Geometrical Method ...........
The Range and Height Equation ....
The General Range Equation .......
The Release Angle in Shot-Putting ..
Height of Release ..................
Velocity of the Shot During the
Final Effort and at Release ......
Review of Literature Related to
Projectile Motion ..................
Projectiles in General ................
Gravitational Force and Air
Resistance ......................
22
22
22
22
29
29
31
32
33
39
40
46
46
46
Table of Contents--Continued
The Shot as Projectile ............. 47
Velocities and Resultant ........... 48
vii
Complementary Angles ............... 49
Path of a Projectile Released from
Ground Level .................... 51
Velocities and Resultant ........ 52
Peak Height ..................... 54
Total Time ...................... 54
Range ........................... 54
Path of a Projectile Released
from a Height (h) ............... 55
Total Time ...................... 56
Range ........................... 56
Forces Affecting Projectiles .......... 56
Gravity ............................ 57
Aerodynamical Forces ............... 59
Drag Coefficient ................... 60
Frontal Sectional Area ............. 61
Fluid Density ...................... 61
Velocity ........................... 62
Relative Acceleration .............. 62
Drag Effect on Different Masses .... 63
Vector Representation in Projectile
Motion ............................. 65
Resultant Vector ................... 65
Vector Components .................. 66
Factors Affecting a Projectile’s
Maximum Horizontal Displacement .... 68
Point of Release at Ground Level ... 69
Point of Release at a Height (h)
Above the Ground ................. 69
Velocity of Release .............. 72
Angle of Release ................. 72
Height of Release ............... 73
Optimum Angle of Release when Projectile
is Released from a Height (h) .... 73
Summary ................................ 75
Chapter 3.
METHODOLOGY ............................. 77
Research Design ....................... 77
Table of Contents--Continued
The Relationship Between the Angle,
Velocity, and the Height
of Release ..........................
Preliminary Investigation .............
Subjects ..............................
Instrumentation .......................
viii
79
79
81
83
Collection of Data .................... 84
Filming Procedures .................. 84
Testing Procedures .................. 86
Analysis of Data ...................... 89
Film Analysis ....................... 89
Statistical Analysis ................ 92
The Theoretical Model ................. 94
Gravity ............................ 94
Velocity and Position Vectors ...... 95
The Range of a Projectile Released
from a Height (h) .............. 100
The Model for the Optimum Angle of
Release ........................ 102
Validation of the Model .............. 111
Comparison with the Range and
Height Equation .................. 112
Result for the 45 Degree Angle ..... 114
Chapter 4.
RESULTS ................................
Introduction .........................
Findings .............................
Relationships and Correlation
Coefficients .....................
Scattergrams .......................
The Real Life Regression Models
and the Significance of the
Regression Equations .............
The Assumptions of the Regression
Models .........................
Validity and Numerical Solution
of the Model .....................
The Real Life vs. The Theoretical
Angles of Release ..............
Obtained Ranges ....................
116
116
119
120
121
124
125
131
137
147
Table of Contents--Continued
Chapter 5.
DISCUSSION ............................. 153
The Relationship Between the Angle and
the Velocity of Release ........... 153
Possible Causes for the Observed
Discrepancy and Suggested
Remedies .......................... 156
The Issue of Collinearity with the
Regression Models ................. 158
The Optimum Angle Model .............. 161
The Assumptions of the Theoretical
ix
Model .......................... 161
Force due to Air Resistance ....... 162
The Projectile is a Point Mass .... 163
The Earth is Non-rotating ......... 163
The Gravitational Force is Constant
and Acts Perpendicularly to the
Earth’s Surface ............... 164
Motion Occurs in a Plane .......... 164
The Construction of the Model ........ 164
The Importance of the Angle of
Release ........................ 165
Chapter 6.
SUMMARY, CONCLUSIONS AND
RECOMMENDATIONS ........................
Summary ..............................
Conclusions ..........................
Recommendations ......................
171
171
176
178
References .............................
Appendix A .............................
Appendix B .............................
Appendix C .............................
Appendix D .............................
179
188
197
199
201
x
LIST OF TABLES
1.
Correlation coefficients between the obtained
parameters for each of the subjects ............ 120
2.
The obtained regression equations for each of the
subjects ....................................... 124
3.
Real and theoretical obtained angles for the first
subject ........................................ 138
4.
Real and theoretical obtained angles for the
second subject ................................. 139
5.
Real and theoretical obtained angles for the third
subject ........................................ 140
6.
Real and theoretical obtained angles for the
fourth subject ................................. 141
7.
Real and theoretical obtained angles for the fifth
subject ........................................ 142
8.
Tests of significance of the regression
coefficient (b) and the constant (a) ........... 143
9.
Ranges obtained using the analysis data and the
actual measured ranges for the first subject ... 148
10. Ranges obtained using the analysis data and the
actual measured ranges for the second subject .. 149
11. Ranges obtained using the analysis data and the
actual measured ranges for the third subject ... 150
12. Ranges obtained using the analysis data and the
actual measured ranges for the fourth subject .. 151
xi
13. Ranges obtained using the analysis data and the
actual measured ranges for the fifth subject ... 152
14. Collinearity diagnostics for the regression
equation for each of the subjects .............. 160
15. Velocities, angles, and heights of release for
the first subject ....................... appendix A
16. Velocities, angles, and heights of release for
the second subject ...................... appendix A
17. Velocities, angles, and heights of release for
the third subject ....................... appendix A
18. Velocities, angles, and heights of release for
the fourth subject ...................... appendix A
19. Velocities, angles, and heights of release for
the fifth subject ....................... appendix A
xii
LIST OF FIGURES
1.
A typical trajectory of a projectile released
from ground level ................................ 7
2.
The right angle vector system in projectile
motion .......................................... 14
3.
The optimum angle of release in shot-put ........ 31
4.
The frontal area and center of gravity of the
shot ............................................ 48
5.
The resultant force and its components .......... 50
6.
Path of a projectile in the absence of air
resistance ...................................... 52
7.
Path of a projectile when released from a height
(h) above the ground ............................ 56
8.
The effects of gravity upon the trajectory of a
projectile ...................................... 58
9.
Velocity components and resultant during shot-put
release ......................................... 65
10. Analysis of vector components from the
resultant ....................................... 68
11. The relationship between release and landing
angles in projectile motion when the point of
release is higher than that of landing .......... 74
12. The proposed experimental arrangement of the
equipment for collection of the data ............ 85
13. The release parameters to be calculated ......... 89
14. The right angle vector system in projectile
xiii
motion .......................................... 97
15. The maximum value of (x) and the tangent of
the angle (a) at peak .......................... 103
16. Scattergram of the relationship between the angle
of release and the velocity of release for the
first subject .................................. 121
17. Scattergram of the relationship between the angle
of release and the velocity of release for the
second subject ................................. 122
18. Scattergram of the relationship between the angle
of release and the velocity of release for the
third subject .................................. 122
19. Scattergram of the relationship between the angle
of release and the velocity of release for the
fourth subject ................................. 123
20. Scattergram of the relationship between the angle
of release and the velocity of release for the
fifth subject .................................. 123
21. Normal probability plot for the data of the first
subject ........................................ 126
22. Normal probability plot for the data of the second
subject ........................................ 127
23. Normal probability plot for the data of the third
subject ........................................ 127
24. Normal probability plot for the data of the fourth
subject ........................................ 128
xiv
25. Normal probability plot for the data of the fifth
subject ........................................ 128
26. Randomly distributed residuals for the data of the
first subject .................................. 129
27. Randomly distributed residuals for the data of the
second subject ................................. 129
28. Randomly distributed residuals for the data of the
third subject .................................. 130
29. Randomly distributed residuals for the data of the
fourth subject ................................. 130
30. Randomly distributed residuals for the data of the
fifth subject .................................. 131
31. Variation of the theoretical optimum angle of
release with changing height and velocity of
release in the shot-put ........................ 133
32. Variation of projectile range with changing angle
and height of release when the velocity is
12 m/sec. ...................................... 135
33. Variation of projectile range with changing angle
and height of release when the velocity is
13 m/sec. ...................................... 136
34. Variation of projectile range with changing angle
and height of release when the velocity is
14.5 m/sec. .................................... 137
35. Variation of the real versus the theoretical angle
of release for the first subject ............... 144
36. Variation of the real versus the theoretical angle
xv
of release for the second subject .............. 145
37. Variation of the real versus the theoretical angle
of release for the third subject ............... 145
38. Variation of the real versus the theoretical angle
of release for the fourth subject .............. 146
39. Variation of the real versus the theoretical angle
of release for the fifth subject ............... 146
40. The extra distance gained by the thrower over the
circle boundaries .............................. 147
xvi
CHAPTER 1
The Problem
Introduction
Descriptive analysis, presently and in the recent
past, constitutes the major contribution of biomechanics
to the understanding and improvement of sports skill
execution.
A review of research in biomechanics, as
related to sport in general and to track and field events
in particular, clearly indicates that by far the majority
of studies are centered on descriptive mechanical
evaluation of a specific performance.
These research
endeavors to apply a “scientific” approach to the
understanding of human movement have led to improved
associations between the researcher and the practitioner.
Indeed, new techniques have emerged as the result of
scientific analysis (Hay, 1993).
Moreover, there is an ongoing process of attempting
to improve performance by developing new techniques in
most sports.
Having established the nature of the
technique that should be used in a given sport, coaches
face the task of both detecting and correcting potential
errors during an athlete’s execution of a particular
sport skill.
Biomechanical analysis here, again, can
help a coach in finding the cause of the error and
finally correcting it.
In biomechanics research, Newtonian mechanics and
basic or advanced mathematics are often used to allow
1
the researcher to arrive at sound and meaningful results.
For example, velocity, acceleration, force, power, work,
energy and other important elements of a biomechanical
analysis of a sport skill are obtained through the use of
simple mathematical formulas/equations.
In analyzing the throwing events of the track and
field athletics, for example, researchers often apply
principles from the kinematics part of mechanics, and in
particular those principles dealing with projectile
motion.
In the shot-put event a spherical iron ball is
the object projected into the air and is called the
projectile.
The shot-putter releases the shot using his
own arm and hand mainly by pushing the shot through a
series of skillful movements.
In this case, the projectile is released at a
certain initial velocity, at a certain angle, from a
certain height, all depending upon the ability, skill,
stature, technique and other characteristics of the
thrower.
For the coach and the thrower him or herself,
an understanding of the factors that govern the behavior
of projectiles is of critical importance (Hay, 1993).
The objective of the shot-put action is to throw a
7.257 kilograms ball as far as possible while following
the rules that govern the event.
The distance the shot
is thrown is determined by its conditions at release
namely, the velocity, height, and angle of release.
Changes in the velocity of release will affect the
2
distance thrown more than equal changes in either the
angle or height of release (Gregor, McCoy, & Whitting,
1990).
Although not as critical as the velocity of release,
the angle of release is important (Gregor et al., 1990).
The theoretical optimum angle of release for the shot-put
event should be between 41 and 43 degrees (Hay, 1993).
Presently, however, analyses of shot-put athletes have
shown release angles between 34 and 42 degrees (McCoy,
1989).
A marginal to significant discrepancy is then
observed between the theoretical optimum angles and the
angles achieved in practice by the athletes.
Researchers and coaches have empirically attempted
to give an explanation for this phenomenon, which
nevertheless has not been studied.
The Relationship Between the Angle and the Velocity of
Release, and the Application of the Theoretical Model
Given the empirical information that the velocity of
release might be dependent on the angle of release, the
examination of the relationship between these two
components is sought.
Moreover, the height of release
and its relationship with the angle and the velocity of
release should be examined.
In addition, given the potential development of a
theoretical model to estimate the optimum angle of
release, a question logically arises whether the model
will be readily applicable and useful for the shot-
3
putter.
A similar well known phenomenon occurs in the
long jump event where, given the average velocities and
the average height of take off the jumpers achieve, the
optimum angle of take off should be in the vicinity of 43
degrees (Hay, 1985).
In practice, however, it has been
shown that the athletes use angles between 15 and 22
degrees, at least 20 degrees below the theoretical
optimum (Hay, 1985).
It is evident that the athletes cannot maintain
their individual high velocities at take off that they
achieve under 20 degrees, if a 40 degree angle is used.
The result is that the long jumpers prefer to use
narrower angles of take off in order to be able to
maintain a higher velocity at take off.
As will be seen
later, the velocity of release or take off, is the single
most important factor affecting a projectile’s range.
In the shot-put event, the phenomenon mentioned
above is not as evident.
The estimated optimum angle of
release may very well fall within the abilities of the
shot-putter while maintaining a high velocity.
At the
present time, however, it is not known whether this is
the case.
Hay (1985) wrote pertaining to this issue:
When considering projectile motion as applied to
the throwing events, it should be understood that
it may not be possible for the thrower to achieve
the same release velocity over a wide range of
release angles.
In the case of a shot-putter for
4
example, if he/she concentrates all his/her strength
in a horizontal direction, gravity will not have a
tendency to reduce the release velocity of the shot.
If, however, the shot-putter attempts to exert
some force vertically, gravity will oppose that
attempt.
While the shot-putter pushes the shot in
an upward direction, gravity tends to pull the shot
downward at the same time.
The result of this
opposition of the gravitational force is a decrease
in the velocity of release that the shot-putter can
impart to the shot.
So it may happen that while a shot-putter,
having found what is the optimum angle of release
for his/her abilities, is trying to increase the
angle with which he/she releases the shot, the
advantage to be gained from this action may be lost
or more than lost by the effect that gravity has
upon the shot, which is mainly a reduction in the
velocity of release the thrower is able to produce.
(p. 43-44).
Hay (1985) also postulated that “presently there is
little information in the literature on the aspect of
projectile motion in sports and until relevant
experimental evidence becomes available there is little
that can be done beyond being aware of the problem”
(p.44).
The Theoretical Model
5
In attempting to mechanically and mathematically
describe projectile motion, specific mathematical
formulas have been derived and presented in the
biomechanics literature (Dyson, 1977; Ecker, 1985; Hay,
1993; Kreighbaum & Barthels, 1990).
Usually, formulas
that describe the principles of vertical motion of the
projectiles, the horizontal motion of the projectiles as
well as formulas that describe the range of a projectile
are presented.
Projectile motion has been traditionally studied in
various courses with the assumption that the point of
release of the projectile is at the same level with the
point of landing (see figure 1).
Under this condition,
the optimum angle of release (a), should always be 45
degrees for maximum horizontal range (Bueche, 1972).
This horizontal range (x) is given by the equation,
X = v2 sin2a / g
(Hay, 1993)
where, v = the velocity of release, a = the angle of
release, and g = the gravitational acceleration.
trajectory
A
B
point of release
Figure 1.
point of landing
A typical (parabolic) trajectory of a
6
projectile when the level of release is the same as that
of landing.
During many events, and during the throwing events
in particular, however, the point of projection is at a
different, in fact higher, level than that of landing.
This is of course due to the nature of the throwing
events which do not allow for a release from a ground
level.
Under these circumstances, the horizontal range
for a particular projection/throw is given by,
X = [v2 sina cosa + vcosa Ö(v sina)2 + 2gh] /g
(Hay,1993)
where, h = the height of release.
The optimum angle of release, however, depends upon
both the height of release and the velocity of release.
A theoretical model to estimate the optimum angle of
release from velocity and angle of release data is then
sought, to allow for a comparison between this model and
a model derived from real life data.
Hay (1993), wrote
characteristically that, “regrettably the optimum angle
of projection cannot be found as readily as in the case
in which release and landing are at the same level” (p.
38).
In addition, Hay (1985, 1993) presented a table with
the optimum angles of release for the shot-put event when
the height of release and the speed are known.
This
table, however, is general and gives a solution of
optimum angles that correspond to some combinations of
height and velocity of release and not to any specific
7
combination.
In short, if one wants to know the optimum
angle of release using values other than those presented
in the table, he/she might not be able to use it.
Other
authors (Dyson, 1977; Ecker, 1985) have presented similar
tables having the same limitation.
In various literature sources (Bowerman & Freeman,
1991; Ecker, 1985; Hay, 1993), the optimum angle of
release for the shot-put event has been determined to be
between 41 and 43 degrees, the exact figure depending
upon the height and velocity of release.
Nevertheless, a
theoretical model could help one obtain a solution for
the optimum angle that corresponds to any specific
combination of height and velocity of release.
The mathematical modeling which has been used to
describe projectile motion involves consideration and
knowledge of basic Newtonian mechanics as well as
application of mathematical procedures in order to arrive
at a satisfactory answer.
According to Berry, Burghes,
Huntley, James, & Moscardini (1986), there are usually
five steps for the development of a theoretical model.
Step 1, involves identifying a physical system of
interest where the object is the estimation of motion.
Steps 2, 3 and 4 require the construction of a
mathematical model for the physical system, the
formulation and solution of appropriate differential and
other equations, followed by an interpretation of the
results in terms of the original problem.
8
In step 5, the
results are compared to direct observations of the
physical system.
The complexity of a theoretical model of a physical
system and the difficulties associated with its solution
increases with the influence of external forces.
In the
projectile motion, however, the influence of external
forces are relatively limited, at least as far as the
projection of the shot is concerned (Greenwood, 1983).
The gravitational force is the major factor, while the
force due to air resistance (drag) could also be
considered but in practice is often ignored in projectile
motion analysis (Timoshenko & Young, 1956).
Statement of the Purpose
The purpose of this study was, a) to examine the
relationship between the angle of release and the
velocity of release and also the relationship between the
angle of release and the height of release in the shotput event, b) based upon the above relationships, to
obtain a real life equation which would estimate the
angle of release from velocity and height of release
data, and c) develop and validate a theoretical model to
estimate the optimum angle of release from velocity of
release and height of release data.
This is for the
purpose of comparing the results from this model with
those obtained from real life throwing.
The agreement
between theory and practice was thus, assessed.
Specifically, it was an attempt,
9
1.
To examine the relationship between the angle of
release and the velocity of release, and also the angle
of release and the height of release in shot put, so as
to determine, primarily, whether any dependence of the
velocity of release on the release angle exists.
Based
on the above relationships, equations predicting the
angle of release from the velocity and the height of
release were obtained.
2.
To mathematically model the motion of the shot
as a projectile with respect to its optimum angle of
release, taking into consideration the mechanical
conditions that are prevalent during the event.
3.
To examine the validity of the model by
comparing it to other models/methods relative to
projectile motion.
4.
To examine the potential application of the
model to real life situations by experimentally testing
it and applying it to a number of shot-putters.
Scope of the Study
The main concern of the study was the examination of
the relationship between the angle of release and the
velocity of release in the shot-put event, and also
between the angle of release and the height of release.
The study was delimited to 5 subjects, 10 throws for
each athlete for each of the 5 different kinds of angles
of release that were studied.
The 5 subjects were
college age shot-putters using the rotational technique
10
of throwing the shot, and were selected from 4
universities in the State of Kansas.
All subjects were
given equal opportunity to achieve maximum performance as
they executed their puts.
All the throws were made in a
non competitive environment.
Subsequently, equations to estimate the angle of
release from velocity of release and height of release
data obtained from real life shot-put throwing, were
derived for each subject.
Another concern was the
development of a theoretical model to estimate the
optimum angle of release from velocity of release and
height of release data and its subsequent application in
real life situations.
The creation of such a model helped in comparing the
obtained real life angle of release with the theoretical
optimum angle of release.
The application of the
theoretical model took place by experimentally measuring
various parameters of a shot-putter’s throwing action,
such as velocity of release, height of release and angle
of release.
Subsequently, data from the real life
throwing situations were compared to the data obtained
from a theoretical model developed for this purpose.
Data from the shot-putters were obtained with the use of
high speed videography.
Theoretical data were obtained
with the use of the theoretical model.
The main concern for the development of the model
was to mathematically examine the various mechanical
11
components present during the release and flight of the
implement, and how they relate to the optimum angle of
release in order to achieve maximum horizontal
displacement/distance.
A system of two axes, a vertical (y) and a
horizontal (x) was assumed (see figure 2).
In this
system, (a) is the angle of release, (0) the point of
release, (j) the vertical vector component, (i) the
horizontal vector component, (v0) the initial velocity,
or velocity of release and, (mg) the weight of the
implement, or gravitational force.
The gravitational
force was assumed to be the only force acting on the
implement after it was released.
y
u0
m g
j
a
0
Figure 2.
i
x
The right angle vector system in projectile
motion.
The primary mathematical simulation of the x, y
system, a first and a second order derivative of the
gravitational force, was derived through Newtonian
mechanics.
Trigonometry principles were also considered
throughout the construction of the model.
12
Finally,
mathematical integrations and differentiations were
employed along with equation solving procedures to arrive
at the desired model.
The model was validated through
simple comparisons where the model was tested in two
conditions at which the result for the optimum angle of
release is known as, for example, is the case when the
level of release of a projectile is the same as the level
of landing.
Assumptions
The following assumptions and idealizations were
made to simplify the equations of motion:
1.
The gravitational force is the only force acting
on the shot after it has been released and it is in
flight trajectory.
2.
The projectile is a body possessing finite
volume and a definite surface configuration.
The concern
would then be with the motion of the mass center.
3.
The earth is non rotating.
4.
The gravitational field is constant and acts
perpendicular to the surface of a flat earth.
5.
The air offers no resistance to motion.
Consequently, motion occurs as it would in a vacuum.
6.
Motion occurs in a plane.
Significance of the Study
Many studies have already been conducted in the area
of track and field and, more specifically, in shotputting.
Up to the present, investigators have been
13
concerned with the kinematic or kinetic parameters of the
throwing action in general, with no attempt to
specifically study the relationship between the angle of
release and the velocity of release in a way that
adequate information would be presented to facilitate one
in assessing the application of a theoretical model on
the shot-put event.
McCoy et al., (1989) wrote that
apparently no studies have ever been conducted to
investigate the effect of the release velocity on the
release angle.
Morris (1971) postulated that the term technique
defines the most rational and economical utilization of
kinematic and dynamic possibilities and relationships.
It then follows that a detailed description of the
kinematic parameters involved in the release action of
shot-putting is necessary, and if basic understanding of
the technique of the activity is to be achieved, special
emphasis must be placed on securing detailed information
on such action.
The study revealed important information pertaining
to the proper delivery of the shot-put.
More
specifically, the study examined and attempted to
determine the relationship between the velocity of
release and the angle of release in real life shotputting.
If velocity at release can be maintained at a
variety of angles of release, then the proposed
14
theoretical model to estimate the optimum angle of
release can be used and applied to real life situations,
particularly for the improvement of a shot-putter’s
performance.
Unlike the long jump event, the theoretical
model can be applicable in practice in the shot-put
event, and this theoretical model then, can indeed form
the foundation of a direct link between the researcher,
coach, and athlete with the object of increased
understanding of human movement and improved sports skill
performance as it relates to the shot-put event.
On the other hand, a disagreement between the
theoretical model and what really happens in practice,
will show that although the mechanical laws prescribe a
certain set of rules to be followed for optimum
performance, the application of such laws to the athlete
will not bring the desired improvement.
As it often
happens in practice, the limitations of the human body
will not have allowed for a direct application of theory
to practice and the model might be what it originally
was, just theory.
Moreover, a discrepancy between the theoretical
model and what happens in real life, showing inability of
the athletes to throw according to the mechanical laws,
will potentially cause researchers, coaches, and
athletes, to begin examining and investigating new ways
of training with the purpose of overcoming the observed
handicap.
15
In addition, the theoretical model presented in this
study will lead to an improved general understanding of
the projectile motion, and will provide information about
the exact fluctuation of the optimum angle of release in
the shot-put event.
To this extent, the application of
this model appears to be unbounded since it can include
all potential combinations of height and velocity of
release.
Furthermore, it can be included in the
compendium of a series of formulas/equations that have
been presented in the literature and describe projectile
motion.
J. Hay (personal communication, February 1995)
addressed the fact that the question involving the
application of a theoretical model to estimate the
optimum angle of release in real life shot-putting in
particular, and to the whole host of sports and physical
activities in general, is an old one, but still one that
has not been addressed in any thorough fashion.
The theoretical model for the optimum angle of
release then, if applicable, could form the foundation of
a direct link between the researcher, coach, and athlete
with the object of increased understanding of human
movement and improved sports skill performance as it
relates to the shot-put event in particular, and
potentially to other throwing activities, in general.
Hypotheses
The following hypotheses were formulated based on
16
the review of the literature and the results anticipated
from the study.
1.
The higher (steeper) the angle of release, the
lower the velocity of release.
2.
The higher (steeper) the angle of release, the
higher the height of release.
3.
The theoretical model will not be applicable to
the shot-put event.
Definition of Terms
Terminology employed in the text of this study is
consistent with common usage in mathematics and physics.
The following are terms defined for this study.
Acceleration.
Time rate of change in velocity.
Angle of release.
Is the angle formed between the
velocity vector of a projectile after its release and a
horizontal reference.
A theoretical model
refers to a mathematical
description through equations, of the relationship
between the angle of release and a) the velocity of
release, b) the height of release, and c) the
gravitational acceleration.
Center of gravity.
Point in the body through
which resultant force of gravity acts.
Derivative.
For a function (f) whose domain
contains an open interval about x 0, m0 is the derivative
of (f) at x0 providing that,
1.
for every m < m0, the function
17
f(x) - [f (x0)+ m(x-x0)]
changes sign from negative to positive at x0.
2.
for every m > 0, the function
f(x) - [f (x0)+ m(x-x0)]
changes sign from positive to negative at x 0.
Differentiation.
The process of finding the
derivative of a function.
Displacement.
Dynamics.
Distance and direction of movement.
Study of motions of bodies and forces
acting to produce the motions.
Gravity.
The attraction that earth has for all
other bodies.
Height of release.
The vertical distance between
the point of release of a projectile and a horizontal
reference through the level of landing.
Kinematics.
Study of the relation between
displacement, velocity, and acceleration.
Mass.
Resistance of change in linear velocity.
The
amount of matter equals weight divided by acceleration of
gravity (m = W / g).
Mechanics.
Study of the action of forces on bodies.
Parallelogram law.
Resultant of two concurrent
forces is the diagonal of a parallelogram whose sides are
the original forces.
Resultant.
Simplest equivalent force system that
will replace any given system.
Vector quantity.
Quantity having both magnitude and
18
direction.
Velocity of release.
The velocity a projectile
possesses exactly at the instant it is released.
19
CHAPTER 2
Review of Related Literature
Introduction
This study encompasses both theoretical and research
aspects of sports biomechanics.
Literature specifically
relevant to the present investigation was reviewed and is
presented in this chapter in two main sections: a) review
of literature related to the shot-put event and more
specifically, to the angle, velocity, and height of
release and, b) review of literature related to
projectile motion.
Review of Literature Related to the Shot-Put Event, with
Emphasis on the Basic Mechanics and Principles and Also
the Angle, Velocity and Height of Release
In this section, literature relevant to the angle of
release in the shot-put event along with information
about the height of release and the velocity of release
will be presented.
Literature pertaining to the basic
biomechanics and principles that apply to the shot-put
event will also be presented here.
Basic Mechanical and Other Principles
Black (1987) postulated that to achieve maximum
range in the shot-put event, the athlete must, a) release
the shot with maximum velocity, b) release it from an
optimum angle and, c) release it from an optimum height.
The thrower should possess two important characteristics
to achieve maximum distance.
20
These characteristics are,
a) technical and, b) speed and strength characteristics.
Because technique is affected by the physical
characteristics of the performer, importance should be
placed in the training of the thrower.
Dunn (1987), postulated that the emphasis during
training should be in proportion with those areas of the
body that are responsible for producing maximum
performance.
According to Dunn (1987), 50% of the
training regimen should be devoted to the muscles of the
legs, 30% to the trunk area and 20% to the arms.
The
strength exercises needed for high level shot-putting are
the inclined press, the squat, the clean and jerk and the
push press exercises (Ward, 1976).
O’Shea (1986), stated that:
A good thrower must be trained to quickly exert
forces in multiple directions utilizing several body
joints simultaneously.
Thus, throwers need to
concentrate on developing explosive reactive
ballistic movement through a full range of movement
and less on absolute strength and muscle mass.
The
greatest transfer of muscular power developed
through weight training to the throws will result
from the execution of explosive torso-rotational
lifting movements, such as full range total body
lifts that require the thrower to concentrate and
think in terms of both strength and speed.
Athletic
type lifts meeting this criteria are power snatches,
21
power cleans, high pulls, squats, and the push
press.
(p. 28).
Brown (1985) hinted that the key to success in shot
putting is leg power.
He suggested that energy flows
from the legs up to the trunk and shoulders.
This
indicates that the use of lifts such as the jerk should
be beneficial in the training of the shot-putter.
Hakkinen, Kauhanen, & Komi (1986), indicated that an
overemphasis on the development of absolute strength may
hinder rather than help performance.
This is because
training using high intensity and slow contraction speed
may cause the neuromuscular system to adapt to a slower
development of force production.
Absolute strength is
necessary for high shot-put performance but only
providing that it contributes to explosive strength.
Hakkinen & Komi (1985), demonstrated that
improvements in explosive force production require
specific training with high contraction velocities.
It
was further indicated that, the Olympic style of weight
training is more appropriate for shot-putters than power
lifting style of training.
Moreover, plyometric
exercises can improve shot-putting performance.
Pagani (1981), summarized some basic mechanical
principles applicable to the shot-put event.
He
addressed that the primary factors affecting the distance
of a throw are the velocity, the angle and, the height of
release, the most important being the velocity of
22
release.
The acceleration of the shot should be gradual, in a
straight line with a direction toward the impact area
(Vigars, 1979).
The force application should cover as
much distance as possible in the least time possible.
The length of the path is largely decided by the
difference of the height of the shot at the start and at
the end of the delivery phase (O’Shea & Elam, 1984).
McDermott (1986), suggested that improved throwing
performance will result from a decrease in the time
required to complete an eccentric/concentric coupling
(from a forced stretching of the muscle to a voluntary
contraction) during the execution of the throw.
According to Pagani (1981), in order to achieve
proper summation of the forces applied to the shot, the
larger and slower muscle groups should be employed
initially followed by the smaller, weaker muscles.
Although the various muscle contractions may start at
different times during the throwing action, they should
nevertheless, end together.
O’Shea & Elam (1984) postulated that maximum
throwing power is generated through the multi-linked
muscle-skeletal system beginning with the body’s power
zone which includes the large muscle groups of the legs,
hips, gluteals, lower back and abdominals.
The resulting
power from these muscles is transferred to the upper
back, the shoulders and finally to the throwing arm and
23
the shot.
Moreover, the application of force over a full
range of multiple joint movement requires timed
coordination of acceleration and deceleration of all the
body segments and also a proper sequence of activation in
order to produce maximum velocity of the throwing hand
(O’Shea & Elam, 1984).
Ariel (1972) addressed that the relationship between
maximum velocities and accelerations is important during
the throwing action.
For the best throws Ariel (1972)
analyzed during an Olympic training camp, he reported
that the velocity of the last segment should be at its
maximum close to the release, not at the release, even
though the deceleration of the segment begins prior to
release.
The rapid deceleration of the arms, for
example, just prior to the moment of release will result
in an increase of the force applied to the shot.
To
achieve this, the various joints of the body should be
stabilized for the instance of release.
For a good throw
the action of the link system should be coordinated thus,
creating the basis for properly timed and coordinated
accelerations and decelerations of all body segments.
The sequence of the action should be from the left foot
to the right throwing hand.
Ariel (1973a) analyzed the top 6 finalists during
the 1972 Olympic games.
He postulated that the better
throwers utilize their body segments to produce maximum
velocity in their throws.
They do that by properly
24
timing the deceleration of the lower body segments.
Excessive “follow through” action, used by the American
throwers in their effort to increase the acceleration of
the shot, may be detrimental to the throw since it does
not allow for summation of the forces towards the
direction of the throw.
The achievement of the summation
of the forces towards the direction of the throw requires
properly timed stopping or deceleration of the front leg,
trunk, free arm and shoulders.
This action is critical
in achieving the maximum resultant force in the direction
of the throw.
The most important muscle groups involved in
generating maximum power in shot-putting are, a) the
quadriceps and the calf muscles, b) the hip flexors and
extensors, gluteals, abdominals, spinal erectors,
obliques and, c) the upper back, shoulders, arms and
chest muscles (O’Shea & Elam, 1984).
The deceleration of
the thighs the trunk and the shoulders is important for a
successful throw.
As these segments decelerate, there is
a transfer of momentum to the shot (Pagani, 1981).
Theoretically, a double contact with the ground,
during the final phase of release, produces a better
release and is more efficient.
In practice, however,
film analyses of world class shot-putters reveal that
they break contact with the ground at least with their
rear foot at the moment of release (Dyson, 1977; Pagani,
1981).
Hay (1993), stated that to date there appear to
25
be no objective basis for an answer to these questions.
It seems that practice shows that if the athlete were to
reduce the vertical forces he/she can exert, by
maintaining contact with both feet on the ground, the
resulting loss in the release velocity of the shot is
probably greater than that due to being in the air during
the phase of delivery.
The velocity of the glide should not be too great.
Even highly skilled shot-putters are unable to utilize
more than 30-40% of their glide velocity.
The glide
velocity of world class throwers ranges from 2.49 m/sec.,
to 2.99 m/sec.
On the other hand, the release velocity
using the glide can reach 13.10 to 13.41 m/sec., and
12.99 m/sec., when throwing from a standing position
(Pagani, 1981).
Because momentum equals mass times velocity, an
increase in functional mass, the muscle mass, is an
effective way of increasing momentum (Pagani, 1981).
The optimum angle of release when the point of
release is at the same level as that of landing is 45
degrees (Hay, 1993).
The optimum angle of release when
the point of release is at a higher level than that of
landing depends on both the height and the velocity of
release (Hay, 1993; McCoy, Gregor & Whiting, 1989).
Generally, as the height of release increases, the
release angle decreases and becomes smaller than 45
degrees.
As the velocity increases, the release angle
26
increases reaching the value of 45 degrees (Hay, 1993).
Methods of Estimating the Optimum Angle of Release
The geometrical method.
To estimate the optimum
angle of release from range and height data, Dyson (1977)
in a paper devoid of mathematics noted that the optimum
angle using this method can be found by bisecting the
angle between a vertical line drawn through the thrower’s
hand at the instant of release, and a straight line
connecting this point and the point of landing (see
figure 3).
This method requires that the distance thrown is
known beforehand along with the height of release.
For
this reason this method of estimating the exact optimum
angle of release cannot be used when only the height of
release and the velocity of release is known.
Although
Dyson (1977) did not offer any proof, this rule of thumb
seems to be exactly true.
Given the above method to estimate the theoretical
optimum angle of release, one could allege that the
optimum angle that a shot-putter should use can be
estimated.
If it is known that an athlete is, for
example, a 50 feet thrower then, by knowing the height
from which he/she releases the shot, one can estimate the
optimum angle of release that corresponds to a 50 feet
throw, using the geometrical method.
Pertaining to this
issue, F. Mirabelle (personal communication, January
1995) postulated that the knowledge of the optimum angle
27
of release in this case is of little use for the athlete.
It is obvious that the athlete can already throw 50 feet;
that is why the 50 feet distance was chosen in the first
place.
Consequently, he/she cares less what the optimum
angle for a distance that he/she can already achieve is.
The athlete’s primary goal is to always achieve maximum
distance.
The geometrical method of estimating the optimum
angle of release can be useful for the military, where
projectiles are to be projected towards a predetermined,
fixed distance.
Of course, the accuracy of this method
will also depend upon the ability of the person to
accurately draw the lines and measure the angles.
optimum angle
of release
Figure 3.
The optimum angle of release in shot-put.
The Range and Height Equation.
Zatsiorsky (1981)
presented an equation to mathematically estimate the
optimum angle of release based on range and height data.
According to Zatsiorsky (1981),
x = h tan2a
28
and hence,
tan2a = x / h
where, a = the optimum angle, x = the distance achieved,
h = the height of release.
Obviously, this method of estimating the optimum
angle of release is more efficient and accurate than the
geometrical method when range and height data are known.
The general range equation.
Given the following
equation for range when the point of release is higher
than that of landing,
X= [v02 sina cosa +v0 cosa Ö(v0 cosa)2 + 2gh] /g(Hay, 1993)
a specific height (h) and a specific velocity (v) can be
assumed, and the optimum angle can be estimated through
repeated solution of the above equation.
Using this
method, one can use various values for angles with their
respective sines and cosines to repeatedly solve the
equation for range.
Finally, the angle is chosen as the
optimum the one which will give the longest/maximum
range.
Obviously, this could be a laborious way to
estimate the optimum angle of release.
This method of
the estimation of the optimum angle of release is used by
the United States military in order to determine the
optimum angle of release of the various military
projectiles (F. Mirabelle, personal communication,
January 1995).
In the literature (e.g., Ecker, 1985; Hay, 1993),
29
there have been tables that present the exact optimum
angle of release.
However, these tables are restricted
to a number of specific combinations of velocities and
heights of release.
The values for the optimum angle
presented in these tables have probably been estimated
using one of the three methods described above.
In a paper by Poole & Bangerter (1981) a similar
table was presented.
More specifically, they attempted
to offer a table with the optimum angle of release for
selected velocities from a height of release of 2.13
meters.
According to estimations, the figures presented
in this table were only a good approximation of the real
theoretical figures.
The Release Angle in Shot-Putting
Although not as important as the velocity of
release, the angle of release is also important and
should be considered in all analyses (Gregor et al.,
1990).
Pertaining to the angle of release Pagani (1981)
postulated that the optimum angle of release should be
approximately 40 to 41 degrees whereas Vigars (1979), was
more liberal suggesting an approximate 45 degree angle of
release.
Bashian, Gavoor, & Clark (1982), suggested that
for the range of typical heights and velocities the
optimum angle turns out to be close to 41 degrees in all
cases.
Zatsiorsky (1981), presented the results from the
30
analysis of 8 competitive shot-putters where it is shown
that the angles of release fluctuated from 39 to 42
degrees.
Similar results were presented by Bashian,
Gaur, & Clarck (1982), Cureton (1939), and Koutiev
(1966).
Zatsiorsky (1990), postulated that modern research
has shown that in real conditions during the shot-put
event, the release angle of elite athletes was about 3637 degrees, much lower than it was initially thought.
Dessureault (1976), analyzed 13 shot-putters of
various abilities.
He reported angles of release
fluctuating between 27.3 and 41 degrees.
Obviously,
greater variation was observed in Dessureault’s study.
McCoy, Gregor, Whiting, Rich, & Ward (1984) found
average release angles of 37 degrees for 37 elite male
throwers, and 36 degrees for 14 elite female throwers.
McCoy (1990), reported angles of release between 35.1 and
40.6 degrees with an average of 37.4 degrees for a total
of 15 throws executed by 6 American top male shotputters.
McCoy (1990) reported a correlation of 0.40
between the angle of release and the distance thrown for
the 6 throwers analyzed.
McCoy (1990), also analyzed 7
elite female shot-putters.
Their angles of release
fluctuated between 29.2 and 42.4 degrees with an average
of 34.6 degrees.
The correlation between the angle of
release and the distance thrown was -0.17.
Susanka and Stepanek (1987) in a rare 3 dimensional
31
analysis of the 8 finalists in the men’s shot-put event
during the 2nd world track and field championships in
Rome, Italy, reported angles of release between 34.1 and
41 degrees with an average of 37.2 degrees.
In the same
analysis, the 8 finalists in the women’s shot-put event
exhibited angles of release between 34.1 and 42.6 degrees
with an average of 38.2 degrees.
Groh, Kubeth, & Baumam (1966) found similar results.
McCoy in a personal communication (February, 1995)
postulated that he had observed angles of release as low
as 32 degrees.
McCoy et al. (1984), indicated that a possible
reason for these lower angles (as compared to the
theoretical optimum angles) is that shot-putters train
predominantly for strength and power using the bench
press movement, which may make the throwers stronger in a
body position that produces lower release angles.
They
further suggested that if this is true then, perhaps the
athletes should train with the arm in a position that
produces the optimum release angle nearer to the vicinity
of 42 degrees.
A similar explanation was given by Gregor et al.,
(1990).
They postulated that the reason for the lower
observed angles of release can be discussed in light of
the performance of the shoulder and muscles surrounding
the shoulder in order to maximize output of this system
and minimize injury.
32
Hay (1993) offered a theoretical reasoning for a
possible deviation of the real angle of release from the
estimated optimum.
He postulated that,
When considering projectile motion as applied to
the throwing events, it should be understood that
it may not be possible for the thrower to achieve
the same release velocity over a wide range of
release angles.
In the case of a shot-putter for
example, if he/she concentrates all his/her strength
in a horizontal direction, gravity will not have a
tendency to reduce the release velocity of the shot.
If, however, the shot-putter attempts to exert
some force vertically, gravity will oppose that
attempt.
While the shot-putter pushes the shot in
an upward direction, gravity tends to pull the shot
downward at the same time.
The result of this
opposition of the gravitational force is a decrease
in the velocity of release that the shot-putter can
impart to the shot.
So it may happen that while a shot-putter,
having found what is the optimum angle of release
for his/her abilities, is trying to increase the
angle with which he/she releases the shot, the
advantage to be gained from this action may be lost
or more than lost by the effect that gravity has
upon the shot, which is mainly a reduction in the
velocity of release the thrower is able to produce.
33
(p. 43-44).
Zatsiorsky (1990), gave a similar explanation. He
stated that,
The angle and velocity are correlated and the putter
can achieve greater velocity only when angles are
less.
The cause of this seems to be at least partly
in the following: the force an athlete applies to
the shot is spent as to accelerating the shot rather
than to compensate for its weight.
The force the
athlete spends to accelerate the shot is always less
than the force applied to the shot.
The greater the
angle of release is, the greater part of force is
spent to compensate its weight and the less to
accelerate it.
That is why it is easier to
accelerate the shot when the angle of release is
less, than when it is greater. (p.120).
McWatt (1982) found that maximum force can be
exerted to the shot in the 10 to 20 degree angle instead
of the generally accepted optimum angle of 40 degrees.
Some limitations, however, in this study were, a) that
the subjects were basically untrained, and b) the
movement used for the testing of the experiment was
isometric in nature as opposed to a dynamic one occurring
in shot-putting.
Dyson (1977) stated that forces exerted during the
throwing action in shot-put give a greater release
velocity when they are directed nearer to the horizontal.
34
It seems that factors responsible for the velocity
conflict with those that give an optimum angle of
delivery.
The differences between the theoretical angle
and the angle observed in practice may be due to the
dependence of the release velocity on the release angle
(McCoy et al., 1984).
Height of Release
The height of the object at release is determined by
the size of the athlete and by his/her body position.
A
large thrower with a low trunk and arm position may have
a lower release height than a smaller thrower who is
fully extended and off the ground at release.
To
optimize the distance of the throw, the height of release
should be as high as possible (McCoy et al., 1989).
Nevertheless, it seems that the height of release is the
least important factor affecting the range of the shot
(Gregor et al., 1990).
McCoy et al. (1984) analyzed the then ten top shotputters in the United States, and found a mean height of
release of 2.29 meters.
These high release heights were
attributed to the stature of the athletes who had a mean
height of 1.92 meters.
Moreover, all the throwers were
off the ground at the time of release.
McCoy (1990), reported heights of release between
1.89 and 2.41 meters with an average of 2.18 for 6 elite
male American shot-putters.
For 7 elite female shot-
putters the height of release fluctuated between 1.71 and
35
1.99 meters with an average of 1.83 meters.
The
correlation between the height of release and the
distance thrown was 0.08 for the male, and -0.42 for the
female shot-putters.
Susanka & Stepanek (1987), found that the height of
release fluctuated between 2.13 and 2.29 meters with an
average of 2.22 meters for the 8 finalists of the men’s
shot-put event during the 2nd world track and field
championships.
During the same championships the 8
finalists in the women’s shot-put event exhibited heights
of release between 1.94 and 2.24 meters with an average
of 2.07 meters.
Dessureault (1976), reported heights of release in
the range of 1.83 to 2.20 meters.
Francis (1948),
reported similar values for the height of release as well
as Koutiev (1966), and Zatsiorsky et al. (1981), in a
classical two dimensional study of the kinematics of the
shot-put event.
Velocity of the Shot During the Final Effort and at
Release
The most important release parameter is the velocity
of release (Atwater, 1979; Hay, 1993).
Small changes in
the release velocity produce more change in the distance
achieved than do similar changes in the height or angle
of release.
McCoy (1990), found a high correlation for
the resultant velocity of release and distance thrown
(r = 0.75).
McCoy (1990), also indicated that increases
36
in the resultant velocity were brought about through
increases in the horizontal, and not the vertical
components of the release velocity.
At the start of the throwing action, the implement’s
velocity increases.
The velocity then decreases as the
body moves into the power position to begin the delivery
phase.
When this phase starts, at the power position,
the velocity is approximately 10% of its release
velocity.
The implement’s velocity increases tenfold
during its delivery (McCoy et al., 1984).
The velocity
of release fluctuates between 13 and 14 meters per second
for high caliber throws (Hay, 1993).
Ariel (1979), studied 13 American and 6 European
shot-putters the latter all finalists during the 1976
Olympic games. The resultant velocity of the shot at
release for the Olympic competitors was between 13.56 and
14.1 m/sec., while for the American shot-putters the
resultant linear velocity fluctuated between 11.58 and
13.33 m/sec.
Ariel (1979) concluded that the most
important factor in shot-putting is the velocity of the
shot at release.
He also postulated that for maximum
velocity at release, there must be a summation of forces
from the various phases of the throw and the various body
segments.
McCoy (1990), reported the velocities of release for
6 elite male American shot-putters.
The values for the
velocity of release fluctuated between 12.05 and 13.72
37
m/sec., with an average of 12.8 m/sec.
Seven elite
American female shot-putters in the same study exhibited
velocities of release between 9.92 and 11.29 m/sec., with
an average of 10.8 m/sec.
The correlations between the
horizontal velocity, the vertical velocity, the resultant
velocity (velocity of release) and the distance thrown
were 0.52, 0.63, and 0.67 respectively for the male
throwers, and 0.18, -0.15, and 0.06 respectively for the
female throwers.
Groh et al., (1966) used cinematography to study the
velocity and the acceleration of the shot during the
final effort.
The maximum velocity was reported to be
11.99 meters per second whereas the maximum acceleration
was 219.9 meters per second 2.
Marhold (1974), reported that technically perfect
athletes were superior to others in demonstrating a
better ability to increase the velocity of the shot
immediately before the beginning of the push-off phase.
Marhold (1974) further added that another feature of a
perfect throw is a relatively high level of velocity at
the beginning of the push-off phase while at the same
time a maximum final acceleration is reached.
For throws
over 21 meters values between 3.5 and 3.7 ms -1 have been
obtained.
Moreover, analyses of top German shot-put
throwers showed that they tend to perform better with
increasingly greater accelerative paths during the pushoff phase.
To achieve this, the German athletes tend to
38
reduce shot-paths during the transitional phase while
they shorten the time during the same period (Marhold,
1974).
Dyson (1977) showed that the shot’s acceleration
dropped off at the arm thrust, whereas it further
increased again when the athlete added a final wristsnapping action as the shot was loosing contact with the
fingers.
Koslov (1969), examined the utilization of
force during the final effort in order to achieve maximum
speed.
He used 12 skilled athletes and a simulation of
the movement at an angle of 40 degrees with a weight
corresponding to that of the shot.
Koslov (1969)
determined three phases: (1) the beginning of the final
effort would be very fast, (2) the end of the final
effort would be very fast, (3) the combination of the two
previous phases.
He concluded that the velocity achieved
by the shot is greater when the shot-putters exerted
their maximum power at the end of the final effort.
Consequently, maximum power should be applied at the end
of the throwing action.
Herman (1962), classified 6 male shot-putters as
good, average, and poor in terms of performance.
He
found that the most effective way to achieve the highest
velocity at release, is by gradually increasing the
acceleration during the movements across the circle, with
an even larger increase during the final shoulder and arm
action.
39
Marhold (1964), studied the role of the feet contact
with the ground upon the release velocity.
He postulated
that it is impossible to put the shot with only the inner
forces of the body once both feet are off the ground.
There should be a decrease in the final velocity of
release.
His estimation of that loss was in the nature
of 0.3048 meters/second.
On the other hand, the height
of release increased from 0.10 to 0.15 meters.
This gain
might have offset the loss in the final velocity.
Ariel (1973b), used one male subject, who was a
former world record holder, to study the significance of
the displacement of the center of gravity as it relates
to the contact with the ground.
For the less successful
throw this particular subject exhibited no vertical
displacement of the center of gravity whereas at the same
time he lost contact with the ground while the shot was
still in his hand.
For the best throw the same athlete
maintained contact with the ground throughout the release
of the shot.
The difference in distance between the two
throws studied was 5 feet.
Ariel (1973) concluded that
contact with the ground is a critical factor in the shotput technique.
Anderson (1972), examined the kinematic parameters
involved in shot-putting.
subjects.
For this purpose he used 4
The main results of his study were, a) that
the best performer exhibited his highest rate of linear
acceleration at the instant of release.
40
The rest of the
subjects exhibited decreasing rates of acceleration at
the instant of release, b) The higher the velocity of
release the longest the throw achieved by each of the
subjects, c) given the height of release, the angle of
release, and the velocity of release, it seems that the
linear and vertical velocities at release are the most
critical in achieving the longest throw.
In the same study Anderson (1972) showed that the
athlete who exhibited the longest throw, also exhibited
the highest height of release, the steepest launch angle,
and moreover, the fastest linear, horizontal and vertical
velocities at the time of release.
The opposite was true
for the thrower who exhibited the worst performance.
Review of Literature Related to Projectile Motion
In this section, literature specifically relevant to
projectile motion was reviewed and is presented here in
five parts: (1) projectiles in general, (2) forces
affecting projectiles, (3) vector representation in
projectile motion, (4) factors affecting a projectile’s
maximum horizontal displacement, and (5) optimum angle of
release.
Projectiles in General
The term projectile describes any body that is
impelled by some force and then continues to move through
the air by its own inertia (Glashow, 1981).
A thrown
ball, an airborne long jumper, and a released shot or
discus are some examples of projectiles.
41
Gravitational Force and Air Resistance
The common element shared by all projectiles is the
gravitational force that constantly acts on them, and is
identical for all cases.
The gravitational force acting
on a projectile causes it to follow a parabolic
trajectory given that the projection is not exactly
vertical (Glashow, 1981).
Air resistance may alter the
parabolic course of the projectile.
The degree to which
air will affect the parabolic course depends upon the
projectile’s size, weight, shape, nature of its surface,
and most important its speed (Ivey & Hume, 1974).
Traditionally, the effects of air resistance have
been ignored in the various attempts to solve and explain
projectile problems.
Timoshenko and Young (1956)
postulated that consideration of the effects of air
resistance greatly complicates the solution of various
projectile problems.
They further added that air
resistance in some cases can have a significant effect
particularly when the velocity of the projectile is high.
The Shot as a Projectile
The study of projectiles encompasses the application
of principles related to both linear motion and free
falling bodies.
The consideration of the shot as a
projectile is a justified one as far as non consideration
of forces due to air resistance are concerned (Otto,
1987).
This is due to the fact that its shape enables it
to maintain a constant center of gravity and constant
42
frontal area in contact with the air (see figure 4).
More details will be presented later in this chapter.
When the effects of air resistance are ignored, the
effects of the gravitational force can then be studied
more accurately.
The gravitational force that acts on a
projectile is entirely independent of the projectile’s
horizontal speed (The M.I.T physics series, 1971).
When
a projectile is released at some angle relative to the
horizontal, the angle of release, there are generally two
velocity components which operate independently and act
perpendicularly to each other.
frontal area
center of gravity
Figure 4.
The frontal area and center of gravity of the
shot.
Velocities and Resultant
The vertical velocity (v y) is affected by gravity,
and as a result, its value will vary from maximum at the
instant of release of the projectile, to zero at the peak
of the projectile’s flight.
The vertical velocity is the
factor determining how high a projectile will rise
(Michels, Correll, & Patterson, 1968).
This peak height
is called the vertex of the trajectory.
An interesting
note here is that the time needed for the projectile to
43
reach its vertex equals the time it takes to reach the
level of landing from the vertex (Michels et al., 1968).
The horizontal velocity (vx) is constant in its
value which is in accordance with Newton’s first law of
motion which states that “every body continues in its
state of rest or motion in a straight line unless
compelled to change that state by external forces exerted
upon it” (Hay, 1993).
Indeed, since the forces due to
air resistance that act on the projectile are not
considered, once a projectile’s horizontal velocity is
known, it can be assumed that it will remain the same
throughout its flight (Michels et al., 1968).
The resultant of a plotting of the v x and vy vectors
represents the projectile’s actual velocity (v0) in the
direction of projection as caused by an unbalanced force
applied upon the projectile up to the instant of release.
It is unbalanced in the sense that it does not have a
purely horizontal or purely vertical direction.
This
resultant can be determined trigonometrically or
graphically.
Figure 5, shows a graphical representation
of the resultant force.
Complementary Angles
When projected at each of two complementary angles
at the same velocity, a projectile’s horizontal
displacement will be the same if air resistance is
ignored (Jones, 1979).
Complementary angles are those
whose sum equals 90 degrees.
44
For example, a projectile
projected at 20 degrees will travel the same distance as
one released at 70 degrees.
The same is the case with
projectiles released at 30 and 60 or 40 and 50 degrees.
vertical
force
Figure 5.
resultant
horizontal force
The resultant force and its components.
In the first case of these examples, the horizontal
velocity is high but the time in the air is short.
In
the latter case, there is considerably more time in the
air, but the horizontal velocity is less.
The phenomenon
of the complementary angles holds true only when the
level of release and landing are the same (Jones, 1979).
In the majority of sports it is only necessary for
an athlete to place emphasis on accuracy as is the case
in the sports of basketball or baseball to mention a few.
Only in some sports is it necessary for the athlete to
achieve maximum distance, height, or velocity.
This is
exactly the case in all the throwing events and of course
for the shot put event.
Here, due to the fact that the point of release is
at a higher level than that of landing, the principle of
the complementary angles mentioned above is not valid.
The angle that is smaller than 45 degrees, when two
45
complementary angles are compared, will always be more
appropriate for maximum distance.
For example, the same
projectile when released under the same velocity and
height, will travel further when the angle of release is
42 degrees than if it is 48 degrees, or the projectile
will travel further when the angle is 40 than if it is 50
degrees and so on (Courant & Hilbert, 1953).
The optimum angle of release in situations where the
point of release is at a higher level than that of
landing, as it happens during the shot-put event, depends
upon the height and the velocity of release (Dyson,
1977).
The exact figure of the optimum angle of release
is currently given in the literature (Bowerman & Freeman,
1991; Ecker, 1985; Hay, 1993), as a particular range of
angles for the shot-put event.
Path of a Projectile When the Point of Release and the
Point of Landing are in The Same Level
Figure 6, shows a typical path of a projectile in
the absence of air resistance.
Traditionally, all
projectile problems seek to find one or more of the
following parameters:
v0 : the velocity of release
vy : the vertical velocity component
vx : the horizontal velocity component
a :
the angle of release
t : time to reach the vertex (peak) of the
trajectory
46
T : total flight time
X : horizontal range
s : peak height reached (vertex)
peak height
trajectory
angle of release
range of projectile
Figure 6.
Path of a projectile in the absence of air
resistance.
Following, basic mathematical formulas will be
presented briefly to illustrate the nature of ways used
to solve projectile problems.
Velocities and resultant.
Because the component
velocities vx and xy always act at right angles to each
other, trigonometry is often used to solve projectile
problems.
The side adjacent to angle (a) is represented
by vx, whereas vy represents the opposite side.
Finally,
v0 represents the hypotenuse of the right triangle.
Again, using trigonometry principles, v x is estimated by
multiplying the velocity at release by the cosine of the
angle formed between the velocity vector and the
horizontal.
Consequently,
vx = v0 cosa
The cosine is equal to the adjacent side divided by
the hypotenuse (Wenworth, 1951).
47
Consequently,
vy = v0 sina
The initial velocity (v 0) can be determined in two
ways.
Using the first way, if the value of either side
and the angle (a) is known then, v0 will be,
v0 = vy / sina
if the vy is known or,
v0 = vx / cosa
if the vx is known.
Using the second way, and if both v y and vx are
known, the Pythagorean theorem can be used as follows,
v0 2 = v x 2 + v y 2
Peak height.
Peak height reached at the top of a
parabola, is a function of the vertical velocity
component and is not influenced by the projectile’s
horizontal velocity.
If the vertical velocity is known
then, the peak height (s) is,
s = vy2 / 2g
(Barford, 1973)
where g = 9.81 m/sec2, or the gravitational acceleration.
If the time is known then peak height (s) is,
s = 1/2 g t2
(t) here represents time taken either to rise or to fall.
Total time.
The time to rise or to fall is found
using the formula,
t = vy / g
if the vertical velocity is known, or by using the
formula,
48
t2 = 2s / g
if the height is known.
The total time is found by
doubling (t) or by using the formula,
T = 2vy / g
Range.
(Barford, 1973)
The range (X) or horizontal displacement of
a projectile can be determined by any of the following
procedures which yield the same results:
If the total time in the air and the horizontal
velocity is known then,
X = vx T
also, because vx = v0 cosa then,
X = v0 cosa T
If the two component velocities are known then,
X = 2vx vy / g
which in essence is the same as the previous equation if
one substitutes for (T) from its equivalent from the
total time formula above.
Subsequently, substituting for
vx and vy we have that,
X = [2 (v0 cosa) (v0 sina)]/g
which becomes,
X = v02 sin2a / g
(Barford, 1973)
and gives the horizontal displacement when only the
projecting velocity and angle of release are known.
It
should be noted here, that the above equations and the
procedures followed to derive them, set the stage for the
derivation of the equations describing projectile motion
49
when the point of release is at a higher level than that
of landing.
Path of a Projectile When the Point of Release is Higher
than the Point of Landing
Figure 7, shows the path of a projectile when the
point of release is at a higher level than that of
landing.
Following, some formulas pertaining to these
conditions will be presented.
Total time.
The total time of flight when the point
of release is higher than that of landing is given by,
T = [v0 sina + Ö(v0 sina)2 + 2 g h] / g (Hay, 1993)
Range.
The range of the projectile can be obtained
by using the formula,
X= [v02 sina cosa +v0 cosa Ö(v0 cosa)2 + 2gh] /g(Hay, 1993)
Interpretation of this formula and how it affects the
potential range of the projectile is given later in this
chapter.
velocity of release (u0)
angle of release (a)
height of release (h)
point of release
Figure 7.
point of landing
Path of a projectile when the point of release
is higher than that of landing.
50
Forces Affecting Projectiles
From the moment a projectile is released and is in
flight, mainly, only two forces can influence its motion
until it lands (Cohen, 1985).
These two forces are, a)
the gravitational force, and b) the force due to air
resistance (drag).
Gravity
Gravity is the force exerted by the earth on other
masses and is always directed toward the center of the
earth.
The magnitude of the gravitational force
depends upon the magnitude of the masses as well as the
distance of their centers of gravity from each other
(Misner, Thorne, & Wheeler, 1973).
If gravity were the only external force acting on a
projectile, the path the projectile would take would be
in the shape of a parabola.
The term parabola and
parabolic course describes a type of curve resembling a
hill (see also figure 6), the steepness of which can vary
with the greater the angle of release, the steeper the
parabola (Misner et al., 1973).
The acceleration (g) of a projectile due to the
gravitational force is approximately 9.81 m/sec 2.
That
means that the velocity of the center of gravity of a
projectile will be altered in the vertical direction by
9.81 meters per second for every second that it is nonsupported.
This acceleration which is due to the force
of gravity is constant on any projectile, regardless of
51
the projectile’s weight.
Figure 8, graphically shows the
change that gravity causes in the magnitude and direction
of the projectile as it hovers along its trajectory
course.
The magnitude of velocity is indicated by the length
of the vector and the shape of the trajectory represents
only the effect of the gravitational force on its
vertical motion after its release.
In figure 8, the
vertical component decreases during ascent and then
increases during descent thus, showing gravity’s
decelerating and accelerating effects upon the
projectile.
As discussed earlier, the horizontal component
remains the same throughout the trajectory path because
the gravitational force can only affect vertical motion,
and the assumption here is that no other forces act upon
the projectile.
Figure 8.
The effects of gravity on the trajectory of a
projectile.
Aerodynamical Forces
As mentioned previously, gravity is not the only
external force acting on a projectile.
52
Forces generated
by the flow of air relative to the moving projectile can
play a decisive role in determining how a projectile will
travel during its flight, particularly if the object is
not massive (Courant & Hilbert, 1953).
In some cases, air forces are so small and their
effect so insignificant that they can be ignored.
For
example, one can overlook air forces when studying the
putting of the shot.
This implement is heavy enough
(7.26 kilograms or 16 pounds) not to be greatly
influenced by air resistance (Sokolnikoff & Redheffer,
1966).
Moreover, its shape, which is described as non
aerodynamic, does not allow the air currents to further
affect its flight since it (the shape), stays the same
under all circumstances.
Projectiles of aerodynamic
nature like the discus or the javelin can be greatly
affected by air direction and magnitude since their
position in relation to the air flow can change thus,
affecting their flight ability.
In these events air
resistance should always be considered (Hay, 1993).
Air resistance depends upon a) the frontal or cross
sectional area exposed to the air flow, b) the velocity
of the projectile relative to that of the air currents,
and c) a coefficient of the air force.
The air force can
have two components/forces, a) the drag and b) the lift.
While the drag force acts on all projectiles, the lift
force acts only on the so called aerodynamical
projectiles (Wong, 1991).
Since the shot is not a
53
predominantly aerodynamical projectile, only the drag
force will be considered here.
Drag Coefficient
The drag coefficient (C D) is found experimentally
using wind tunnels.
A projectile’s (CD) gives
information about how streamlined it is, which depends on
how the shape of the projectile relates to the air flow.
Ganslen (1964) found that the C D for a discus changes
depending upon the tilt of the discus relative to the air
flow.
An 11 degree inclination gives a CD of 0.10 and a
35 degree inclination gives
C D of 0.68.
Similar changes
were found for the javelin.
Generally, a blunt, non streamlined object will have
a large value for the drag coefficient (e.g., a box),
while long streamlined projectiles will have smaller
values for the drag coefficient (e.g., a discus or a
javelin).
Finally, depending upon the nature of movement
and the event, fluid forces can be important and must be
given more than mere mention if the motion of a
particular projectile is to be fully understood
(Armstrong & King, 1970).
Frontal Sectional Area
The drag coefficient which represents the degree
of streamlining of a projectile’s shape, is different
than the area of the projectile facing the flow.
If the
area facing the flow, for example, increases because of a
change to an unstreamlined position, the air force will
54
increase due to the greater area, irrelevantly how
streamlined the projectile is.
A double increase in the
area will bring about a double increase in the drag force
acting upon the projectile’s surface as it hovers through
the air and it is in flight trajectory (Armstrong & King,
1970).
Fluid Density
The density of a fluid (r) is the fluid’s mass
divided by a specific volume of it such as a liter.
It
is a representation of how closely the fluid’s atoms and
molecules are arranged to form the fluid under
consideration.
To this respect, water is close to one
thousand times denser than air.
Consequently, movement
through air will be much easier as compared to that
through water.
Although water density does not change,
air density does change depending upon atmospheric
pressure, air temperature, humidity, or altitude.
Air
density, for example, at 35 degrees Celsius is 12% less
than at zero degrees Celsius (Armstrong & King, 1970).
Velocity
The drag coefficient and the frontal area (A)
become more important at faster flow velocities because
they are multiplied by the velocity in the second power.
In the drag equation,
Drag = 1/2 CD A r v2
the increase in drag force is proportional to the square
55
of the velocity.
If at a velocity of 10 m/sec., there is
drag of 30 Newtons, a double increase in velocity to 20
m/sec., will bring about a four time increase in the drag
force from 30 Newtons to 120 Newtons.
Relative Acceleration
In those cases where the projectile could travel for
very long distances, the relative acceleration, also
called the Coriolis acceleration, should be taken into
account.
Generally, a projectile can travel further if
projected towards the east rather than towards the west
(F. Mirabelle, personal communication, January 1995).
Drag Effect on Different Masses
Due to their nature, some projectiles cannot be
streamlined.
The shot is such a projectile.
This
projectile is spherical in shape and ball-like looking
and generally allows for a great air resistance in the
form of drag (Kibble, 1973).
The mass, however, of a
spherical projectile which moves through the air is
important in determining the amount of deceleration which
is caused by a particular force applied on the
projectile.
An increase or a decrease in the mass of the
projectile does not alter the magnitude of the drag force
acting against it.
However, the mass of the projectile
determines to some extend how the projectile’s motion
will be influenced by the drag force (Kibble, 1973).
For example, if one were to drop a tennis ball and
an iron ball of the same size from a high elevation, they
56
would notice that the iron ball will reach the ground
sooner than the tennis ball.
In this case, although the
size of the balls is the same, the mass is significantly
greater in the case of the iron ball.
Obviously, the
tennis ball is affected more by the drag force than the
iron ball.
It seems then that the same amount of drag
force affects the tennis ball more than the iron ball.
This is because the drag force easier matches the weight
of the tennis ball whereas, there is a need for more time
if the drag force were to match the weight of the iron
ball (Kibble, 1970).
An explanation for this can be found by applying
Newton’s second law of motion.
If force equals mass
times acceleration,
F = m * a, and,
a = F / m
then, the acceleration (or deceleration) is inversely
proportional to that object’s mass.
Consequently, the
smaller the mass the more the effect of the same drag
force.
When motion in a horizontal plane is considered,
where acceleration due to gravity does not affect the
deceleration of the object, the deceleration of the
object will be equal to the resistive force divided by
the mass of the object (Kibble, 1973).
This means that
the retarding effect is directly proportional to the drag
force on the object which depends upon the area facing
the air flow, and also that this retarding effect is
57
inversely proportional to the mass of the object.
In
this fashion, a tennis ball can travel faster than a
ping-pong ball.
The difference in the area facing the
wind currents partially explains why a skier prefers to
descent the mountain slopes in a crouched position
(reduced frontal area) than in a standing position.
Vector Representation in Projectile Motion
Resultant Vector
When a shot-putter imparts velocity to the shot in
the direction of the throw (see figure 9), the thrower
essentially imparts both a vertical force represented by
vector AB, and a horizontal force represented by vector
AC.
The net effect or the resultant of these two
velocities vectors can be obtained by completing the
parallelogram of which AC and AB are adjacent sides and
then construct the diagonal through the point A.
diagonal represents in magnitude and direction the
This
resultant velocity of the shot (Hay, 1993).
D
B
resultant
A
C
58
Figure 9.
Velocity components and resultant during shot-
put releasing.
Consequently, the combination of the effects of the
vertical and horizontal forces causes the shot to move in
the direction indicated by the diagonal AD and with a
velocity represented by the length of that diagonal.
This type of parallelogram is called the parallelogram of
vectors.
To find the resultant force when the angle between
the two vectors AB and AC is a right angle and, the
magnitude of the vertical and horizontal velocities is
known, the theorem of Pythagoras may be used to find the
magnitude of the resultant AD as follows,
AD = ÖAB2 + AC2
The direction of the resultant can be obtained by,
tana = AB / AC
where (a) is the angle formed between the vector AC and
the resultant (Wenworth & Smith, 1951).
When the angle formed between the two vectors AB and
AC is not a right angle, the process of finding the
magnitude and direction of their resultant is more
complicated and beyond the scope of this review.
Vector Components
In some situations information about the vectors AB
and AC is desired rather than about the resultant.
this case, and providing that only the magnitude and
In
59
direction of the resultant is known, the resultant must
be broken down into two components thus, obtaining
horizontal and vertical components, which may represent
velocity, acceleration, displacement, and so on.
These
components can be determined using two ways: a) they can
be determined graphically by following a procedure that
reverses the construction of a parallelogram of vectors
and, b) they can be determined through the use of
trigonometry (Basic systems Inc., 1962).
Using the first way, it is assumed that the angle
between the resultant and the horizontal is 40 degrees,
and also that the length of the resultant is 10 cm.,
representing 10 m/sec., velocity.
For this example then
(see figure 10), a line is drawn from the top of the
resultant force directly vertically until this line
crosses the horizontal line.
A horizontal line is then
drawn again from the top of the resultant.
The parallelogram that is formed includes the
horizontal and the vertical components of the resultant.
A measurement of the two vector components will give a
measure of their relative magnitude.
The parallelogram
that is formed includes the horizontal and the vertical
components of the resultant.
A measurement of the two
vector components will give a measure of their relative
magnitude.
Using trigonometry to find the two component
vectors, it is known that,
60
cosa = AC / resultant
The horizontal component is then,
AC = resultant cosa
Similarly,
AB = resultant sina
The trigonometric method is faster and more important
more accurate than the graphic method.
B
10 m/sec.
AB=10x0.642=
=6.42 m/sec.
40 degrees
A
C
AC=10x0.766 = 7.66 m/sec.
Figure 10.
Analysis of vector components from the
resultant.
Factors Affecting a Projectile’s Horizontal Displacement
Earlier in this chapter, the forces that affect a
projectile during its flight were examined.
The
conditions, however, under which a projectile is
released play a paramount role in determining the
maximum range achieved.
Point of Release is at Ground Level
A review of the equation for the range of a
projectile when the point of release is at the same level
as that of landing,
61
X = v2 sin 2a / g
reveals that the range is directly proportional to the
square of the velocity of release and the angle of
release, and inversely proportional to the gravitational
force.
It is obvious from that equation that the
velocity of release is the single most important factor
here, because it is squared thus, its weight in the
equation is significantly higher than that of the angle
of release.
Due to the fact that this formula is not
directly applicable to the shot-put event, a numerical
solution of the equation will not be pursued here in
order to validate what was stated above.
Instead, a
numerical solution will be sought for the more applicable
to the throwing events equation next.
Release Occurs at a Higher Level Than That of Landing
The equation which gives the range of a projectile
when it is released from a height above the ground is as
follows:
X = [v2 sina cosa + v cosa Ö(v sina)2 + 2gh] / g
Again it can be seen that the value of the velocity
is squared and this declares the paramount importance of
the velocity of release in achieving maximum range, when
the point of release is at a higher level than that of
landing as it happens during the throwing events.
A
numerical example can help for a more clear explanation
of the relative importance of the velocity of release,
62
the angle of release, and the height of release in the
effort to achieve the higher possible range.
Let us assume that a projectile is released at a
velocity of 13 m/sec., at an angle of 38 degrees, from a
height of 1.9 meters above the ground.
Under these
conditions and if the equation for range presented above
is solved using these numbers, the final value for the
range will be 18.87 meters.
Now let us see how a 5% increase in each of these
characteristics at the moment of release will affect the
distance reached.
It is important here to note that for
comparison reasons, while an increase of 5% is taking
place in one of the three components, the other two will
remain as they were initially in this example.
First, an increase of 5% in the velocity of release
from 13 m/sec., to 13.65 m/sec., (the angle is 38 degrees
and the height is 1.9 meters), will increase the range to
20.60 meters, a net increase of 1.73 meters as compared
to the original range of 18.87 meters.
Second, an
increase of 5% in the angle of release from 39 to 39.9
degrees (the velocity is 13 m/sec., and the height is 1.9
meters) will increase the range to 18.98 meters, an
increase of 0.11 meters.
Third, an increase of 5% in the
height of release from 1.9 meters to 1.995 meters (the
velocity is 13 m/sec., and the angle is 39 degrees) will
increase the range to 18.97 meters, an increase of 0.10
meters.
63
It is clear that an increase in the velocity of
release brought about more dramatic results than similar
increases in either the angle or the height of release.
It should be emphasized here, however, that in practice
it happens all too often that the difference between the
winner and the loser is a mere 10 or 20 centimeters, a
reality that declares the need for careful consideration
of every aspect of the throw if a maximum range is to be
achieved (Papageorgiou, 1986).
The following then, can
be summarized about the velocity of release, the angle of
release and the height of release.
Velocity of release.
In projectile motion, speed of
release is the most important factor.
As shown above, a
small percentage of increase in release speed will always
bring about a greater percentage of increase in distance,
if all the other factors remain constant.
While the
athlete must continually attempt to increase the
implement’s speed at release, he/she must avoid
increasing one velocity component (horizontal or
vertical) without also increasing the other.
Otherwise,
the angle of release is likely to be too high or too low
and the distance of the throw may be reduced, even though
the release speed has been increased (Papageorgiou,
1986).
Angle of release.
No matter which kind of
projectile is being considered, there is a particular
optimum projecting angle for every attempt, no matter
64
what ability the individual thrower happens to possess
(Bowerman & Freeman, 1991).
However, it is not
necessarily the same angle for each thrower in an event,
or even the same angle for an individual athlete’s
attempts in the same competition.
The contribution of
the angle of release to the overall achieved distance is
not as important as that of the velocity, but
nevertheless, it may make the difference between winning
and losing in a competition.
Height of release.
The height of release of a
projectile can also be of importance, and as is the case
with the angle of release, it may make the difference
between winning or losing.
Generally, there is not an
optimum height of release.
Traditionally, athletes have
been trying to release their implements from a height as
high as possible while at the same time they try to
adjust the angle (optimum) at which the implement should
be released to achieve maximum distance (Dyson, 1977;
Hay, 1985).
Optimum Angle of Release When the Projectile is Released
at a Level Higher Than That of Landing
If a thrower is to obtain maximum distance, it will
not be sufficient to give the missile maximum release
velocity.
It must also be thrown at an appropriate
angle.
When the point of release and landing are at the
same level and the aerodynamic forces are ignored, the
65
optimum angle for the projection of a projectile should
be 45 degrees regardless of its velocity of release.
In
this case, vertical and horizontal component velocities
are equal and the missile also lands at a 45 degrees
angle.
However, in all four throwing events and of course
in the shot-put event, the implement is released from a
point above the ground and this affects the optimum
release angle.
Then, the optimum angle depends upon the
height and the velocity of the projection (Dyson, 1977).
In the shot-put event where aerodynamic factors are
of no account (Dyson, 1977), the optimum angle will be
less than 45 degrees.
In the literature, the optimum
angle of release has been reported to be in the range of
41-43 degrees, depending upon the height and the velocity
of release.
49.400
40.200
2.1 meters
49.400
19.18 meters
Figure 11.
The relationship between release and landing
angles in projectile motion when the point of release is
higher than that of landing.
In theory, a projectile thrown at 40 degrees and 20
66
minutes, will land at 49 degrees and 40 minutes (see
figure 11) if released from 2.1 meters above the ground
at a velocity of 13 m/sec., for a distance of 19.18
meters (Dyson, 1977).
Summary
The release parameter data confirm that the velocity
of release is the single most important factor in
obtaining maximum distance in shot-put.
The angle of
release should always be accounted for, if a successful
throwing performance is to be achieved.
A discrepancy
has been shown to exist between the theoretical optimum
angle of release in shot-put and the actually obtained
angle of release, from real life data analyses.
A number
of viable explanations have been given to explain this
phenomenon.
Three main factors along with the gravitational
force will affect the range a projectile can achieve when
in fact it is released from a specific height (h) above
the ground.
First, the velocity of release (v0) of the
projectile, second, the angle of release of the
projectile (a) and third, the height of release of the
projectile (h).
Of the three factors, the velocity at the time of
release is the most important one and more dramatically
affects the range achieved, much more than the angle of
release and the height of release can.
However, both the
angle of release and the height of release can contribute
67
to a perfect skillful projection/throw.
In the throwing
events the selection of the optimum angle of release can
contribute to winning or losing in a competition, both in
the beginning and in the advanced levels.
Given the relative importance of the angle of
release in achieving maximum range, a theoretical model
is sought to estimate the exact optimum angle of release
for the shot-put event.
The theoretical model will
assume that the shot as a projectile obeys the principles
governing projectile motion when the point of release is
at a higher level than that of landing.
The theoretical
model estimating the optimum angle should be a function
of the velocity of release and the height of release
while it will take into account the gravitational force.
68
CHAPTER 3
Methodology
Research Design
The purpose of this study was, a) to examine the
relationship between the angle of release and the
velocity of release and also the relationship between the
angle of release and the height of release in the shotput event, b) based upon the above relationships, to
obtain a real life equation to estimate the angle of
release from velocity and height of release data, and c)
to develop and validate a theoretical model to estimate
the optimum angle of release from velocity of release and
height of release data.
This, for the purpose of
comparing the results from this model with those obtained
from real life throwing.
The agreement between theory
and practice was thus, assessed.
Specifically, it was an
attempt,
1.
To examine the relationship between the angle of
release and the velocity of release, and also the angle
of release and the height of release in shot put, so as
to determine, primarily, whether any dependence of the
velocity of release on the release angle exists.
Based
on the above relationships, equations predicting the
angle of release from the velocity and the height of
release were obtained.
2.
To mathematically model the motion of the shot
as a projectile with respect to its optimum angle of
69
release taking into consideration the mechanical
conditions that are prevalent during the event.
3.
To examine the validity of the model by
comparing it to other models/methods relative to
projectile motion.
4.
To examine the potential application of the
model to real life situations by experimentally testing
it and applying it to a number of shot-putters.
To determine the relationships between the angle,
the velocity, and the height of release, correlational
techniques were used.
To obtain a real life model,
multiple regression techniques were used.
The research
design to obtain the proposed theoretical model fell into
the realm of descriptive research.
To achieve the
development of the theoretical model, the process of
mathematically deriving the model that estimates the
exact optimum angle of release for a projectile when its
point of release is at a higher point than that of
landing will be explored and described in detail in this
chapter.
The elements and procedures that help construct the
model were described here.
Finally, the methods to be
used to validate the model and the potential application
of the model will be discussed.
The Relationship Between the Angle, Velocity, and Height
of Release
To achieve the objective of the study to establish
70
the relationship between the angle, velocity and height
of release, the following procedures were followed: a)
preliminary investigation, b) selection of subjects, c)
instrumentation, d) collection of data and, e) analysis
of data.
Preliminary Investigation
A preliminary investigation was undertaken to (1)
determine the type of instrumentation that would be
needed to adequately investigate the kinematics of the
release phase in shot-putting, (2) identify the major
problems encountered in the filming of the movement, (3)
familiarize the investigator with the experimental
equipment, and (4) develop methods and procedures
suitable for conducting this type of experiment.
The preliminary investigation revealed that a frame
rate of at least 120 frames per second would be adequate
in capturing the movement of the shot after its release.
It was found that a sufficient number of frames would be
available for the accurate estimation of the velocity,
height and angle of release of the shot at release.
No
blurring occurred.
It was decided that the filming of the subjects
would take place over a five day period to allow for rest
between workouts.
be used.
Moreover, two assistants would have to
One would help measure the actual distances
thrown by the shot-putters, the other would record the
identification for each subject and the progress of the
71
various trials.
Correlational analysis of the preliminary
investigation data showed that a significant correlation
existed between the height of release and the angle of
release.
The correlations between the velocity of
release and the angle of release and also the velocity of
release and the height of release were not significant.
There was no attempt to create a regression equation to
predict the angle of release from velocity and height of
release data.
The non significant correlation between the velocity
of release and the angle of release could be attributed
to the fact that the subject used for the preliminary
investigation was able to achieve angles of release
between 17 and 35 degrees only thus, making it possible
to achieve similar velocities of release throughout the
angle range.
It is expected that with steeper angles the
velocity will decrease.
As it was observed in this study
this, indeed, was the case.
Selection of Subjects
Subjects were 5 healthy, male collegiate shotputters using the rotational technique, volunteers, ages
18-22, from the track and field teams of 4 different
universities in the State of Kansas.
The investigator
approached each of the 5 subjects and solicited their
participation.
Selection of the subjects was made based
upon their experience in the shot-put event.
72
An
experienced shot-putter was someone who was able to
demonstrate much above average technical skill in
throwing the shot, as assessed by the investigator.
The
main technique elements the investigator attempted to
evaluate were,
1.
Slow/controlled preparatory movements in the
back of the circle as the thrower was ready to enter the
turn.
2.
Slow/controlled transition from double support
to single support over the left foot (right hand thrower)
with lowering of the center of gravity and proper trunk
position (no excessive bending).
During the same phase a
relaxed left arm, along with an active right leg which
would lead the thrower towards the center of the circle.
Moreover, the thrower was to use only his legs and hips
to acquire momentum, during this phase, while the trunk
and upper body were to be inactive.
3.
After the athlete landed somewhere in the middle
of the circle, a good separation between the shoulder and
the hip axes with the right shoulder well back to his
right.
Exactly at the moment the athlete landed his
right foot, all the body weight was to momentarily be
over that foot.
At the same moment, as the athlete
completed the double support phase, his left foot was to
be quickly placed near the stop-board.
4.
After the athlete had assumed the power
position, a coordinated movement starting from the feet
73
legs and hips and finally up to the trunk, shoulders and
throwing arm was sought.
The action of legs and hips was
to be given priority at the start of the final effort.
The direction of the final effort was to be from back
(over the right foot) to front (over the left foot) and
from down, up.
During this phase, the left (free) arm
was to be very active, actually helping the trunk to
swiftly rotate towards the left.
At the instant the
athlete released the shot, the free (left) arm was to
bend in the elbow and its action was to block the
tendency of the trunk to over-rotate to the left.
5.
The overall flow of the throw was to show a well
controlled speed of execution with a significantly faster
final effort as compared to the turning phase of the
throw.
Subjects were required to read and sign a consent
form as shown in appendix B, before participating in the
study.
An approval for the use of human subjects was
also obtained from the advisory committee on human
experimentation (A.C.H.E) of the University of Kansas and
is shown in appendix C.
Instrumentation
A Pulinix high speed video camera mounted on a
sturdy tripod was used.
A Citizen 5.7 x 4.3 cm monitor
was used to help the investigator see what was actually
recorded.
A Panasonic AG-1960 Proline SVHS video
cassette recorder was used in conjunction with the camera
74
to record the performance of each of the subjects.
The
tapes used to record the throws were high quality SVHS
videotapes.
The digital clock used to assess the real
speed of the camera was from Lafayette Instrument Co.,
model 54419-A.
For the film analysis, a Panasonic AG-7350 SVHS
video cassette recorder was used whereas the monitor for
the analysis was a Sony Trinitron PVM-1341.
The computer
unit that was used to perform the analysis was a 486/33
from Advanced Logic Research.
The computer monitor was a
Panasonic Panasync C-1395.
Collection of Data
Filming Procedures
Subjects were filmed both indoors and outdoors, some
indoors some outdoors, depending upon the weather
conditions and also the availability of an indoor
facility.
All subjects threw from a regular shot-put
throwing circle.
The camera was set at an angle of 90
degrees (perpendicular) to the subject’s right sagittal
plane as he faces the direction of the throw (right hand
thrower).
The filming procedure was 2 dimensional (2D).
The f-stop was set at 1.2 during filming indoors while it
was set at 16 during filming outdoors.
The shutter
factor was 1/160 in most cases whereas it was set at
1/500 in some of the indoor filming.
The speed of the
film was set at 120 pictures per second.
To determine the reproducibility of the camera
75
speed, a digital clock was filmed.
By counting the
number of frames in one second, the exact and actual
number of frames per second was determined.
It was found
that the camera was running at the exact picture rate.
To convert film distances to real life distances, a
meter stick with a length of 4 feet (1.22 meters) was
used and filmed perpendicularly to the camera exactly in
the location where the throwing action was to take place.
Knowing that the filmed stick represents real life
distance, a scale factor was automatically derived by the
computer and it was used for the analysis.
It was also
determined that the amount of light was always adequate
and consequently, no artificial light was used.
B
D
C
A:
B:
C:
D:
E:
F:
F
E
Camera
Plumb bob
Clock
Trial number
Throwing circle
Throwing sector
13.9 m
Figure 12.
A
The experimental arrangement of the equipment
for collecting the data.
76
The distance between the camera and the plane of
action was 13.9 meters, whereas the distance between the
camera lens and the ground was 1.18 meters in all cases.
A plumb bob in the background was used during filming so
that the investigator could determine exactly where the
vertical direction was.
Cards identifying the order of
the attempts were also used.
Testing Procedures
For each attempt, the athlete tried to make the
throw in one of five different ways as follows:
1.
The athlete threw at his normal angle,
2.
The athlete threw slightly higher than his
normal angle,
3.
The athlete threw slightly lower than his normal
angle,
4.
The athlete threw much higher than his normal
angle and,
5. The athlete threw much lower than his normal
angle.
Based on the angle they usually throw, the athletes
were instructed to make the proper adjustments to achieve
the slightly higher and the slightly lower angles.
The
terms “slightly higher” and “slightly lower” above,
implied the use of an angle that was approximately 5
degrees above or below the “normal” angle.
However, for
the “much lower” and the “much higher” angles the
athletes were instructed to throw as low as possible and
77
as high as possible, respectively.
The purpose of using the five kinds of throwing
angles was not to achieve accuracy in the angle of
release used rather, to achieve a wide variety of angles
of release filmed.
As Bashian, Gavoor, & Clark (1982)
speculated, it would be credulous for anyone to claim
that an athlete can purposely throw the shot at a
specific angle of release.
Each athlete attempted 10
trials, as suggested by Bates, Dufek & Davis (1992), in
each of the above 5 kinds of throwing thus, resulting in
the filming of a total of 50 throws for each thrower.
The throws were “all-out” with the athlete trying to
achieve his best potential each time as if he were in a
real competitive environment.
The athlete was told immediately before each throw,
under which of the 5 kinds of angle of release he was
supposed to throw.
The order of the throws was randomly
determined beforehand by the investigator.
To achieve
that, each kind of angle was written on a piece of paper
and drawn at random.
The throwers practiced the high and
low throws in several workouts before the filming
session.
This took place in an effort to avoid an unfair
advantage of throwing under the normal angle as opposed
to throwing over the higher and lower angles.
Since the athletes had to throw a total of 50
throws, to avoid fatigue from occurring, the athletes
were tested on 5 different days thus, executing 10 throws
78
for each day.
Based on the personal experience of the
investigator as a shot-putter, and also on personal
communications with a number of athletes and coaches, it
was deemed appropriate to ease the daily workload.
This
was even more imperative considering that the attempts
were to be “all-out” and energy would be quickly lost.
Foul throws were not used.
repeated.
A foul throw had to be
The athletes were encouraged to perform as in
competition where, among others, foul throws do not
count.
Each throw was measured with a tape; this, in an
effort to assess the accuracy of the obtained from the
film velocity, angle and height of release data.
Using
the data obtained from the analysis and the equation for
range, when a projectile is released from a height (h)
above the ground, the range based on these data can be
estimated.
Results from the solution of the equation
should approximately agree with those of the actually
measured range.
Finally, the implement used by the athletes was
according to the International Amateur Athletic
Federation (I.A.A.F) guidelines, weighing 7.257 kilograms
with a maximum diameter of 13 centimeters.
Analysis of Data
Film Analysis
To calculate the desired parameters, a video
analysis system along with computer software from Peak
Performance Corporation was used.
79
Figure 13 shows the
release parameters that were calculated.
velocity
angle
height
Figure 13.
The release parameters calculated.
The Peak Performance computer analysis system uses a
central processing unit, a computer monitor, and a video
tape monitor in which the film is actually viewed.
A
menu of various options shown on the screen of the
computer monitor aids the user in choosing the proper
actions to be taken for the analysis.
Moreover, a
computer mouse is always used to locate the various
points of interest on the image shown on the video
monitor.
To calculate the velocity and the angle of the shot
at release, the horizontal and vertical location of the
shot in the air in each frame of approximately the first
2 meters of its path after release was determined.
Following, a straight line to the horizontal location
versus time values was fitted.
A parabola of second
derivative equal to 9.81m/sec 2 to the vertical location
versus time values was also fitted.
To achieve the fit
of the straight line and of the parabola, a FORTRAN
computer program provided by Dr. Dapena was used (J.
80
Dapena, personal communication, January 1995).
This
computer program estimated the horizontal and vertical
velocities at the time of release and is shown in
appendix D.
Based on the equations of the line and of the
parabola, and also of the best estimate of when (i.e.,
the frame number) the shot was released, it was possible
to get a very good estimate for the location, the
horizontal and vertical velocities, and therefore for the
angle of release and the magnitude of the velocity vector
at release.
Trigonometry principles were used to
calculate the velocity vector from the horizontal and the
vertical velocities.
More specifically, using the
Pythagorean theorem,
v = Öa2 + b2
where, v = the velocity vector, a = the vertical velocity
vector, b = the horizontal velocity vector.
To calculate the angle of release, which is the
angle formed between the velocity vector and the
horizontal velocity vector again, trigonometry was used.
We know that,
tana = a / b
where, a = the angle of release, a = the vertical
velocity vector, b = the horizontal velocity vector.
In both cases, a computer program was used to
quickly and accurately obtain the results for the
81
velocity and the angle of release.
The goal was to have at the end a release velocity
versus release angle relationship/scattergram for each
thrower.
Taking this relationship into account and also
the height of release parameter, a multiple regression
equation was obtained for each thrower.
Thus, the true
optimum angle of release was calculated for each thrower
and subsequently compared with the so-called theoretical
optimum angle obtained from the theoretical model.
The
result from the theoretical model naturally assumes that
the velocity is not dependent upon the release angle.
Statistical Analysis
To examine whether a linear relationship exists
between the angle of release and the velocity of release,
and also between the angle of release and the height of
release, the Pearson product moment correlation was used.
To estimate the angle from velocity and height data,
an equation of the form,
Y’ = a + b1x1 + b2x2 + e
was obtained using multiple regression procedures, with
the angle being the dependent and the velocity and the
height being the independent variables.
In the above equation, Y’ = the predicted angle of
release, b1, b2 = the regression coefficients associated
with the velocity and height of release variables (x 1,
x2), a = the constant of the regression equation, and e =
error or residual.
Finally, in an effort to initially
82
detect the presence of collinearity, a Pearson product
moment correlation was performed between the velocity and
height of release variables.
Other collinearity
diagnostics were also used.
To examine whether a significant difference exists
between the results of the theoretical model and the real
life model, simple regression techniques were used.
Twenty random pairs of velocity and height of release
data were assumed.
To do this, all the possible values
for velocity of release and height of release, achieved
during throwing under the normal angle of release for
each subject, were written on a piece of paper and then
drawn at random one pair at a time.
Based on these data,
both the theoretical and the real life models were solved
numerically and the resulting angles of release were
obtained.
Finally, the regression equation for the two models
with the real model being the dependent and the
theoretical model being the independent variable, was
obtained.
As a result of the analysis, the
intercept/constant (a) and the regression coefficient (b)
of the corresponding regression equation were also
obtained.
For a perfect agreement between the two models
a correlation of one should be observed with the constant
(a) being equal to zero, and the regression coefficient
(b) being equal to one.
Since it was impossible for the
two models to agree perfectly, the regression coefficient
(b) was tested against the hypothesis that it was
83
significantly different from one, and the constant (a)
was tested against the hypothesis that it was
significantly different from zero.
To achieve this, the
(t) values for both parameters were obtained and their
significance was determined.
The Theoretical Model
The procedures followed to obtain the theoretical
model are described in detail below:
Gravity
When a small body is projected near a much larger
body, its trajectory is not straight but curves back
toward the larger body.
Newton’s law of gravitation
specifies that, when both bodies are spherically
symmetric and the small projectile is outside the larger
body, the force acting by the larger mass (m 1) on the
smaller mass (m2) is given by,
F=
-
G m1 m 2
r2
®
r
(1)
where G is the gravitation constant, r is the vector from
the center of mass of the larger body to the center of
mass of the smaller body and r®, is the corresponding
unit vector (Bergman, 1987).
When the larger body is the Earth (m 1 = me) and the
small projectile (m2 = m) is close enough to a point
fixed on the earth’s surface, the Earth may be considered
as having spherical symmetry with r » re which is the
radius of the Earth.
In this case, equation (1) can be
84
written as,
F=
®
G me m
-
j
re2
where, j® is a unit vector in the upward vertical
direction, which is considered to be constant in both
direction and magnitude.
The assumption that (F) is constant is called the
“flat Earth” assumption (Bergman, 1987) and then,
®
F=
where g = G me / r2e.
- m g j
= m g
Since G = 6.67 x 10-11, me = 5.98 x
1024 (kilograms) and re = 6.38 x 106 (meters) then, g =
9.81 m/sec2, and (g) is called the acceleration due to
gravity.
Its magnitude varies by less than 1% for
projectiles within 30 kilometers of the Earth’s surface.
The force, m g, is referred to as the weight of any body
of mass (m).
Velocity and Position Vectors
The simplest approximation for a projectile’s motion
is to consider that the only force acting on it, after it
is released, is its weight.
Then, for motion in free
space Newton’s second law of motion pertaining
to acceleration gives,
m
d2 r
d t2
d2 r
d t2
=
85
= m g
®
- g j
(2)
This is simply the second derivative of (r) in respect to
time and is called the acceleration vector.
In this
equation, (r) is the position vector with respect to a
fixed origin on the Earth’s surface.
Figure 14, which is
a reproduction of figure 2, shows the coordinate and
vector system.
If initially, that means t = 0, the projectile is
traveling at velocity u0 at an angle (a) to the
horizontal, the initial velocity vector will be,
®
v 0 = u0 cosa i
®
+ u0 sina j
(3)
(v) here, represents the velocity vector, while (u0) is
the initial velocity, (i®) is a unit vector in the
horizontal direction which forms a right angle with (j®)
(see figure 14).
If equation (2) is integrated with respect to time
(t) we initially have,
d2 r
d t2
®
=
- g j
which is the acceleration vector as explained earlier,
and is the same as,
d
d t
d r
d t
=
d u
d t
which is the first derivative, or the velocity vector
which is the same as,
d u
d t
=
®
- g j
86
y
u0
m g
j
a
0
Figure 14.
i
x
The right angle vector system in projectile
motion.
This then is integrated and becomes (in respect to
time),
v (t) =
®
- g j (t) + c
where (c) is a constant used in this expression to
potentially explain information not included in it.
To
continue, the expression above becomes,
v (t) =
®
- g t j
+ c
(4)
However, we have defined earlier that the value of t
=0.
So substituting for t we have,
v (0) =
®
- g (0) j
+ c
Now, because v(0) = velocity when time is zero (0)
it is the same as saying that the initial velocity u0 is
the velocity v0.
Consequently,
v (t) =
®
- g t j
+ v0
From this expression we can see that because the
expression - g (0) = 0 the constant (c) will be equal to
87
v0 .
So equation (4) finally becomes,
®
v = v0 - g t j
and substituting for v0 from equation (3) we finally
have,
®
v = v0 - g t j
®
v = u0 cosa i
®
+ (u0 sina - g t) j
(5)
where, v = dr / dt which is the first derivative of the
vector (r) in respect to time, or the velocity vector at
any time (t).
When equation (5) is integrated with respect to
time, and the assumption r = 0 when t = 0 is made, we
have,
d r
d t
®
= v = v0 - g t j
Keeping in mind that: d/dt(t 2/2) = 2t/2 = t, we have,
t2
r (t) = v 0 t - g
2
®
j
+ c
then, substituting for t = 0 we have,
r (0) = v 0 (0) - g
02
2
®
j
+ c = 0
where c = a constant as explained previously.
However, if the above expression should equal to
zero (0) then the constant (c) should also be zero since
the expression v0(0) - g 02 / 2 = 0.
after eliminating (c) is,
88
The final equation
g t2
2
r (t) = v 0 t -
®
j
Substituting from equation (3) for v0 above we have,
®
®
(u0 cosa i
®
1
g t2 j =
2
+ u0 sina j) t ®
u0 cosa t i
®
+ u0 sina t j
®
u0 cosa t i
+
( u0
-
sina t -
®
1
g t2 j
2
1
g t2
2
)
®
j
=
(6)
If the projectile is at the point (x, y) then,
r = x i® + y j®
and the horizontal component of the projectile’s
displacement will be,
x = u0 t cosa
(the first part of equation 6)
and the vertical component will be,
y = u0 sina t - 1/2 g t2
(the second part of equation 6).
When (t) is eliminated from the two equations, after
arrangements we have,
x = u0 t cosa
and
t = x / u0 cosa
y = u0 t sina - 1/2 g t2
Substituting for (t) in the equation above we
obtain,
89
y = u0
u0
2
æ
ö
1
x
g ç
÷
2
è u0 cosa ø
x
sina cosa
The u0’s cancel each other out, and keeping in mind that,
seca = 1 / cosa, and also,
x
sina
cosa
-
sina / cosa = tana,
1
x2
g 2
2
u0
1
cos2a
1
x2
x tana g 2 sec2a
2
u0
y = x tana -
g x2
2 u02
sec2a
we have,
=
=
(7)
The Range of a Projectile Released From a Height (h)
For the problem, the point of release is at a higher
level from the landing point.
It is assumed then, that
the landing point is a vertical distance below the
projection point which is also the origin of the
projection.
So it is assumed that h < 0.
In the effort to estimate the optimum angle of
release, some other aspects should be studied first.
These factors are the determination of, a) the time of
flight and, b) the horizontal range.
To begin, we will consider (h) as the (y) component
of the vector system.
From equation (6) then, because,
y = (u0 t sina - 1/2 g t2) we have that,
h = u0 t sina -
1
g t2
2
This is a quadratic equation of the form,
90
ax2 + bx + c = 0
with solution,
b2 - 4 a c
2 a
- b ±
x =
Thus the expression,
1/2 g t2 - u0 t sina + h = 0
becomes,
t =
u20 sin2a - 2 g h
u0 sina ±
(8)
g
We have already seen that x = u0 cosa t.
Substituting for (t) then in this equation we obtain,
x =
{
u20 sin2a - 2 g h
u0 cosa u0 sina ±
}
g
(9)
which gives us the horizontal range of the projectile
released from a height (h), at an angle (a), and an
initial velocity (u0).
The Optimum Angle Model
For a given initial velocity (u0), the maximum
horizontal range is obtained by considering,
dx / da = 0
This is the first derivative of the range (x) with
respect to the angle (a).
To explain how this happens, let us consider for a
moment the following:
1.
The derivative is assumed to explain the slope
91
of the tangent of the angle (a) (Marsden, & Weinstein,
1981).
2.
In figure 15, we see how (x) is fluctuating
depending upon the change in the angle (a).
In this figure, when (x) has its maximum value, the
tangent of the angle (a) has a zero (0) slope because it
is exactly horizontal.
So, for a maximum (x) in this
case, the slope of the tangent is zero.
Consequently,
the derivative of the range (x) with respect to the angle
(a) should be zero for a maximum range (x).
That is,
dx / da = 0 for maximum range (x).
maximum x
x
slope = 0
a
Figure 15.
The maximum value of x, and the tangent of
the angle (a) at peak.
Considering now equation (7) and, as mentioned
earlier, assuming that y = h, then,
h = x tana -
92
g x2
2
u20
sec2a
Differentiating the above equation with respect to
(a), we first differentiate the first part of the
equation which is the expression, x tana.
d
( x tana)
d a
d
= x
tana + tana
d a
= x sec2a + tana
d x
d a
Thus,
d x
d a
=
(10)
Following, the second part of the equation (gx 2 /
2u0) sec2a, is differentiated while keeping in mind the
following rules applying to differentiation:
1.
d/dx [ f(x) g(x)] = f(x) d/dx g(x) = g(x) d/dx
f(x).
2.
d/dx [f(x)]2 = 2 f(x) df/dx.
3.
d/dx (tanx) = sec2x
and,
d/dx (secx) = secx
tanx.
We then have for the second part,
d
d a
g
=
g
2
u20
2
2 x
=
u20
æ g x2
ç
è 2 u 20
d x2
sec2a +
d a
d x
sec2a +
d a
g
u20
x
g
2
u20
d x
sec2a +
d a
ö
sec2a÷
ø
g
2
u20
x2
=
d
sec2a =
d a
x2 2 seca seca tana =
g
u20
x2 sec2a tana
(11)
Incorporating now expressions (10) and (11) together
93
we obtain,
0 = x sec2a + tana
ég
d x
- ê 2 x sec2a
d a
ë u0
-
ù
x2 sec2a tana ú
û
g
+
d x
d a
u20
As mentioned earlier, for a maximum (x), dx/da = 0.
The above expression then becomes,
g x2
2
0 = x sec a -
u20
sec2a tana
which reduces to,
é
0 = x ê1 ë
g x
u20
ù
tana ú
û
Considering again equation (7) we see that x = 0
does not satisfy that equation.
maximize (x) not to be zero.
We simply want to
That means that the maximum
value of x satisfies (continue from above),
1 g x
1 =
tana = 0
u20
tana =
u20
x
g x
=
u 20
g
u20
g
= x tana
1
tana
(12)
Keeping in mind that,
1 / tana = cota
we obtain,
x =
u20
cota
g
94
(13)
Equations (7) and (13) can be solved simultaneously
to obtain the maximum value for (x), again considering
that y = h.
The two equations are,
g x2
h = x tana -
2
sec2a
u20
and,
x =
u20
cota
g
Keeping in mind that, cota = 1 / tana, we have from
equation 12,
tana =
Now, sec2a = 1 + tan2a.
h = x
u
2
-
x g
h =
h =
u02
g
=
g x2
2
(1
u02
g x2
2 u02
é
ê1 +
ë
g x2
2 u02
é x2 g2 + u04 ù
ê
ú =
x2 g2
êë
úû
g x2
2 u02
x2 g2 + u04
x
2
g
2
95
+ tan2a
)
=
2
é
æ u02 ö ù
ê1 + ç
÷ ú =
ê
è x gø ú
ë
û
g x2
2 u02
-
-
u20
x g
Equation (7) then becomes,
0
x u02
x g
1
cota
=
u04 ù
ú =
x2 g2 û
u02
g
u02
g
- h =
-
h g
g
=
2
x
2
g
+
u04
u02
= 2
x2 g2 = 2 u02
( u02
é u02 - h g ù
g ê
ú =
g
ë
û
)
- h g
- u04 g =
x2 g2 = 2 u04 - 2 h g u02 - u04 =
x2 g2 = u04 - h g 2 u02 =
2
x
x =
=
u04 - h g 2 u02
g2
u04 - 2 h g u02
g2
x =
u0
=
u02 ( u02 - 2 h g)
=
g2
u02 - 2 h g
g
=
(14)
Equation (14), presents the solution for maximum
(x).
In essence, it gives the maximum range that a
projectile can achieve, when it is released with an
initial velocity (u0), from a height (h) above the
ground.
As mentioned earlier,
tan a =
u20
x g
Substituting in this equation for (x) from equation
(14) we obtain,
96
tana =
u20
u20
u20 - 2 h g
g
tana =
=
g
u0
u20 - 2 h g
(15)
Equation (15) gives the optimum angle of projection
when a projectile is released with an initial velocity
(u0), from a height (h), and is proposed by this study as
a theoretical model to obtain the optimum angle of
release when the point of release is at a higher level
than that of landing.
The above model can be rewritten to reflect a
solution for the optimum angle of release as (already
shown above),
a opt
ìï
üï
u0
ý
= arctan í
ïî u20 - 2 h g ïþ
(16)
It is obvious, that the solution of the model will
produce the tangent of the optimum angle which can be
subsequently found.
However, because the above
expression is a trigonometric function, there should be
another model derived from the model above (15).
More specifically, we know from trigonometry that,
tana = opposite / adjacent
Considering then the triangle ABC below,
97
B
a
A
C
tana = BC / AC
but from (15) we have that,
tana =
u0
u20 - 2 h g
then, BC = u0, and AC = Öu02 - 2 g h.
Using the
Pythagorean theorem the AB (hypotenuse) will be,
AB2 =
(
u20 - 2 g h
AB =
)
2
+ u20 =
u20 - 2 g h + u20
=
2 u20 - 2 g h
We also know from trigonometry that,
sina =
opposite
hypotenuse
Substituting then we have,
sina =
u0
2 u20 - 2 g h
(17)
which also gives the optimum angle of release when a
projectile is released with a velocity (u0), from a
height (h), and is also proposed as a theoretical model
in this study to produce the optimum angle of release
when the point of release is at a higher level than that
of landing.
To further explore the above model, we multiply
98
denominator and numerator by 1 / u0, and we get,
u0
1
u0
=
2 -
1
2 h g
2
u20
=
1
u0
=
- 2 g h
(2
1
u20
1
æ
2 ç1 è
g hö
÷
u20 ø
é æ
sina = ê2 ç1 êë è
h gö ù
÷ú
u20 ø úû
u20
1
)
=
- 2 gh
1
=
-
u20
é
ê2
ë
æ
ç1 è
1
2
h g öù
÷ú
u20 øû
-
1
2
=
(18)
or,
a opt
é æ
= arcsin ê2 ç1 ë è
h g öù
÷ú
u20 øû
-
1
2
(19)
which also gives the optimum angle of release when a
projectile is released with a velocity (u0), from a
height (h), and is also proposed as a theoretical model
in this study to produce the optimum angle of release
when the point of release is at a higher level than that
of landing.
It is important to note here, that in the equations
(8), (9), (15), (16), (17), (18) and, (19) careful
consideration to the positive and negative signs should
be given in respect to height (h).
Thus, in equation (8), the positive (+) sign should
always be used after the expression u0 sina.
99
Under the
square root, however, if a minus (-) sign is to be used
as it is proposed after the expression u02 sin2a, then the
value of (h) should be negative when trying to
numerically solve the equation.
For example, a height of
release of 2 meters above the ground should be written as
(-2).
This is because the whole work to obtain the
equations has assumed that (h) is smaller than zero (h <
0).
It should also be noted that the same result can be
obtained if a positive sign (+) is used after the
expression u02 sin2a under the square root, and a positive
(h) is assumed.
The same holds true for equation (9).
The same also is true for equations (15), (16) and, (17)
for the sign under the square root in the denominator,
and in equations (18) and (19), for the sign after the
number 1.
Again, a positive sign will go with a positive
height (h), and a negative sign will go with a negative
height (h).
Validation of the Model
After arriving at the formulation of the model to
estimate the optimum angle of release (see equations 15,
17, 19), it was thought that a validation was necessary
to see if the models are estimating what they are
supposed to estimate.
Comparison With the Range and Height Equation
A method that can be used to validate the model is
100
with a comparison to the range and height equation.
This
method requires that the distance or range and the height
of release for a particular projection be known.
Using this method, the following equation should be
solved,
tan2a = x / h
To solve then the equation, a randomly selected
distance (x) of 17 meters and height (h) of 2.1 meters
were considered.
To achieve this, 10 commonly observed
distances and heights of release were written on a piece
of paper and a pair was drawn at random.
For this
information, the equation produces,
tan 2a = 17 / 2.1 = 8.095
The angle with a tangent of 8.095 is that of 82.96
degrees.
Dividing by 2, we obtain that the optimum angle
for the presented range and height is 41.48 degrees.
For the purpose of comparing the theoretical model
with the range and height equation of estimating the
optimum angle, the velocity of release should be known.
To achieve this, equation 9 from chapter 3 is considered.
Equation 9 can be solved for (u0) and thus, one can
substitute for the velocity of release in equation 14.
Solving for velocity, equation 9 becomes,
u0 =
x * g
(2 g h cos2a) + (2 x g sina cosa)
Substituting in the above equation the various
101
values as given earlier we obtain,
u0
=
17 * 9.81
= 12.14 m /sec
(2*9.81*2.1*0.56)+(2*17*9.81*0.66 *0.74)
The theoretical model given the above velocity will
estimate the optimum angle to be as follows:
12.14235
tana =
2
1214235
.
- 2 * 9.81 * (-2.1)
=
12.14235
13.7345
=
tana = 0.8840766
The angle that has a tangent of 0.8840156 is that of
41.48 degrees, and this is the optimum angle proposed by
the model.
Using the other model we obtain,
é æ
sina = ê2 ç1 ë è
9.81 * (- 2.1)ö ù
÷ú
øû
12.142352
-
-
1
2
=
1
sina = (2.2794556) 2
= 0.6623452
The angle with a sine of 0.6623452 is that of 41.48
degrees.
Result for the 45 Degrees Optimum Angle
Another simple way to validate the model is to
assume that (h) has the value of zero when trying to
solve it.
Since a height of zero essentially means that
the projectile is released from the ground level, and
consequently, the point of release is at the same level
as the point of landing, we know that the optimum angle
is 45 degrees.
If the models then can answer this
102
problem correctly, then it can be assumed that they will
be able to correctly estimate the optimum angle for any
other potential value of (h).
A random value for
velocity was determined to be 13 m/sec, while the height
remained zero.
To achieve this, 10 commonly observed
velocities of release were written on a piece of paper
and one was drawn at random.
Using the first model (see equation 14) we obtain,
tana =
13
132 - 2 * 9.81 * 0
13
13
=
=
tana = 1
The angle with a tangent of 1, is that of 45
degrees.
Using the second model we obtain,
é æ
sina = ê2 ç1 ë è
-
sina = (2)
1
2
9.81 * (- 0)ö ù
÷ú
øû
132
-
1
2
=
= 0.7071067
Again, a sine of 0.7071067 belongs to 45 degrees.
103
104
CHAPTER 4
Results
Introduction
The purpose of this study was, a) to examine the
relationship between the angle of release and the
velocity of release and also the relationship between the
angle of release and the height of release in the shotput event, b) based upon the above relationships, to
obtain a real life equation which would estimate the
angle of release from velocity and height of release
data, and c) develop and validate a theoretical model to
estimate the optimum angle of release from velocity of
release and height of release data.
This was for the
purpose of comparing the results from this model with
those obtained from real life throwing.
The agreement
between theory and practice was thus, assessed.
Specifically, it was an attempt,
1.
To examine the relationship between the angle of
release and the velocity of release, and also the angle
of release and the height of release in shot put, so as
to determine, primarily, whether any dependence of the
velocity of release on the release angle exists.
Based
on the above relationships, equations predicting the
angle of release from the velocity and the height of
release were obtained.
2.
To mathematically model the motion of the shot
as a projectile with respect to its optimum angle of
105
release taking into consideration the mechanical
conditions that are prevalent during the event.
3.
To examine the validity of the model by
comparing it to other models/methods relative to
projectile motion.
4.
To examine the potential application of the
model to real life situations by experimentally testing
it and applying it to a number of shot-putters.
The main concern of the study was the examination of the
relationship between the angle of release and the
velocity of release in the shot-put event, and also
between the angle of release and the height of release.
The study was delimited to 5 subjects, 10 throws for
each athlete for each of the 5 different kinds of angles
of release to be studied.
The 5 subjects were college
age shot-putters using the rotational technique of
throwing the shot, and were selected from 4 universities
in the state of Kansas.
All subjects were given equal
opportunity to achieve maximum performance as they
executed their puts.
All the throws were made in a non
competitive environment.
Subsequently, equations to estimate the angle of
release from velocity of release and height of release
data obtained from real life shot-put throwing, were
derived for each subject.
To achieve this, multiple
regression techniques were used.
Another concern was the
development of a theoretical model to estimate the
106
optimum angle of release from velocity of release and
height of release data and its subsequent application in
real life situations.
The creation of such a model helped in comparing the
obtained real life angle of release with the theoretical
optimum angle of release.
The application of the
theoretical model took place by experimentally measuring
various parameters of a shot-putter’s throwing action
such as velocity of release, height of release and angle
of release.
Subsequently, data from the real life
throwing situation were compared to the data obtained
from a theoretical model developed for this purpose.
Data from the shot-putters were obtained with the use of
high speed videography.
Theoretical data were obtained
with the use of the theoretical model.
The main concern for the development of the model
was to mathematically examine the various mechanical
components present during the release and flight of the
implement, and how they relate to the optimum angle of
release in order to achieve maximum horizontal
displacement/distance.
A system of two axes, a vertical (y) and a
horizontal (x) was assumed.
The gravitational force was
assumed to be the only force acting on the implement
after it was released.
The primary mathematical
simulation of the x, y system, a first and a second order
derivative of the gravitational force, was derived
107
through Newtonian mechanics.
Trigonometry principles
were also considered throughout the construction of the
model.
Mathematical integrations and differentiations
were employed along with equation solving procedures to
arrive at the desired model.
Finally, the model was
validated using two different methods and it was found to
be valid.
Findings
The information presented below pertains to, a) the
relationship between the angle of release and the
velocity of release, b) the predicting the real life
angle of release models and their significance and
assumptions, c) the theoretical model and its application
to the shot-put event and, d) the real life versus the
theoretical model.
Relationship and Correlation Coefficients
Table 1 depicts the correlation coefficients between
the parameters measured in this study based on the data
obtained (tables 15, 16, 17, 18, and 19 in appendix A
show a detailed reference to these data).
108
Table 1
The Correlation Coefficients Between Velocity, Height and
Angle of Release for Each of the Subjects
_________________________________________________________
Subject
ANGLE-VELOCITY
ANGLE-HEIGHT
VELOCITY-HEIGHT
_________________________________________________________
1.
- .51***
.70***
- .37**
2.
- .65***
.93***
- .54***
3.
- .76***
.91***
- .75***
4.
- .86***
.92***
- .79***
5.
- .89***
.84***
- .74***
_________________________________________________________
**p < .01. ***p < .001.
All three release parameters were significantly
correlated with each other for all the subjects of the
experiment.
Of particular interest for this study was
the significance of the negative relationship between the
velocity of release and the angle of release.
This,
possibly suggests a dependency of the angle of release on
the velocity of release, with the higher the velocity of
release the lower the angle of release.
The observation
of the raw data for the velocity of release as well as
the angle and the height of release for the 50 throws of
each of the subjects (see appendix A), showed that in all
cases an increase in the angle of release brought about a
decrease in the velocity of release.
The throws are
reported in the order in which they were thrown.
109
Scattergrams.
Figures 16, 17, 18, 19, and 20,
depict scattergrams showing the negative relationship
between the angle of release and the velocity of release
for each of the subjects.
A best fit line is also shown.
12.0
11.5
V
e
l
o
c
i
t
y
m
/
s
11.0
10.5
10.0
9.5
9.0
20
30
40
50
Angle (degrees)
Figure 16.
Scattergram of the relationship between the
angle and the velocity of release for the first subject.
110
13.0
12.5
V
e
l
o
c
i
t
y
m
/
s
12.0
11.5
11.0
10.5
10.0
9.5
9.0
10
20
30
40
50
60
Angle (degrees)
Figure 17.
Scattergram of the relationship between the
angle and the velocity of release for the second subject.
13.0
12.5
V
e
l
o
c
i
t
y
m
/
s
12.0
11.5
11.0
10.5
10.0
9.5
10
20
30
40
50
Angle (degrees)
Figure 18.
Scattergram of the relationship between the
angle and the velocity of release for the third subject.
111
12.5
12.0
11.5
V
e
l
o
c
i
t
y
m
/
s
11.0
10.5
10.0
9.5
9.0
8.5
10
20
30
40
50
Angle (degrees)
Figure 19.
Scattergram of the relationship between the
angle and the velocity of release for the fourth subject.
13.5
13.0
V
e
l
o
c
i
t
y
m
/
s
12.5
12.0
11.5
11.0
10.5
10.0
10
20
30
40
50
60
Angle (degrees)
Figure 20.
Scattergram of the relationship between the
angle and the velocity of release for the fifth subject.
112
The Real Life Regression Equation Models and The
Significance of the Regression Equations
Table 2, shows the obtained regression equations for
each of the subjects.
Table 2
The Obtained Regression Equations for Each of the
Subjects
_________________________________________________________
Subject
Equation
_________________________________________________________
1.
3.55 + (-2.90 * velocity) + (29.82 * height)
2.
-46.16 + (-2.65 * velocity) + (51.81 * height)
3.
-53.84 + (-1.88 * velocity) + (50.95 * height)
4.
-12.61 + (-3.06 * velocity) + (37.15 * height)
5.
57.13 + (-6.93 * velocity) + (28.19 * height)
_________________________________________________________
For the first subject the velocity and the height of
release accounted for 56% of the variance in the angle of
release (R square = .56; F = 29.31, p<.0001).
For the
second subject the velocity and the height of release
accounted for 90% of the variance in the angle of release
(R square = .90; F = 203.66, p<.0001).
For the third
subject the velocity and the height of release accounted
for 83% of the variance in the angle of release (R square
= .83; F = 116.06, p<.0001). For the fourth subject the
velocity and the height of release accounted for 89% of
the variance in the angle of release (R square = .89; F =
113
200.21, p<.0001).
For the fifth subject the velocity and
the height of release accounted for 87% of the variance
in the angle of release (R square = .87; F = 157.5,
p<.0001).
In all cases, the regression coefficients (b)
were significantly different from zero.
The Assumptions of the Regression Models
To test the degree to which the assumption for
normality of the obtained regression models was met, each
observed value of the standardized residuals was paired
with its expected value from the normal distribution.
These plots are shown in figures 21, 22, 23, 24, and 25
for the data of each of the subjects.
From these
figures, it seems that no violation of this assumption
took place.
Furthermore, the values for the K-S
Lilliefors test for normality were, .0597, p>.20, .0598,
p>.20, .0585, p>.20, .0566, p>.20, and .0694, p>.20
respectively, for each of the subjects.
To test whether the assumption for linearity of the
regression model was met, the standardized residuals were
plotted against the standardized predicted scores.
These
plots are shown in figures 26, 27, 28, 29, and 30 for
each of the subjects.
Again, it seems that the linearity
assumption was met in all cases since there is no
particular systematic clustering of the points.
114
3
2
E
x
p
e
c
t
e
d
N
o
r
m
a
l
1
0
-1
-2
-3
-3
-2
-1
0
1
2
3
Observed Value
Figure 21.
A normal probability plot for the data of the
first subject.
3
2
E
x
p
e
c
t
e
d
N
o
r
m
a
l
1
0
-1
-2
-3
-3
-2
-1
0
1
2
3
Observed Value
Figure 22.
A normal probability plot for the data of the
second subject.
115
3
2
E
x
p
e
c
t
e
d
N
o
r
m
a
l
1
0
-1
-2
-3
-3
-2
-1
0
1
2
3
Observed Value
Figure 23.
A normal probability plot for the data of the
third subject.
3
2
E
x
p
e
c
t
e
d
N
o
r
m
a
l
1
0
-1
-2
-3
-3
-2
-1
0
1
2
3
Observed Value
Figure 24.
A normal probability plot for the data of the
fourth subject.
116
3
2
E
x
p
e
c
t
e
d
N
o
r
m
a
l
1
0
-1
-2
-3
-3
-2
-1
0
1
2
3
Observed Value
Figure 25.
A normal probability plot for the data of the
fifth subject.
2
1
R
e
s
i
d
u
a
l
s
0
-1
-2
-3
-3
-2
-1
0
1
Regression Standardized Predicted Value
Figure 26.
Randomly distributed residuals for the data
of the first subject.
117
2
3
2
R
e
s
i
d
u
a
l
s
1
0
-1
-2
-2
-1
0
1
2
3
Regression Standardized Predicted Value
Figure 27.
Randomly distributed residuals for the data
of the second subject.
3
2
R
e
s
i
d
u
a
l
s
1
0
-1
-2
-3
-2.0
-1.5
-1.0
-.5
0.0
.5
1.0
1.5
Regression Standardized Predicted Value
Figure 28.
Randomly distributed residuals for the data
of the third subject.
118
2.0
2
1
R
e
s
i
d
u
a
l
s
0
-1
-2
-3
-1.5
-1.0
-.5
0.0
.5
1.0
1.5
2.0
2.5
Regression Standardized Predicted Value
Figure 29.
Randomly distributed residuals for the data
of the fourth subject.
2
1
R
e
s
i
d
u
a
l
s
0
-1
-2
-3
-2
-1
0
1
2
Regression Standardized Predicted Value
Figure 30.
Randomly distributed residuals for the data
of the fifth subject.
119
3
Validity and Numerical Solution of the Theoretical Model
In the methodology section of this study, the
validity of the theoretical model presented here was
examined.
It was found that the theoretical model
accurately estimated the optimum angle of release as
compared to the criteria.
The theoretical optimum angle
of release as estimated by the present model was 41.48
degrees and it was in exact agreement with the criterion
method of the range and height equation.
Furthermore,
when a height of zero was assumed, the theoretical model
correctly estimated that the optimum angle should be 45
degrees.
Given then that the model passed both tests for
validity, we do not have any reason to reject it.
It is
by all means an accurate theoretical model estimating the
optimum angle of release of a projectile when its
velocity and height of release are known.
The theoretical model was applied to the shot put
event.
A range of initial velocities similar to those
achieved during competition, by beginners as well as
advanced throwers, were chosen as well as a range of
various heights of release.
Subsequently, the model for
the optimum angle of release was applied and solved for
various combinations of velocity of release.
A computer program was employed throughout this
process for the calculations of the optimum angles of
release and/or ranges achieved by the implement.
120
Figure
31, shows the variation of the optimum angle of release
with changing height and velocity of release.
Velocities
from a low of 11 m/sec., to a high of 15 m/sec., were
examined in combination with heights from 1.8 to 2.2
meters above the ground.
From figure 31, the following conclusions were made:
1.
The optimum angle of release is never 45 degrees
and is always less than 45 degrees.
2.
In the shot-put, the optimum angle of release
will generally fluctuate between 40.4 degrees, for the
lower velocities of release in combination with the
higher heights of release, and 43 degrees, for the higher
velocities of release in combination with the lower
heights of release.
4.
For a particular velocity of release, the higher
the height of release, the smaller the optimum angle.
5.
For a particular height of release, the higher
the velocity of release, the more the optimum angle
approaches the value of 45 degrees.
121
43
Angle of release (degrees)
42.5
11 m/sec
42
12 m/sec
41.5
13 m/sec
13.5 m/sec
41
14 m/sec
40.5
14.5 m/sec
15 m/sec
40
39.5
39
1.8
1.9
2
2.1
2.2
2.3
Height of release (meters)
Figure 31.
Variation of the theoretical optimum angle of
release with changing height and velocity of release in
the shot-put.
Theoretically speaking then for the shot-put event,
a) the lower the velocity of release, the more the
optimum angle of release should deviate (be smaller) from
45 degrees, and b) the higher the height of release, the
more the optimum angle of release also deviates from 45
degrees.
To examine the effect of the use of the proper
angle of release in the shot-put event, a variety of
combinations of angles of release and heights of release
usually observed, were used.
Three main release velocities in the range of which
most athletes release their implement in the shot-put
122
event were used, along with a combination of usually
observed angles and heights of release.
were those of 12, 13, and 14.5 m/sec.
were 1.9, 2.0, and 2.1 meters.
These velocities
The heights used
Following, equation 9,
which gives the range of a projectile when the point of
release is higher than that of landing was repeatedly
solved numerically using a computer program to achieve
accuracy of the results each time.
Results of the solutions of this equation are
graphically shown in figures 32, 33, and 34.
16.65
16.6
16.55
Range (m)
16.5
1.9 m
16.45
2.0 m
16.4
2.1 m
16.35
16.3
16.25
16.2
38
38.5
39
39.5
optimum
43
Angle (degrees)
Note: The Theoretical Optimum Angles of Release for the Heights
are: h = 1.9 angle = 41.71, h = 2 angle = 41.56, and h = 2.1
angle = 41.40
Figure 32.
Variation of projectile range with changing
angle and height of release when the velocity of release
is 12 m/sec.
In all of these figures, where results were examined
123
with the velocity of release constant for each of them,
it is seen that generally, as the angle increases, the
range of the projectile also increases up to the optimum
angle (as estimated by the model) while it drops as the
angle further increases beyond the optimum angle.
Obviously, the degree to which the range increases
depends upon the proximity of the angle under examination
to the optimum angle of release.
The further away from
the optimum angle the more dramatic the increase in the
range and the opposite.
19.3
19.2
Range (m)
19.1
1.9 m
19
2.0 m
2.1 m
18.9
18.8
18.7
38
38.5
39
39.5
optimum
43
Angle (degrees)
Note: The Theoretical Optimum Angles of Release for the
Heights are: h = 1.9 angle = 42.15, h = 2 angle = 42.01, and h
= 2.1 angle = 41.88
Figure 33.
Variation of projectile range with changing
angle and height of release when the velocity of release
is 13 m/sec.
124
23.5
23.4
Range (m)
23.3
23.2
1.9 m
23.1
2.0 m
23
2.1 m
22.9
22.8
22.7
38
38.5
39
39.5
optimum
43
Angle (degrees)
Note: The Theoretical Optimum Angles of Release for the
Heights are: h = 1.9 angle = 42.66, h = 2 angle = 42.55, h =
2.1 angle = 42.44
Figure 34.
Variation of projectile range with changing
angle and height of release when the velocity of release
is 14.5 m/sec.
The Real Life Versus the Theoretical Angles of Release
Tables 3, 4, 5, 6, and 7, show the angles of release
obtained via the regression models (real life angles) and
the angles obtained via the theoretical model
(theoretical optimum angles), for each of the subjects
tested.
It was found that for all the subjects the
regression coefficients (b) were significantly different
from one, and the constants (a) significantly different
from zero (see table 8).
Consequently, there was no
agreement between the real life and the theoretical
model.
Table 3
Real and Theoretical Obtained Angles for the First
125
Subject
_________________________________________________________
Real angle (degrees)*
Theoretical angle (degrees)
_________________________________________________________
1. 39.17
1. 40.41
2. 31.05
2. 41.31
3. 38.00
3. 40.63
4. 27.95
4. 41.60
5. 32.69
5. 41.24
6. 33.20
6. 40.71
7. 38.17
7. 40.75
8. 35.14
8. 40.62
9. 33.97
9. 40.84
10. 37.30
10. 40.89
11. 38.43
11. 40.48
12. 33.00
12. 41.27
13. 33.10
13. 40.98
14. 34.17
14. 41.08
15. 32.22
15. 41.12
16. 35.07
16. 41.08
17. 32.41
17. 41.33
18. 35.51
18. 41.03
19. 32.49
19. 40.97
20. 28.98
20. 41.49
_______________________________________________________
* Note.
Real life angles were obtained via the
regression model for this subject.
Table 4
Real and Theoretical Obtained Angles for the Second
Subject
_________________________________________________________
Real angle (degrees)*
Theoretical angle (degrees)
_________________________________________________________
126
1. 29.02
1. 41.50
2. 25.52
2. 41.67
3. 33.42
3. 41.42
4. 40.32
4. 40.91
5. 38.53
5. 40.80
6. 39.01
6. 41.10
7. 38.36
7. 41.14
8. 25.53
8. 41.58
9. 35.12
9. 41.26
10. 32.43
10. 41.07
11. 36.30
11. 41.10
12. 32.21
12. 40.69
13. 29.18
13. 41.30
14. 28.42
14. 41.17
15. 29.85
15. 41.05
16. 33.72
16. 41.08
17. 29.40
17. 41.60
18. 26.18
18. 41.54
19. 24.74
19. 41.65
20. 32.82
20. 41.08
_______________________________________________________
* Note.
Real life angles were obtained via the
regression model for this subject.
Table 5
Real and Theoretical Obtained Angles for the Third
Subject
_________________________________________________________
Real angle (degrees)*
Theoretical angle (degrees)
_________________________________________________________
1. 32.59
1. 41.23
2. 39.35
2. 41.13
3. 31.22
3. 40.77
4. 28.46
4. 41.49
127
5. 39.44
5. 40.29
6. 39.76
6. 40.35
7. 38.05
7. 41.24
8. 29.98
8. 40.72
9. 26.20
9. 41.49
10. 38.67
10. 40.91
11. 31.90
11. 41.30
12. 33.76
12. 41.01
13. 35.18
13. 41.27
14. 36.86
14. 41.04
15. 26.75
15. 41.59
16. 31.44
16. 41.16
17. 37.08
17. 40.31
18. 36.65
18. 40.65
19. 31.08
19. 40.98
20. 28.21
20. 41.02
_______________________________________________________
* Note.
Real life angles were obtained via the
regression model for this subject.
Table 6
Real and Theoretical Obtained Angles for the Fourth
Subject
_________________________________________________________
Real angle (degrees)*
Theoretical angle (degrees)
_________________________________________________________
1. 35.57
1. 40.88
2. 34.28
2. 40.65
3. 35.83
3. 40.63
4. 32.25
4. 40.90
5. 31.16
5. 40.82
6. 31.18
6. 40.69
7. 35.44
7. 40.31
8. 27.24
8. 41.26
128
9. 29.54
9. 41.20
10. 27.92
10. 41.19
11. 31.26
11. 40.60
12. 36.38
12. 40.89
13. 31.59
13. 40.85
14. 31.20
14. 40.99
15. 27.53
15. 41.31
16. 33.25
16. 40.62
17. 29.25
17. 40.73
18. 37.14
18. 40.25
19. 36.27
19. 40.67
20. 27.64
20. 41.13
_____________________________________________________
* Note.
Real life angles were obtained via the
regression model for this subject.
Table 7
Real and Theoretical Obtained Angles for the Fifth
Subject
_________________________________________________________
Real angle (degrees)*
Theoretical angle (degrees)
_________________________________________________________
1. 37.24
1. 41.17
2. 34.51
2. 41.34
3. 31.95
3. 41.45
4. 33.91
4. 41.42
5. 39.41
5. 41.03
6. 30.63
6. 41.54
7. 38.78
7. 41.07
8. 33.26
8. 41.43
9. 39.26
9. 41.01
10. 35.44
10. 41.21
11. 30.70
11. 41.53
12. 36.79
12. 41.14
129
13. 34.04
13. 41.39
14. 34.54
14. 41.38
15. 38.89
15. 41.00
16. 35.07
16. 41.20
17. 39.71
17. 40.91
18. 39.19
18. 41.03
19. 38.95
19. 40.99
20. 34.90
20. 41.27
_______________________________________________________
* Note.
Real life angles were obtained via the
regression model for this subject.
Table 8
Tests of Significance for the Regression Coefficients
(b), and constants (a), in Determining the Agreement
Between the Theoretical and the Real Life Models
_________________________________________________________
Subject
Regression coefficient (b)
Constant (a)
_________________________________________________________
1.
t = -8.29, p <.05
t = 8.14, p = .0001
2.
t = -4.96, p <.05
t = 4.88, p = .0001
3.
t = -3.65, p <.05
t = 3.56, p = .0022
4.
t = -5.53, p <.05
t = 5.39, p = .0001
5.
t = -20.71, p <.05
t = 20.53, p = .0001
_________________________________________________________
Figures 35, 36, 37, 38, and 39, show a graphical
representation of the results obtained using the real
life versus the theoretical model, for each of the
subjects.
The velocities and the heights used were those
exhibited during normal throwing.
130
As expected, the real
life angle trend is the exact opposite of that exhibited
by the theoretical optimum angle.
Higher velocities
correspond to lower angles of release in real life shotputting, whereas the opposite is the case in theory
where, for a given height, the higher the velocity the
higher the optimum angle.
Moreover, there is only a
minor fluctuation of the theoretical optimum angle of
release as opposed to the real life angle which presents
greater variation.
45
43
Angle of release (degrees)
41
39
10.6 R
11
R
11.3 R
37
11.6 R
11.8 R
35
10.6 T
11
33
T
11.3 T
11.6 T
31
11.8 T
29
27
25
2
2.05
Note.
Figure 35.
2.1
2.2
Height (meters)
R = Real angle data, T = Theoretical angle data
Variation of the real life versus the
theoretical angle of release for the first subject.
131
44
42
Angle of release (degrees)
40
38
11.3
11.6
11.9
12.1
12.3
11.3
11.6
11.9
12.1
12.3
36
34
32
30
R
R
R
R
R
T
T
T
T
T
28
26
24
22
2
2.05
Note.
2.1
2.15
Height of release (meters)
R = Real angle data, T = Theoretical angle data
Figure 36. Variation of the real life versus the
theoretical angle of release for the second subject.
43
41
Angle of release (degrees)
39
10.6
R
11
R
35
11.4
11.8
12.1
R
R
R
33
10.6
11
T
T
31
11.4
11.8
T
T
12.1
T
37
29
27
25
2
2.05
Note.
2.1
2.2
Height of release (meters)
R = Real angle data, T = Theoretical angle data
Figure 37. Variation of the real life versus the
theoretical angle of release for the third subject.
132
43
41
Angle of release (degrees)
39
37
35
10.4
R
10.8
R
11.1
11.3
R
R
11.65 R
33
31
10.4
T
10.8
T
11.1
T
11.3
T
11.65 T
29
27
25
2
2.05
Note.
2.1
Height (meters)
R = Real angle data, T = Theoretical angle data
2.15
Figure 38. Variation of the real life versus the
theoretical angle of release for the fourth subject.
43
41
Angle of release (degrees)
39
11.4
R
37
11.7
R
35
11.9
12.1
R
R
12.25 R
33
31
11.4
T
11.7
T
11.9
T
12.1
T
12.25 T
29
27
25
2
2.05
Note.
2.1
Height (meters)
R = Real angle data, T = Theoretical angle data
2.2
Figure 39. Variation of the real life versus the
theoretical angle of release for the fifth subject.
133
Obtained Ranges
Tables 9, 10, 11, 12 and 13, show the ranges
obtained using the data from tables 15, 16, 17, 18, and
19 in the equation for range (equation 9 in the
methodology section) and also the ranges actually
measured.
The ten “normal” throws for each subject were
considered for that purpose.
It should be noted here
that the equation for range does not account for the
distance in front or, for that case, behind the stopboard (see figure 40).
The equation will account for the
distance between the point of release and the point of
landing, whereas the actual measurement in the shot-put
event takes place from the inside of the stop-board to
the point of landing.
As a result, a certain degree of
discrepancy should be expected as the values of the two
differently obtained ranges are evaluated.
extra distance
Figure 40.
The extra distance gained by the thrower over
the circle boundaries.
It can then be seen that, as a rule, the ranges
obtained using the data and the equation for range, are
in approximation with the actually measured ranges.
134
Consequently, there was no reason to seriously doubt the
accuracy of the data obtained from the analysis.
Table 9
Ranges Obtained Using the Data From the Analysis and the
Equation for Range, and also Ranges Actually Measured
During Throwing, for the 10 Normal Throws of the First
Subject
_________________________________________________________
Throw
Actually measured range
(meters)
Estimated range
(meters)
_________________________________________________________
1.
13.86
13.38
2.
13.91
13.61
3.
13.77
13.26
4.
14.70
14.49
5.
15.11
14.62
6.
15.01
15.03
7.
14.63
14.62
8.
15.66
15.64
9.
14.87
14.91
10.
14.68
15.29
_______________________________________________________
Table 10
Ranges Obtained Using the Data From the Analysis and the
Equation for Range, and also Ranges Actually Measured
During Throwing, for the 10 Normal Throws of the Second
Subject
_________________________________________________________
135
Throw
Actually measured range
(meters)
Estimated range
(meters)
_________________________________________________________
1.
16.17
15.69
2.
16.34
16.30
3.
16.28
16.37
4.
15.62
14.93
5.
15.39
14.89
6.
15.46
15.45
7.
15.57
15.31
8.
15.90
15.71
9.
14.29
14.23
10.
15.41
14.62
_________________________________________________________
Table 11
Ranges Obtained Using the Data From the Analysis and the
Equation for Range, and also Ranges Actually Measured
During Throwing, for the 10 Normal Throws of the Third
Subject
_________________________________________________________
Throw
Actually measured range
(meters)
Estimated range
(meters)
_________________________________________________________
136
1.
15.29
14.83
2.
14.93
15.57
3.
14.50
14.64
4.
15.26
14.70
5.
15.39
15.89
6.
15.72
16.15
7.
16.16
16.61
8.
15.88
16.15
9.
14.93
15.69
10.
13.28
12.81
_________________________________________________________
Table 12
Ranges Obtained Using the Data From the Analysis and the
Equation for Range, and also Ranges Actually Measured
During Throwing, for the 10 Normal Throws of the Fourth
Subject
_________________________________________________________
Throw
Actually measured range
(meters)
Estimated range
(meters)
_________________________________________________________
1.
13.74
13.47
2.
13.33
12.67
3.
14.22
14.21
137
4.
13.89
14.60
5.
13.97
14.19
6.
13.71
13.79
7.
14.47
14.94
8.
13.58
13.19
9.
13.48
13.83
10.
13.73
14.22
_________________________________________________________
Table 13
Ranges Obtained Using the Data From the Analysis and the
Equation for Range, and also Ranges Actually Measured
During Throwing, for the 10 Normal Throws of the Fifth
Subject
_________________________________________________________
Throw
Actually measured range
(meters)
Estimated range
(meters)
_________________________________________________________
1.
15.23
14.96
2.
16.54
17.00
3.
16.40
16.55
4.
14.78
15.14
5.
16.94
16.51
6.
16.02
16.14
138
7.
14.67
15.15
8.
15.85
16.30
9.
15.63
15.97
10.
16.82
16.65
_________________________________________________________
139
CHAPTER 5
Discussion
The investigation was performed utilizing five
experienced shot-putters.
The distances they put varied
between 13.30 and 16.65 meters depending upon the angle
of release under investigation.
Fifty throws for each
subject were filmed.
This resulted in the analysis of a
total of 250 throws.
In this chapter, the findings of
the study and also other issues relevant to the present
investigation will be discussed.
The Relationship Between the Angle and the Velocity of
Release
A significant relationship was observed between the
velocity of release and the angle of release for all the
shot-putters tested in this study thus, showing a
dependency of the angle of release on the velocity of
release.
As hypothesized by Zatsiorsky (1990) and also
by Hay (1985, 1993) there should be a correlation between
the angle of release and the velocity of release.
Consequently, the hypothesis that the angle of release is
dependent on the velocity of release is accepted.
Initially it was thought that the shot-putters would
be able to project the shot with the appropriate velocity
in spite of the release angle.
The theoretical optimum
angle of release then, would potentially allow the shotputters to achieve maximum range.
In the last few years,
however, biomechanical analyses have demonstrated that
140
athletes in real life competition prefer to throw using
angles lower or much lower than the theoretical optimum
(McCoy, 1990; Susanka & Stepanek, 1987; Zatsiorsky, 1990)
It is evident then from the results of this study
that apparently shot-putters intuitively choose as the
angle of release that which will result in the least loss
of velocity of release possible while at the same time
they maintain the angle of release within “reasonable”
levels.
It should be repeated here that since release
velocity is the release parameter most dramatically
affecting the range of a projectile, it seems wise and
logical to “sacrifice” another release parameter, like
the angle of release, instead of the most valuable one of
velocity.
From a mechanical point of view it should also be
evident that as the angle of release increases and
becomes steeper, the athlete works more and more against
the gravitational force, a fact that apparently results
in the loss of velocity at the moment of projection.
Moreover, less appreciated is the assumption made during
the derivation of the theoretical model and its
implications.
The theoretical model presented in this
study holds true only when the velocity of release (u0)
is assumed to be an independent variable and not a
function of the angle of release (a) which would be the
case had the thrower been able to achieve the same
141
release velocity independent of the release angle.
On the other hand, if the thrower can throw faster
at lower angles, which is probably the case as shown in
this study, then the optimum release angle may be found
using optimization techniques.
In this case the optimum
angle of release will be somewhat or considerably smaller
than that predicted by the theoretical model.
The degree
to which the angle will be smaller should be thrower
specific.
Given the above, the hypothesis that the
theoretical model is not applicable to the shot-put
event, is accepted.
Vigars (1977) discussed that during the delivery of
the shot the throwing arm is forcefully extended with the
shot projected at approximately a 45 degree angle.
Although there was no discussion as to the nature of the
approximation, presently, figures in the order of 45, 42
or even 40 degrees should be too great to be suggested as
real optimum angles of release.
Certainly, for the
purpose of this study, all the shot-putters were able to
project the shot at angles greater than 45 degrees, the
highest being 54.14 degrees.
Again, for the purpose of
the experiment the lowest angle recorded was 15.97
degrees.
Possible Causes for the Observed Differences Between Real
and Theoretical Angles and Suggested Remedies
Mc Watt (1982) discussed that the observed
differences should, at least in part, be attributed to
142
the anatomical and physiological character of the shotputter.
He discussed two ways that might help the shot-
putter achieve higher angles of release.
the first as the shoulder roll.
He described
According to this
method, the athlete does not push the shot from a purely
vertical position, because naturally the trunk bends
backwards slightly during the release of the shot, a
movement that allows the shoulders to roll into a
position that increases the angle of release without
changing too much the ability of the arm to deliver
maximum power.
The second way is described as the leg thrust.
It
is possible that further elevation can occur from a well
coordinated forceful upward extension of the legs at the
moment of release exactly at the time the arm extends to
push the shot.
This might add a vertical component to
the horizontal component of the arm thrust thus,
resulting in the production of a resultant force closer
to the theoretical optimum.
Judge (1994) speculated that the shot-putter he
analyzed exhibited a lower release angle because there
was a difference in bench press strength as opposed to
overhead strength.
He postulated that more incline bench
presses and overhead movements would develop the muscles
necessary for a projection close to the theoretical
optimum.
The same speculation was made by McCoy, Gregor
and Whitting (1989).
143
Most certainly, all these speculations pertaining to
the various actions to be taken to enable the shot-putter
to throw according to his/her theoretical optimum,
deserve attention and are worth researching in the near
future.
Still, some world class shot-putters have been
reported (Dessureault, 1976; Zatsiorsky, 1981) to have
achieved angles of release very close to the theoretical
optimum.
This fact leads to further speculation whether
highly trained shot-putters are able to throw at angles
of release similar to those estimated by the theoretical
model.
Given the observations mentioned above and also the
results obtained from this study, it should be evident
that in real life shot-putting the velocity of release
depends crucially on the angle of release.
This
phenomenon might be generalized and may lead us to the
concept that the optimum release conditions other than
the velocity of release depend dramatically on how the
highest obtainable release velocity is functionally
related to the other release conditions.
As expected, there were significant correlations
between the angle of release and the height of release.
Obviously, as the athlete tries to increase or decrease
the angle of release he/she can most conveniently do that
by projecting the shot from a higher or lower point,
respectively.
Consequently, the hypothesis that the
angle of release is dependent on the height of release is
144
accepted.
The Issue of Collinearity During the Construction of the
Regression Models
The phenomenon of collinearity was considered in the
present investigation.
As a rule, this phenomenon occurs
when two or more predictor variables correlate highly
with each other.
Since, in this study, two predictor
variables (the velocity and the height of release) were
used to estimate another variable (the angle of release),
it was deemed appropriate to examine whether the two
variables were highly collinear.
Pedhazur (1982) discussed three main elements to aid
in the detection of the presence of collinearity.
First,
a decrease in the overall R square, showing a redundancy
between the variables that interrelate.
Second, a
reduction in the magnitude of the betas (b) for the
variables that interrelate and third, an increase in the
standard error of the betas (b) for the same variables.
Stevens (1992) addressed that the presence of
multicollinearity severely limits the size of the
multiple R, because the predictors are going after the
same variable.
The addition of the other collinear
variable will not increase (R) over and above the
magnitude of the simple correlation between one of the
collinear variables and the dependent variable.
Stevens
(1992) added that highly correlated variables do not
145
always indicate the extend of multicollinearity.
Another indicator of the presence of collinearity is
the examination of the variance inflation factor (VIF).
It is believed that a value of the VIF that exceeds 10
should cause concern.
Norusis (1992) added that the
tolerance of a variable is closely related to the
variance inflation factor.
A low tolerance level would
indicate the presence of collinearity.
Table 14 shows the values of the various parameters
that might indicate the presence of collinearity.
These
values were observed before and after the height variable
was entered into the equation as the second predictor.
Table 14
Collinearity Diagnostics for the Regression Equation of
Each of the Subjects
_________________________________________________________
Subject
Mult.R
b
S.E.b
VIF
Tolerance
_________________________________________________________
1.
2.
3.
4.
5.
Before:
.50
-.50
1.25
1.00
1.00
After:
.74
-.28
1.05
1.16
.86
Before:
.64
-.64
1.45
1.00
1.00
After:
.94
-.20
.73
1.41
.70
Before:
.75
-.75
1.04
1.00
1.00
After:
.91
-.17
1.00
2.31
.43
Before:
.85
-.85
.63
1.00
1.00
After:
.94
-.35
.63
2.60
.38
Before:
.89
-.89
.75
1.00
1.00
146
After:
.93
-.59
.91
2.21
.45
_________________________________________________________
From table 14 it can be seen that the multiple R’s
present significant increases, the standard error of the
coefficients (b) does not increase, the variance
inflation factors present rather minor increases, and the
tolerance levels remain generally high.
Given these
observations and also the observation that the two
predictor variables (velocity and height) correlate
higher with the criterion (angle) than with each other
(see table 1), it was decided that both predictor
variables were to be retained for the regression
analysis.
The Theoretical Optimum Angle Model
A theoretical model to estimate the exact
theoretical optimum angle of release in the shot-put has
been presented.
This model was created mainly for the
purpose of comparing the data from real life to the
theoretical data obtained with the aid of this model.
It was found that the appropriate solution of the
resulting equation can accurately estimate the
theoretical optimum angle of release.
The Assumptions of the Theoretical Model
The main assumption that dominated the process of
constructing the theoretical model was that gravity was
the only force acting on the projectiles after their
release.
Consequently, some other factors that could
147
affect projectile motion were not considered.
Following,
some reasons as to why these factors were not considered
are given.
Discussion pertaining to all the assumptions
made during the construction of the model is made here.
Force due to air resistance.
These forces were
ignored during the process of model construction.
There
were mainly four reasons for this decision (Kibble,
1973).
First, and probably most important, the effect of
the drag force on the mass of the shot is minimal.
This
is because the drag force is rather small, small enough
to minimally affect the much larger mass of the shot
weighing 7,26 kilograms or 16 pounds.
As mentioned
earlier, the mass of the implement determines how motion
will be affected by the drag force.
Second, the velocity achieved in the shot-put event
is not high.
The maximum velocity that has been achieved
in the shot-put event is in the vicinity of 14.5 m/sec.
These velocities are small enough as compared to the
velocities achieved by other projectiles such as military
projectiles which can achieve velocities up to 1000
m/sec.
The low velocities of release coupled with a low
drag force result in minimal effects of the drag force
upon the implements.
Third, the air density is low, almost one thousand
times lower than that of water.
Because density
determines how compact a fluid is, movement through a low
148
density fluid is much easier than through a high density
fluid.
Finally, the area facing the air flow of the shot is
constant due to its spherical shape.
Although this
implement is not considered aerodynamical, its shape
allows it to maintain the drag force acting on it also
constant.
The projectile is a point mass or particle.
In an
accurate analysis, it is considered that the projectile
possesses finite volume and a definite surface
configuration.
The concern would then be with the motion
of the mass center.
The shape of the shot, indeed,
allows one to conveniently examine the motion of the mass
center since it remains unchanged throughout its flight
(Wallace & Fenster, 1969).
The earth is non-rotating.
If greater accuracy is
required, the accelerated or non-inertial motion
(Coriolis acceleration) of the earth beneath the
projectile must be taken into account.
In the
construction of the model, the earth is used as a
reference for which Newton’s laws are assumed valid.
This is a very good approximation for short ranges.
The gravitational field is constant and acts
perpendicularly to the surface of a flat earth.
For
distances small in comparison with the earth’s radius,
the flat earth assumption will yield good results.
Motion occurs in a plane.
149
This is, indeed, the case
during the projection of the shot.
Finally, it should be noted that it is possible to
account for the effects of all the aforementioned factors
upon the shot during its projection.
F. Mirabelle
(personal communication, January 1995), postulated that
all things considered, the change in the optimum angle of
release in shot-put is a mere 0.1 of a degree less than
what a vacuum model would estimate.
The Construction of the Model
Gravity then, as the only force acting on the
projectile was first considered toward constructing the
model and the Newton’s second law of motion was explored
in relation to the x, y right angle system depicting the
projectile’s range.
During these first steps, the point
of release and the point of landing were assumed to be at
the same level with an (h) of zero.
Following a series
of procedures it was possible to obtain a solution which
gave the vertical component (y) and how it functions with
the other elements of the vector system, particularly
with the angle of release.
Following, the implementation of the (y) component
to the whole projectile problem (given that the
projectile is released from a height (h) above the ground
as it happens in a real life projection of the shot), was
considered.
It was thus assumed for this purpose that y
= h and using a series of mathematical functions the
theoretical model which described how the optimum angle
150
of release functions in respect to the height of release
(h), the velocity of release (u0), and the gravitational
force was obtained.
The intention in the process of constructing this
model was to always manipulate the various equations and
find ways to see how the angle of release can be thought
as a function of the height (h) and the velocity of
release (u0).
The Importance of the Angle of Release
Although it has been concluded that it should be
futile to attempt to apply the theoretical/mechanical
principles in estimating the optimum angle of release in
the shot-put event, still, a discussion of the
implications of the theoretical model can be useful in
assessing the importance of the proper angle of release.
Theoretically, as shown in the results section of
this study, the relative importance of the angle of
release in the shot-put event can be extremely
significant always depending upon how close to the
optimum angle of release the achieved angle is, and also
assuming that all the other factors that affect the
projectile’s range are constant for comparison purposes
only.
In the shot-put, an increase of one degree in the
angle of release from 38 to 39 degrees will bring an
increase of a minimum of 5 centimeters up to a maximum of
151
10 centimeters depending upon the velocity of release,
the higher the velocity, the higher the increase.
A
further increase in the angle from 39 degrees to the
optimum angle will bring an increase of a minimum of 6,
up to a maximum of 16.5 centimeters again depending upon
the velocity of release.
A total increase between 10 and
26.5 centimeters resulted when the angle of release
increased from 38 degrees to the optimum angle of
release.
Once again the higher the velocity of release, the
greater the effect of increasing the angle of release
toward the optimum.
It should be repeated here that the
importance of the angle of release achieved depends upon
its proximity to the optimum angle of release.
Thus, if
the optimum angle is, say, 42 degrees, a change from 40
to 42 degrees will bring about greater changes in the
range than a change from 41.5 to 42 degrees.
The
magnitude of these changes will be in the order presented
above for minimum or maximum increases, or even smaller
if the change in the angle is minimal.
It is expected then, that minor increases in the
angle of release toward the optimum angle, will also
bring minor increases in the range.
More important, it
can be of interest how much, in practice, the increase in
the range is.
A difference/increase of, say, 5
centimeters might not sound significant, however, it
could be significant in real life where 5 centimeters
152
might make the difference between winning or losing.
For example, during the 1980 Olympic games the
three first winners in the shot-put event were within 5
centimeters of a meter from each other.
Other examples
like this, are numerous during shot-put competition.
Therefore, the results should be evaluated only
quantitatively.
One can argue here, that the differences in the
projectile range are not necessarily due to differences
in the angle of release.
Of course, one may never know
what exactly causes these differences unless a detailed
analysis is performed and thus, the differences between
throwers can be located.
The point here, however, is
again, that even minor differences in distance achieved
between throwers can mean the difference between winning
or losing and that the angle of release can contribute
more or less towards a thrower’s goal in achieving the
perfect throw.
In today’s highly competitive environment
of sports, more and more athletes strive for perfection.
In the literature, the theoretical optimum angle
of
release for the shot-put event has been reported to be in
the range of 41 to 43 degrees.
In this study, absolutely
speaking, the results for the theoretical optimum angle
of release fluctuated between 40.4 degrees for low
velocities of release (11 m/sec.) to 43 degrees for high
velocities of release (14.5 m/sec.).
It was estimated that to achieve throwing distances
153
of 19.12 meters up to 20.5 meters, the theoretical angle
of release should in most cases, where the height of
release does not exceed 2 meters, exceed the value of 42
degrees.
For distances between 20.5 meters up to 21.88
meters, the theoretical optimum angle should always be
greater than 42 degrees, fluctuating between 42 and 42.6
degrees, depending upon the height of release, with the
lower the height the higher the angle.
For distances of
21.88 meters and above, the optimum angle of release is
also always greater than 42 degrees, fluctuating between
42.2 and 43 degrees if distances more than 23 meters are
to be achieved.
Optimum angles of release below 40.4 degrees
correspond to distances achieved in a low level of
competition and reach values of 39.9 degrees for
velocities of release of about 9 m/sec., which correspond
to approximately 10 meters distance in shot put throwing.
It is also known that the gravitational acceleration
is not constant throughout the world.
differences do exist.
Some minor
For example, at the equator, the
acceleration due to gravity is 9.78 m/sec 2., whereas in
the poles is 9.832 m/sec2.
These changes have been found
to have insignificant effects in the optimum angle of
release.
In the shot-put for example, the optimum angle of
release for a shot released at 13 m/sec., from a height
of 1.9 meters above the ground, under normal
154
gravitational acceleration conditions (9.81 m/sec 2) is
42.15 degrees.
Under a gravitational acceleration of
9.832 m/sec2., this angle would minimally decrease to
42.14 degrees, whereas under a gravitational acceleration
of 9.78 m/sec2., this angle would minimally increase to
42.155 degrees.
Consequently, the effects of gravity on the optimum
angle of release can be considered non-existent in
determining the theoretical optimum angle of release.
155
CHAPTER 6
Summary, Conclusions, and Recommendations
Summary
The purpose of this study was, a) to examine the
relationship between the angle of release and the
velocity of release and also the relationship between the
angle of release and the height of release in the shotput event, b) based upon the above relationships, to
obtain a real life equation which would estimate the
angle of release from velocity and height of release
data, and c) develop and validate a theoretical model to
estimate the optimum angle of release from velocity of
release and height of release data.
This was for the
purpose of comparing the results from this model with
those obtained from real life throwing.
The agreement
between theory and practice was thus, assessed.
Specifically, it was an attempt,
1.
To examine the relationship between the angle of
release and the velocity of release, and also the angle
of release and the height of release in shot put, so as
to determine, primarily, whether any dependence of the
velocity of release on the release angle exists.
Based
on the above relationships, equations predicting the
angle of release from the velocity and the height of
release were obtained.
2.
To mathematically model the motion of the shot
as a projectile with respect to its optimum angle of
156
release taking into consideration the mechanical
conditions that are prevalent during the event.
3.
To examine the validity of the model by
comparing it to other models/methods relative to
projectile motion.
4.
To examine the potential application of the
model to real life situations by experimentally testing
it and applying it to a number of shot-putters.
The release parameter data available confirm that
the velocity of release is the single most important
factor in obtaining maximum distance in shot-put.
The
angle of release should always be accounted for, if a
successful throwing performance is to be achieved.
A
discrepancy has been shown to exist between the
theoretical optimum angle of release in shot-put and the
actually obtained angle of release, from real life data
analyses.
A number of viable explanations have been
given to explain this phenomenon.
Three main factors along with the gravitational
force will affect the range a projectile can achieve when
in fact it is released from a specific height (h) above
the ground.
First, the velocity of release (v 0) of the
projectile, second, the angle of release of the
projectile (a) and third, the height of release of the
projectile (h).
Of the three factors, the velocity at the time of
release is the most important one and more dramatically
157
affects the range achieved, much more than the angle of
release and the height of release can.
However, both the
angle of release and the height of release can contribute
to a perfect skillful projection/throw.
In the throwing
events the selection of the optimum angle of release can
contribute to winning or losing in a competition, both in
the beginning and in the advanced levels.
Given the relative importance of the angle of
release in achieving maximum range, a theoretical model
is sought to estimate the exact optimum angle of release
for the shot-put event.
The theoretical model assumed
that the shot as a projectile obeys the principles
governing projectile motion when the point of release is
at a higher level than that of landing.
The theoretical
model estimating the optimum angle was a function of the
velocity of release and the height of release while it
took into account the gravitational force.
The main concern of the study was the examination of
the relationship between the angle of release and the
velocity of release in the shot-put event, and also
between the angle of release and the height of release.
The study was delimited to 5 subjects, 10 throws for
each athlete for each of the 5 different kinds of angles
of release to be studied.
The 5 subjects were college
age shot-putters using the rotational technique of
throwing the shot, and were selected from 4 universities
in the State of Kansas.
All subjects were given equal
158
opportunity to achieve maximum performance as they
executed their puts.
All the throws were made in a non
competitive environment.
Subsequently, equations to estimate the angle of
release from velocity of release and height of release
data obtained from real life shot-put throwing were
derived for each subject.
Another concern was the
development of a theoretical model to estimate the
optimum angle of release from velocity of release and
height of release data and its subsequent application in
real life situations.
The creation of such a model helped in comparing the
obtained real life angle of release with the theoretical
optimum angle of release.
The application of the
theoretical model took place by experimentally measuring
various parameters of a shot-putter’s throwing action
such as velocity of release, height of release and angle
of release.
Subsequently, data from the real life
throwing situation were compared to the data obtained
from a theoretical model developed for this purpose.
Data from the shot-putters were obtained with the use of
high speed videography.
Theoretical data were obtained
with the use of the theoretical model.
The main concern for the development of the model
was to mathematically examine the various mechanical
components present during the release and flight of the
implement, and how they relate to the optimum angle of
159
release in order to achieve maximum horizontal
displacement/distance.
A system of two axes, a vertical (y) and a
horizontal (x) was assumed.
The primary mathematical
simulation of the x, y system, a first and a second order
derivative of the gravitational force, was derived
through Newtonian mechanics.
Trigonometry principles
were also considered throughout the construction of the
model.
Finally, mathematical integrations and
differentiations were employed along with equation
solving procedures to arrive at the desired model.
The
model was validated through simple comparisons where the
model was tested in two conditions at which the result
for the optimum angle of release is known as, for
example, is the case when the level of release is the
same as the level of landing.
For all five subjects there was a significant
relationship between the angle of release and the
velocity of release thus, showing dependency of the angle
of release on the velocity of release.
For all five
subjects there was no agreement between the real life
angles and those estimated by the theoretical model.
It was found that at present, the theoretical model
is not applicable in the shot-put event.
The optimum
release conditions other than velocity depend crucially
on how the maximum achievable release velocity is
functionally related to the other release conditions.
160
Conclusions
Based on the results of the present investigation, a
significant relationship exists between the angle of
release and the velocity of release in real life shotputting.
This phenomenon suggests that there is a
dependency of the velocity of release on the angle of
release.
Shot-putters do experience a decrease in the
maximum achievable velocity of release as the angle of
release increases.
From the data of the present
investigation it seems that the relationship between the
angle of release and the velocity of release follows a
linear trend.
The observed dependency of the velocity of release
on the angle of release, shows a disagreement between the
theoretical optimum angle of release and the real life
optimum angle of release.
This is because, contrary to
the real life model, the theoretical model assumes that
the velocity of release is an independent function of the
angle of release.
Indeed, it was observed that there was
no agreement between the results obtained based on the
real life and the theoretical models.
The findings of the present study verify the
untested hypothesis that the velocity of release depends
on the angle of release.
Coaches should encourage (or
continue to encourage) their shot-put throwers to
practice the release of the shot under a variety of
angles.
In the effort to maximize the distance thrown,
161
the optimum angle of release should be thrower specific.
Obviously, more capable throwers will be able to achieve
higher angles of release, perhaps angles according to the
theoretical model for the optimum angle of release.
As it has been speculated by some authorities, the
development of new ways of training may help bridge the
gap between theory and practice at least for the shot-put
event.
The issue of proper training could be worthwhile
researching in the future.
Recommendations
Experimentation with world-class shot-putters might
be useful to detect any change in the relationship
between the velocity and angle of release, as compared to
those observed in the present study and exhibited by the
collegiate shot-putters.
Perhaps an even greater number of throws, although
laborious, might again reveal additional information
pertaining to the nature of the relationship between the
angle of release and the velocity of release.
Other than regression mathematical models might also
be useful in estimating the real life, thrower specific,
optimum angle of release.
These models will probably be
constrained optimization models.
162
References
Anderson, J. (1972).
Analysis of selected kinematic data
for the shot-put.
Paper of the school of health,
physical education and recreation, Indiana
University, Bloomington, IN.
Ariel, G. (1972).
Computerized biomechanical analysis of
track and field athletics utilized by the Olympic
training camp for throwing events.
Track and Field
Quarterly Review, 72, 99-103.
Ariel, G. (1973a).
Computerized biomechanical analysis
of the worlds best shot-putters.
Track and Field
Quarterly Review, 73, 199-206.
Ariel, G. (1973b).
Biomechanical analysis of the shot
put technique utilizing the center of gravity
displacement.
Track and Field Quarterly Review, 73,
207-210.
Ariel, G. (1979).
Biomechanical analysis of shotputting.
Track and Field Quarterly Review, 79, 30-37.
Armstrong, L., & King, J. (1970).
thermal physics.
Atwater, E. (1979).
Mechanics, waves, and
Englewood Cliffs: NJ.
Biomechanics of overarm throwing and
of throwing injuries.
Exercise and Sport Sciences
Reviews, 7, 43-85.
Barford, N. (1973).
Mechanics.
New York: John Wiley.
Bashian, A., Gavoor, N., & Clark, B. (1982).
Some
observations on the release in the shot put.
and Field Quarterly Review, 82, 12.
163
Track
Basic systems, Inc. (1962).
Vectors: a programmed text
for introductory physics.
New York:
Appleton-
Century-Crofts.
Bates, B., Dufek, J., & Davis, H. (1992).
trial size on statistical power.
The effect of
Medicine and
Science in Sports and Exercise, 24, 1059-1068.
Bergman, P. (1987).
York:
The riddle of gravitation.
New
Charles Scribner’s Sons.
Berry, J., Burghes, D., Huntley, I., James, D., &
Moscardini, A. (1986). Mathematical modeling,
methodology, models and micros.
New York: John Wiley
& Sons.
Black, B. (1987).
the shot put.
The scientific bases of training for
Track and Field Quarterly Review, 87,
14-17.
Bowerman, W., & Freeman, W. (1991).
training for track and field.
High performance
Champaign, IL: Leisure
Press.
Bueche, F. (1972).
Principles of physics.
New York: Mc
Graw-Hill.
Cohen, B. (1985).
The birth of a new physics.
New York:
W.W Norton & Co.
Courant, R., & Hilbert, D. (1953).
mathematical physics.
New York:
Methodology of
Interscience
Publishers, Inc.
Dessureault, J. (1976).
Selected kinetic and kinematic
164
factors involved in shot-putting (Doctoral
dissertation, University of Indiana, 1976).
Microfiche of typescript, University of Oregon,
Eugene, OR.
Dunn, G. (1987).
Current trends in shot put training.
Track Technique, 98, 3118-3123.
Dyson, G. (1977).
The mechanics of athletics.
New York:
Holmes & Meier Publishers.
Ecker, T. (1985).
Basic track and field biomechanics.
Los Altos, CA: Tafnews Press.
Francis, F. (1948).
Mechanical analysis of the shot-put.
Athletic Journal, 28, 34-50.
Ganslen, V. (1964).
Aerodynamical and mechanical forces
in discus flight.
Glashow, S. (1981).
Athletic Journal, 68, 88-89.
The charm of physics.
New York:
The American Institute of Physics.
Greenwood, S. (1983).
discovery.
Physics: the excitement of
Belmont, CA:
Wadsworth Publishing Co.
Gregor, R., McCoy, R., & Whitting, W. (1990).
Biomechanics of the throwing events in athletics. In
G-P. Bruggeman and J.K. Ruhl (Eds), Techniques in
Athletics, Conference Proceedings (pp. 100-117).
Köln, Federal Republic of Germany.
Groh, H., Kubeth, A., & Baumann, W. (1966).
De la
cinetique et de la dynamique des mouvements corporels
rapids, etude concernat les phases finales du lancer
du poids et du javelot.
Sportarzt, 10.
165
Hakkinen, K., Kauhanen, H., & Komi, P. (1986).
Comparison of neuromuscular performance capabilities
between weightlifters, powerlifters and bodybuilders.
International Olympic Lifter, 4, 24-26.
Hakkinen, K., & Komi, P. (1985).
Effect of explosive
type strength training on electromyographic and force
production characteristics of leg extensor muscles
during concentric and various stretch shortening
cycle exercises.
Scandinavian Journal of Sports
Science, 7, 65-76.
Hay, J. (1985).
(3rd ed.).
The biomechanics of sports techniques
Englewood Cliffs, NJ:
Prentice-Hall,
Inc.
Hay, J. (1993).
(4th ed.).
The biomechanics of sports techniques
Englewood Cliffs, NJ:
Prentice-Hall,
Inc.
Ivey, D., & Hume, J. (1974).
Physics.
New York:
Ronald
Press Company.
Jones, M. (1979).
An introduction to mathematical
methods of physics.
Menlo Park, CA: The
Benjamin/Cummings Publishing Company Inc.
Judge, L. (1994).
coaching tool.
Herman, G. (1962).
Using biomechanical analysis as a
Moderne Athlete and Coach, 32, 15-20.
An electromyographic study of
selected muscles involved in the shot-put.
166
Research
Quarterly,
33, 85-93.
Kibble, T. (1973).
Classical mechanics.
New York:
John
Wiley & Sons.
Koslov, N. (1969).
L’effort final du lanceur.
Liegkaya
Atletika, 9.
Koutiev, N. (1966).
poids.
Comment les champions lancent le
Legkaya Atletika, 9.
Kreighbaum, E., & Barthels, K. (1990).
New York:
Biomechanics.
Mc-Millan Co.
McDermott, A. (1986).
Plyometric training for the
throwing events. National Strength and Conditioning
Association Journal, 8, 52-55.
Marhold, G. (1974).
shotput.
Biomechanical analysis of the
In R. Nelson & C. Morehouse (Eds.),
Proceedings of the 4th International Symposium in
Biomechanics.
Marhold, G. (1964).
du poids.
University Park: PA.
Phase finale en suspension au lancer
Theorie und Praxis der Korpercultur, 8,
695-699.
Marsden, J., & Weinstein, A. (1981). Calculus unlimited.
Menlo Park, CA:
The Benjamin/Cummings Publishing Co,
Inc.
McCoy, R. (1990).
Kinematic analysis of the United
States’ elite shot-putters.
Technical report
presented to the Athletics Congress, elite athlete
program.
McCoy, R., Gregor, R., & Whiting, W. (1989).
167
The
athletics congress’s track and field coaching manual:
Fundamental Mechanics.
Champaign, IL: Leisure
Press.
McCoy, R., Gregor, R., Whiting, W., & Rich, R. (1984).
Kinematic analysis of elite shot-putters.
Track
Technique, 90, 2868-2905.
Mc Watt, B. (1982).
Angles of release in the shot-put.
Modern Athlete and Coach, 20 (4), 17-18.
Michels, W., Correll, M., & Patterson, A. (1968).
Foundations of physics.
New Jersey:
Dovan, Nostrand
Company, Inc.
Misner, C., Thorne, K., Wheeler, J. (1973).
San Francisco, CA:
Morris, W. (1971).
Norusis, M. (1992).
O’Shea, P. (1986).
W.H Freeman and Company.
Concerning technique and method.
Track Technique,
user’s guide.
Gravitation.
43, 1358-1362.
SPSS for windows.
Base system
Chicago, IL: SPSS Inc.
The push press: an alternative to the
bench press. National Strength and Conditioning
Association Journal, 8, 28-31.
O’Shea, P., & Elam, R. (1984).
The shot put-mechanics,
techniques, strength and conditioning programs.
National Strength and Conditioning Association
Journal, 5, 4-5: 74-76: 78-79.
Otto, M. (1987).
throw.
Biomechanical analysis of the hammer
Scientific report on the II world
168
championships in athletics.
Pagani, T. (1981).
Rome: Italy.
Mechanics and technique of the shot.
Track Technique, 82, 2601-2602.
Papageorgiou, P. (1986).
The athletic throws.
Athens,
Greece: University of Athens Press.
Pedhazur, E. (1982).
research.
Multiple regression in behavioral
New York, NY: Harcourt Brace Jovanovich
College Publishers.
Poole, C., & Bangerter, B. (1981). The use of velocity
and angle of release in shot-putting.
Women’s
coaching Clinic, 5(1), 15-18.
Sokolnikoff, S., & Redheffer, M. (1966).
physics and modern engineering.
Mathematics of
New York:
Mc Graw-
Hill, Inc.
Stevens, J. (1992).
Applied multivariate statistics for
the social sciences.
Hillsdale, NJ:
Lawrence
Erlbaum Associates.
Susanka, P., & Stepanek, J. (1987).
analysis of the shot-put.
Biomechanical
Scientific report on the
II world championships in athletics.
Rome: Italy.
The M.I.T Introductory Physics Series. (1971).
mechanics.
New York:
Thirring, W. (1978).
New York:
W.W Norton & Co., Inc.
Classical dynamical systems v.1.
Springer-Verlag.
Timoshenko, S., & Young, D. (1956).
mechanics.
Tokyo:
Newtonian
Engineering
Mc Graw - Hill.
169
Vigars, R. (1979).
throws.
Analysis of critical factors in the
Track Technique, 77, 2447-2449.
Wallace, A., & Fenster, S.(1969).
GA:
Mechanics.
Atlanta,
Holt, Rinehart & Winston, Inc.
Wells, K., & Lutgens, K. (1976).
Philadelphia, PA:
Ward, P. (1976).
Kinesiology.
W.B Saunders Co.
U.S.A Olympic shot put camp.
Track
and Field Quarterly Review, 76, 11-16.
Wentworth, G., & Smith, D. (1951).
and tables.
Wong, C. (1991).
New York:
Boston, MA:
Plane trigonometry
Ginn & Company.
Introduction to mathematical physics.
Oxford University Press.
Zatsiorsky, V., Lanka, G., & Shalmanov, A. (1981).
Biomechanical analysis of shot-putting technique.
Exercise and Sport Sciences Reviews, 9, 353-389.
Zatsiorsky, V. (1990).
technique.
The biomechanics of shot-putting
In G-P. Bruggeman and J.K. Ruhl (Eds),
Techniques in Ahtletics, Conference Proceedings (pp.
118-125).
Köln, Federal Republic of Germany.
170
Appendix A
The Obtained Data for Each
of the Subjects
Table 15
Horizontal and Vertical Velocities, Resultant Velocity
(Velocity of Release), Angle of Release and Height of
Release for the First Subject
_________________________________________________________
171
Velocity
_______________________________
Horizontal Vertical Resultant
Angle
Height
(m/s)
(m/s)
(m/s)
(deg)
(m)
_________________________________________________________
8.63
5.55
10.26
32.77
2.16
8.97
6.35
10.99
35.29
2.18
8.36
6.61
10.66
38.34
2.33
7.99
6.45
10.27
38.88
2.24
8.57
5.72
10.30
33.73
2.25
7.93
6.44
10.22
39.10
2.26
8.53
6.19
10.54
35.97
2.20
8.04
7.00
10.66
41.05
2.32
8.69
6.33
10.75
36.07
2.23
9.05
6.47
11.12
35.58
2.18
8.55
6.30
10.62
36.38
2.15
8.34
6.65
10.67
38.55
2.21
8.07
6.79
10.55
40.08
2.09
8.82
6.72
11.09
37.30
2.22
9.23
6.47
11.27
35.01
2.07
8.80
7.06
11.27
38.73
2.27
8.97
6.74
11.22
36.93
2.06
9.90
6.47
11.83
33.17
2.01
9.29
6.55
11.36
35.20
2.13
9.80
6.34
11.67
32.90
2.06
8.58
6.63
10.84
37.70
2.03
8.71
5.82
10.47
33.78
1.95
9.26
5.54
10.79
30.88
2.09
8.18
5.74
9.99
35.07
2.16
8.89
6.41
10.96
35.77
1.99
9.13
5.69
10.76
31.94
2.13
8.06
6.10
10.11
37.11
2.10
8.04
8.85
9.95
36.05
2.00
8.97
5.52
10.53
31.60
2.01
7.90
6.46
10.21
39.26
2.00
7.14
6.47
9.64
42.19
2.31
6.70
7.10
9.76
46.68
2.30
6.56
6.94
9.54
46.61
2.25
6.14
7.11
9.40
49.21
2.27
6.83
7.24
9.95
46.65
2.25
6.64
7.16
9.76
47.16
2.22
6.80
7.62
10.22
48.27
2.33
7.08
7.26
10.14
45.69
2.27
6.58
7.21
9.77
47.63
2.35
7.06
7.03
9.97
44.86
2.27
10.26
4.49
11.20
23.63
2.08
8.35
5.58
10.04
33.74
2.06
8.38
6.02
10.32
35.68
2.10
172
8.67
5.49
10.26
32.35
2.05
8.39
6.07
10.35
35.89
2.14
8.66
5.44
10.23
32.13
2.10
8.70
5.87
10.49
34.02
2.08
9.14
4.76
10.30
27.52
2.05
9.23
5.14
10.56
29.11
2.08
_________________________________________________________
Note. The throws are reported in the order in which they
were thrown.
Table 16
Horizontal and Vertical Velocities, Resultant Velocity
(Velocity of Release), Angle of Release and Height of
Release for the Second Subject
_________________________________________________________
Velocity
_______________________________
Horizontal Vertical Resultant
Angle
Height
(m/s)
(m/s)
(m/s)
(deg)
(m)
_________________________________________________________
10.33
6.35
12.12
31.57
2.08
9.78
6.49
11.74
33.57
2.15
10.09
6.09
11.78
31.13
2.21
10.84
6.46
12.62
30.80
2.13
9.95
6.60
11.94
33.53
2.15
10.45
6.49
12.30
31.85
2.08
10.47
6.72
12.43
32.72
2.17
10.07
6.65
12.07
33.46
2.11
9.88
6.36
11.75
32.76
2.10
9.76
6.16
11.54
32.28
2.21
11.37
3.84
12.00
18.67
1.91
11.16
4.29
11.96
21.04
1.91
11.52
3.30
11.98
15.98
1.84
11.17
3.15
11.60
15.77
1.87
11.15
4.03
11.86
19.87
1.91
10.07
4.77
11.14
25.35
1.90
9.82
3.81
10.53
21.20
1.80
10.61
3.35
11.13
17.50
1.91
10.31
4.64
11.30
24.22
1.95
10.31
4.64
11.30
24.22
1.95
6.72
7.98
10.43
49.93
2.35
6.27
7.45
9.74
49.95
2.41
5.68
7.26
9.21
51.97
2.46
7.20
7.55
10.43
46.34
2.28
6.86
7.63
10.26
48.03
2.29
7.17
7.87
10.64
47.65
2.32
6.71
7.87
10.34
49.58
2.29
6.93
7.74
10.39
48.17
2.38
173
7.12
7.27
10.18
45.60
2.32
6.12
7.76
9.89
51.75
2.25
10.32
6.06
11.97
30.41
2.13
10.73
6.08
12.33
29.55
2.08
10.16
6.62
12.12
33.08
2.08
9.98
6.09
11.52
31.89
2.14
9.66
6.24
11.50
32.87
2.04
9.75
6.52
11.73
33.80
1.99
10.12
6.04
11.78
30.86
2.09
10.71
5.79
12.18
28.41
2.06
9.82
5.74
11.38
30.28
1.99
10.22
5.60
11.65
28.72
2.03
10.73
3.91
11.42
20.00
1.93
10.10
4.86
11.21
25.68
2.04
10.70
4.51
11.61
22.85
2.04
11.00
4.97
12.07
24.32
1.95
10.47
5.21
11.70
26.48
1.91
10.04
5.86
11.62
30.30
2.04
10.51
4.78
11.55
24.47
1.98
9.95
4.64
10.98
24.99
1.96
10.11
4.97
11.27
26.18
2.03
9.72
5.17
11.00
28.01
2.00
_________________________________________________________
Note. The throws are reported in the order in which they
were thrown.
Table 17
Horizontal and Vertical Velocities, Resultant Velocity
(Velocity of Release), Angle of Release and Height of
Release for the Third Subject
_________________________________________________________
Velocity
_______________________________
Horizontal Vertical Resultant
Angle
Height
(m/s)
(m/s)
(m/s)
(deg)
(m)
_________________________________________________________
9.13
6.98
11.49
37.41
2.13
10.28
6.91
12.38
33.90
2.17
10.02
5.71
11.53
29.69
2.09
9.86
6.08
11.58
31.64
2.23
9.99
5.98
11.64
30.89
2.04
10.33
5.65
11.77
28.70
2.02
11.47
5.04
12.53
23.73
2.09
10.32
5.55
11.72
28.26
2.04
10.93
5.13
12.08
25.15
2.01
10.08
5.34
11.41
27.92
2.03
174
8.05
5.93
10.00
36.41
2.31
7.90
6.29
10.10
38.52
2.24
7.23
6.54
9.75
42.14
2.26
7.77
6.24
9.97
38.80
2.28
7.60
6.76
10.17
41.67
2.22
7.44
5.98
9.55
38.79
2.27
7.27
7.51
10.45
45.95
2.25
6.74
7.88
10.37
49.46
2.28
7.10
7.36
10.23
46.04
2.27
7.15
7.27
10.19
45.48
2.31
9.26
6.48
11.30
34.99
2.20
9.30
6.90
11.58
36.58
2.21
8.53
7.16
11.13
40.00
2.18
9.21
6.43
11.23
34.93
2.24
9.90
6.48
11.83
33.20
2.20
9.97
6.66
11.99
33.76
2.09
9.98
6.89
12.13
34.60
2.13
9.42
7.17
11.84
37.28
2.13
9.85
6.51
11.81
33.48
2.07
9.13
5.35
10.58
30.39
2.16
7.89
7.65
10.99
44.12
2.33
7.66
7.40
10.65
44.03
2.39
6.89
7.98
10.54
49.17
2.25
6.73
7.83
10.32
49.32
2.37
6.88
7.24
9.99
46.44
2.25
7.18
7.72
10.54
47.10
2.32
6.76
7.65
10.21
48.51
2.40
6.91
7.57
10.25
47.60
2.34
6.94
7.34
10.11
46.62
2.35
7.45
7.89
10.85
46.66
2.32
10.54
4.73
11.55
24.17
1.97
10.74
5.00
11.85
24.97
1.93
10.30
4.87
11.39
25.33
2.03
10.58
5.40
11.88
27.04
1.99
10.26
5.28
11.54
27.22
1.97
10.78
4.54
11.69
22.85
1.98
11.04
4.77
12.02
23.36
2.00
11.05
5.02
12.14
24.41
2.03
11.02
4.36
11.85
21.60
2.03
10.99
3.94
11.68
19.74
1.91
_________________________________________________________
Note. The throws are reported in the order in which they
were thrown.
Table 18
Horizontal and Vertical Velocities, Resultant Velocity
(Velocity of Release), Angle of Release and Height of
Release for the Fourth Subject
175
_________________________________________________________
Velocity
_______________________________
Horizontal Vertical Resultant
Angle
Height
(m/s)
(m/s)
(m/s)
(deg)
(m)
_________________________________________________________
10.24
5.84
11.79
29.69
2.02
9.42
6.11
11.23
32.99
2.07
10.99
5.17
12.15
25.19
1.98
10.17
5.69
11.65
29.21
2.12
10.06
5.73
11.58
29.67
2.01
10.07
5.87
11.66
30.24
2.11
10.51
5.65
11.93
28.28
2.09
10.82
5.11
11.96
25.28
2.08
9.85
5.95
11.51
31.13
2.10
10.09
5.35
11.42
27.92
2.05
6.01
6.90
9.15
48.95
2.35
7.05
6.78
9.78
43.87
2.31
6.04
6.68
9.01
47.84
2.40
6.39
6.66
9.23
46.18
2.36
6.18
6.85
9.23
47.97
2.42
5.88
6.78
8.98
49.06
2.29
6.58
6.74
9.41
45.69
2.28
6.47
6.83
9.41
46.58
2.38
5.88
6.72
8.93
48.83
2.29
6.29
6.76
9.23
47.07
2.33
8.35
6.05
10.31
35.93
2.26
9.32
5.24
10.69
29.36
2.17
8.30
5.42
9.91
33.15
2.13
7.28
5.40
9.06
36.56
2.14
7.84
6.47
10.17
39.51
2.11
8.55
6.12
10.51
35.58
2.21
8.71
5.56
10.33
32.56
2.09
9.19
5.64
10.78
31.57
2.07
8.46
5.77
10.24
34.30
2.07
8.30
6.33
10.44
37.32
2.15
8.81
6.12
10.73
34.80
2.11
8.77
5.60
10.40
32.56
2.15
9.86
5.62
11.35
29.69
2.10
9.74
5.96
11.42
31.44
2.09
9.41
5.98
11.15
32.47
2.14
9.26
5.86
10.96
32.33
2.15
10.02
5.97
11.66
30.81
2.01
9.11
5.67
10.73
31.91
2.07
9.40
5.82
11.05
31.77
2.07
10.12
5.46
11.50
28.34
2.02
10.15
4.75
11.21
25.09
1.93
9.17
4.47
10.20
25.99
1.98
176
10.12
4.30
11.00
23.03
2.03
9.67
4.72
10.76
26.04
2.06
10.31
5.10
11.50
26.32
1.90
10.14
4.61
11.14
24.47
1.92
10.16
4.98
11.32
26.09
1.95
10.08
4.75
11.14
25.22
1.92
10.58
3.93
11.29
20.36
1.91
10.56
4.66
11.54
23.82
1.95
_________________________________________________________
Note. The throws are reported in the order in which they
were thrown.
Table 19
Horizontal and Vertical Velocities, Resultant Velocity
(Velocity of Release), Angle of Release and Height of
Release for the Fifth Subject
_________________________________________________________
Velocity
_______________________________
Horizontal Vertical Resultant
Angle
Height
(m/s)
(m/s)
(m/s)
(deg)
(m)
_________________________________________________________
9.50
6.37
11.44
33.86
2.13
9.97
7.13
12.26
35.57
2.11
9.91
6.94
12.10
35.03
2.09
9.61
6.43
11.56
33.79
2.04
9.86
6.98
12.08
35.31
2.07
10.35
6.33
12.13
31.47
2.06
9.51
6.47
11.50
34.22
2.16
10.35
6.38
12.16
31.67
2.12
9.64
6.84
11.82
35.36
2.15
9.91
6.94
12.10
35.03
2.19
10.89
5.99
12.42
28.84
2.24
11.28
6.58
13.06
30.27
2.18
10.47
6.10
12.11
30.22
2.22
10.63
6.29
12.36
30.62
2.25
11.14
6.00
12.66
28.31
2.23
10.49
7.03
12.63
33.81
2.07
10.48
6.78
12.48
32.89
2.02
9.48
7.80
12.27
39.44
2.14
10.18
6.64
12.15
33.12
2.13
10.28
6.76
12.31
33.34
2.12
9.97
7.37
12.40
36.48
2.30
10.21
7.11
12.45
34.86
2.33
10.30
6.84
12.36
33.59
2.31
9.86
7.36
12.31
36.74
2.26
10.18
7.46
12.62
36.24
2.23
177
9.64
7.56
12.25
38.12
2.23
9.91
7.37
12.35
36.64
2.24
9.42
7.76
12.20
39.50
2.18
9.71
7.44
12.23
37.48
2.28
9.50
7.68
12.21
38.96
2.20
6.40
8.14
10.36
51.83
2.36
6.25
8.56
10.60
53.88
2.34
6.52
8.58
10.78
52.76
2.33
7.35
7.75
10.68
46.53
2.34
6.76
8.01
10.48
49.83
2.44
6.56
8.05
10.38
50.86
2.34
6.36
8.18
10.36
52.12
2.35
6.88
7.96
10.52
49.16
2.38
6.05
8.36
10.32
54.14
2.41
7.21
8.10
10.84
48.32
2.38
11.92
4.62
12.78
21.20
1.99
11.65
4.99
12.67
23.19
2.03
12.17
4.09
12.84
18.58
2.00
12.16
4.06
12.82
18.48
1.95
11.79
5.39
12.96
24.56
2.04
12.08
4.90
13.04
22.08
1.90
12.28
3.58
12.79
16.26
2.01
11.19
6.42
12.89
29.84
2.05
12.36
4.39
13.11
19.56
1.88
12.03
5.15
13.08
23.19
1.99
_________________________________________________________
Note. The throws are reported in the order in which they
were thrown.
178
Appendix B
The Consent Statement
CONSENT STATEMENT
179
The department of physical education at the University of Kansas
supports the practice of protection for human subjects participating in
research. The following information is provided for you to decide whether
you wish to participate in the present study. You should be aware that
even if you agree to participate, you are free to withdraw at any time
without penalty.
We are interested in investigating the relationship between the
velocity and the angle of release in the shot-put event, and also the
potential application of a theoretical model to estimate the optimum angle
of release. In essence, we are trying to establish whether the theoretical
optimum angle of release is really the optimum angle of release athletes
should use to improve their performance. You will be participating in a
total of approximately 5 sessions. You will have to throw the shot as you
would in a competition. However, for the purpose of the experiment, you
will have to throw with angles of release approximating those of 20, 35,
45, and 60 degrees, as well as the angle of release you usually use in
practice and competition. You will have to throw 10 times for each of
these five angles for a total of 50 throws.
In order to prevent unfair advantage of your normal angle over the
other angles you will be asked to practice for about a week the throwing of
the “other” angles. All your throws will be measured as in competition.
Foul throws, however, will not count, will not be measured and, they will
have to be repeated.
Your participation will be extremely active during this experiment.
It is estimated that each session will take no more than one hour of your
time. You will not feel any serious discomfort during your participation.
All your throws will be filmed so data from them can later be analyzed. By
the completion of the study, all the tapes used to record your throwing
attempts will be erased. Although participation will not directly benefit
you, we believe that the information will be useful in determining the
relationship between the angle of release and the velocity of release in
the shot-put event as well as the potential application of a theoretical
model to estimate the optimum angle of release.
Your participation is solicited although strictly voluntary. We
assure you that all data will be confidential and that your name will not
be associated in any way with the research findings. The information will
be identified only by a code number. You may also request that your
protocol be destroyed along with the data.
If you would like additional information concerning this study before
or after it is complete, please feel free to contact me by phone or mail.
Sincerely,
Andreas Maheras
Principal Investigator
305 East 20th
Hays, KS 67601
(785) 628 8456
____________________________________
Signature of subject agreeing to participate.
By signing, the subject certifies that he is at least 18 years of age.
Has received a copy of this consent form.
180
Appendix C
The A.C.H.E Form
181
182
Appendix D
The FORTRAN Computer Program
Used in the Analysis
SUBROUTINE SLFIT(T,XLOC,NFIT,S0,V)
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C THIS SUBROUTINE FITS A STRAIGHT LINE TO A SERIES OF
C NFIT PAIRS OF
C VALUES OF TIME (T, IN SEC) AND HORIZONTAL DISPLACEMENT
C (XLOC, IN
C METERS), AND CALCULATES THE HORIZONTAL DISPLACEMENT AT
C TIME ZERO
C (S0) AND THE HORIZONTAL VELOCITY (V).
C
C THIS SUBROUTINE WAS DEVELOPED BY JESUS DAPENA.
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
DIMENSION
T(30),XLOC(30),A(30,2),B(30),X(2),WK(2),IWK(2)
183
DO 1 I=1,NFIT
A(I,1)=1.
A(I,2)=T(I)
B(I)=XLOC(I)
1
CONTINUE
TOL=-1.0
KBASIS=2
CALL LLSQF(A,30,NFIT,2,B,TOL,KBASIS,X,WK,IWK,IER)
S0=X(1)
V=X(2)
RETURN
END
SUBROUTINE PBFIT(T,ZLOC,NFIT,S0,V0)
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C THIS SUBROUTINE FITS A PARABOLA OF SECOND DERIVATIVE C
C =-9.81 M/SEC2 TO A SERIES OF NFIT PAIRS OF VALUES OF
C TIME (T, IN SEC) AND VERTICAL DISPLACEMENT
C (ZLOC, IN METERS), AND CALCULATES THE VERTICAL
C DISPLACEMENT(S0)AND VERTICAL VELOCITY (V0) AT TIME
C ZERO.
C
C THIS SUBROUTINE WAS DEVELOPED BY JESUS DAPENA.
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
DIMENSION
T(30),ZLOC(30),A(30,2),B(30),X(2),WK(2),IWK(2)
DO 1 I=1,NFIT
A(I,1)=1.
A(I,2)=T(I)
1
B(I)=ZLOC(I)+4.905*(T(I)**2)
CONTINUE
TOL=-1.0
KBASIS=2
CALL LLSQF(A,30,NFIT,2,B,TOL,KBASIS,X,WK,IWK,IER)
S0=X(1)
V0=X(2)
RETURN
END
SUBROUTINE LLSQF
(A,IQ,NEQS,NUNK,B,TOL,KBASIS,X,WX,IWK,IER)
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C THIS SUBROUTINE SOLVES A SYSTEM OF LINEAR EQUATIONS.
C IT CALLS SUBROUTINES "QR" AND "OVER" OF THE NAPACK
C PACKAGE (THEY ARE CONTAINED IN GRAFBM). THIS
C LLSQF SUBROUTINE MIMICS THE ACTIONS OF SUBROUTINE
C LLSQF FROM THE IMSL PACKAGE.
C
184
C THIS SUBROUTINE WAS DEVELOPED BY JESUS DAPENA.
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
REAL A(IQ,NUNK),B(IQ),X(NUNK),WX(NUNK)
DIMENSION IWK(NUNK)
DOUBLE PRECISION DA(2043),DB(200),DX(200)
ICOUNT=0
DO 24 I=1,NEQS
DB(I)=B(I)
24
CONTINUE
DO 22 J=1,NUNK
DO 23 I=1,NEQS
ICOUNT=ICOUNT+1
DA(ICOUNT)=A(I,J)
23
CONTINUE
22
CONTINUE
LA=NEQS
M=NEQS
N=NUNK
CALL QR(DA,LA,M,N)
CALL OVER(DX,DA,DB)
DO 25 J=1,NUNK
X(J)=DX(J)
25
CONTINUE
RETURN
END
SUBROUTINE QR(A,LA,M,N)
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C THIS SUBROUTINE BELONGS TO THE NAPACK PACKAGE. IT WAS
C OBTAINED FROM [email protected] THROUGH ELECTRONIC MAIL.
C MADE D.P. BY J. DAPENA.
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
________________________________________________________
C QR FACTOR A GENERAL MATRIX WITH COLUMN PIVOTING
C INPUT:
C A
--ARRAY CONTAINING MATRIX
C (LENGTH AT LEAST 3+N+MN + MIN(M,N))
C LA
--LEADING (ROW) DIMENSION OF ARRAY A
C M
--NUMBER OF ROWS IN COEFFICIENT MATRIX
C N
--NUMBER OF COLUMNS IN COEFF. MATRIX
C OUTPUT: A
--FACTORED MATRIX
C BUILTIN FUNCTIONS: DABS,DSQRT
C PACKAGE SUBROUTINES: RPACK
________________________________________________________|
C
DOUBLE PRECISION A(1),R,S,T,U,V,W
INTEGER B,C,D,E,F,G,H,I,J,K,L,LA,M,N,O,P,Q,Z
IF ( LA .GT. M ) CALL RPACK(A,LA,M,N)
U = 1.
185
10
C
C
C
20
30
40
50
60
70
80
U = .5*U
T = 1 + U
IF ( T .GT. 1. ) GOTO 10
U = DSQRT(40.*U)
O = M + 1
I = 2 + N*O
F = I
IF ( M .LT. N ) F = 2 + M*O
D = I - M
J = 2 + N
H = I + 1
C = H
L = H + N
G = H
V = 0.
--------------------------------------|*** COMPUTE 2-NORM OF EACH COLUMN ***|
--------------------------------------S = 0.
R = S
K = I - M
T = A(I-J)
A(I) = T
I = I - 1
IF ( T .NE. 0. ) GOTO 40
IF ( I .GT. K ) GOTO 30
GOTO 70
S = DABS(T)
R = 1.
IF ( I .EQ. K ) GOTO 70
T = A(I-J)
A(I) = T
I = I - 1
IF ( DABS(T) .GT. S ) GOTO 60
R = R + (T/S)**2
IF ( I .GT. K ) GOTO 50
GOTO 70
R = 1 + R*(S/T)**2
S = DABS(T)
IF ( I .GT. K ) GOTO 50
W = S*DSQRT(R)
A(G) = W
A(L) = U*W
IF ( W .LT. V ) GOTO 80
V = W
P = G
L = L - 1
G = I
186
C
C
C
90
C
C
C
100
110
C
C
C
120
I = I - 1
J = J - 1
K = I - M
IF ( K .GT. 2 ) GOTO 20
A(1) = 3230
A(2) = M
A(3) = N
K = 4
G = M
----------------------------|*** START FACTORIZATION ***|
----------------------------E = K + G
L = P - M
J = P - E
H = H + 1
A(H+J/O) = A(H)
A(H) = (P-3)/O
----------------------------|*** INTERCHANGE COLUMNS ***|
----------------------------DO 100 I = L,P
T = A(I)
Q = I - J
A(I) = A(Q)
A(Q) = T
S = 0.
T = A(E)
IF ( T .EQ. 0. ) GOTO 240
IF ( K .EQ. F ) RETURN
E = E - 1
DO 110 I = K,E
S = S + (A(I)/T)**2
IF ( S .EQ. 0. ) GOTO 240
S = T*DSQRT(S)
T = A(K)
A(K) = S
IF ( T .GE. 0. ) A(K) = -S
R = 1./DSQRT(S*(S+DABS(T)))
I = E
---------------------------------|*** STORE HOUSEHOLDER MATRIX ***|
---------------------------------A(I+1) = R*A(I)
I = I - 1
IF ( I .GT. K ) GOTO 120
IF ( T .LT. 0. ) S = -S
187
130
140
C
C
150
160
170
C
C
C
180
190
200
210
A(K+1) = R*(T+S)
IF ( K .GT. D ) RETURN
J = K
E = -1
Z = H
G = G - 1
V = 0.
IF ( J .GT. D ) GOTO 230
J = J + O
E = E + O
L = J + G
Z = Z + 1
B = L + 1
S = A(B)
T = 0.
IF ( S .EQ. 0. ) GOTO 160
DO 140 I = J,L
T = T + A(I)*A(I-E)
-----------------------------|*** UPDATE FACTORIZATION ***|
-----------------------------DO 150 I = J,L
A(I) = A(I) - T*A(I-E)
T = S*DSQRT(DABS(1.-(A(J)/S)**2))
A(B) = T
IF ( T .LT. A(Z) ) GOTO 170
IF ( T .LT. V ) GOTO 130
V = T
P = B
GOTO 130
I = J + 1
------------------------------|*** COMPUTE COLUMN 2-NORM ***|
------------------------------S = 0.
R = S
DO 180 Q = I,L
IF ( A(Q) .NE. 0. ) GOTO 190
GOTO 220
S = DABS(A(Q))
DO 210 I = Q,L
T = DABS(A(I))
IF ( T .GT. S ) GOTO 200
R = R + (T/S)**2
GOTO 210
R = 1 + R*(S/T)**2
S = T
CONTINUE
188
220
230
240
T = S*DSQRT(R)
A(B) = T
A(Z) = U*T
IF ( T .LT. V ) GOTO 130
V = T
P = B
GOTO 130
K = K + O + 1
IF ( K .LT. F ) GOTO 90
IF ( M .GE. N ) GOTO 90
RETURN
A(H) = 0.
A(1) = -3230
RETURN
END
SUBROUTINE RPACK(A,LA,M,N)
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C THIS SUBROUTINE BELONGS TO THE NAPACK PACKAGE. IT WAS
C OBTAINED FROM
C [email protected] THROUGH ELECTRONIC MAIL. MADE D.P. BY
C J. DAPENA.
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
________________________________________________________
C
REARRANGE THE ELEMENTS OF A REAL ARRAY SO THAT THE
C
ELEMENTS OF A RECTANGULAR MATRIX ARE STORED
C
SEQUENTIALLY
C
INPUT:
C
A
--REAL ARRAY CONTAINING MATRIX
C
LA
--LEADING (ROW) DIMENSION OF ARRAY A
C
M
--ROW DIMENSION OF MATRIX STORED IN A
C
N
--COLUMN DIMENSION OF MATRIX STORED IN A
C
OUTPUT: A --MATRIX PACKED AT START OF ARRAY
_______________________________________________________
C
DOUBLE PRECISION A(1)
INTEGER H,I,J,K,L,LA,M,N,O
H = LA - M
IF ( H .EQ. 0 ) RETURN
IF ( H .GT. 0 ) GOTO 10
WRITE(6,*) 'ERROR: LA ARGUMENT IN RPACK MUST BE
.GE. M ARGUMENT'
STOP
10
I = 0
K = 1
L = M
O = M*N
20
IF ( L .EQ. O ) RETURN
I = I + H
189
30
K = K + M
L = L + M
DO 30 J = K,L
A(J) = A(I+J)
GOTO 20
END
SUBROUTINE OVER(X,A,B)
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C THIS SUBROUTINE BELONGS TO THE NAPACK PACKAGE. IT WAS
C OBTAINED FROM
C [email protected] THROUGH ELECTRONIC MAIL. MADE D.P. BY
C J. DAPENA.
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
________________________________________________________
C
SOLVE A GENERAL OVERDETERMINED LINEAR SYSTEM OF M
C
EQS.
C
INPUT:
C
A
--QR'S OUTPUT
C
B
--RIGHT SIDE
C
OUTPUT:X --SOLUTION (AT LEAST M ELEMENTS, CAN BE
C
IDENTIFIED WITH B BUT RIGHT SIDE DESTROYED)
C
BUILTIN FUNCTIONS: DABS
________________________________________________________
DOUBLE PRECISION A(1),B(1),X(1),T
INTEGER H,I,J,K,L,M,N,O
T = A(1)
IF ( DABS(T) .EQ. 3230 ) GOTO 10
WRITE(6,*) 'ERROR: MUST QR FACTOR COEFFICIENT
MATRIX'
WRITE(6,*) 'BEFORE SOLVING SYSTEM'
STOP
10
IF ( T .GT. 0. ) GOTO 20
WRITE(6,*) 'SINGULAR SYSTEM - COMPUTE REGULARIZED
SOLUTION'
WRITE(6,*) '(SEE SECTION 6-11)'
STOP
20
M = A(2)
N = A(3)
IF ( M .GE. N ) GOTO 30
WRITE(6,*) 'ERROR: THE NUMBER OF EQUATIONS IS LESS
THAN'
WRITE(6,*) 'THE NUMBER OF UNKNOWNS. FOR AN
OVERDETERMINED'
WRITE(6,*) 'SYSTEM, THERE MUST BE MORE EQUATIONS
THAN UNKNOWNS'
STOP
30
IF ( M .GT. 1 ) GOTO 40
190
40
50
C
C
C
60
70
80
C
C
C
90
100
110
C
C
C
120
X(1) = B(1)/A(4)
RETURN
DO 50 I = 1,M
X(I) = B(I)
O = M + 1
L = N
IF ( M .EQ. N ) L = N - 1
K = 4
----------------------------------------|*** APPLY ORTHOGONAL TRANSFORMATION ***|
----------------------------------------DO 80 J = 1,L
T = 0.
DO 60 I = J,M
T = T + X(I)*A(I+K)
DO 70 I = J,M
X(I) = X(I) - T*A(I+K)
K = K + O
J = N
K = 3 + O*N
H = K
--------------------------------|*** SOLVE TRIANGULAR SYSTEM ***|
--------------------------------K = K - O
T = X(J)/A(J+K)
X(J) = T
IF ( J .EQ. 1 ) GOTO 110
J = J - 1
DO 100 I = 1,J
X(I) = X(I) - T*A(I+K)
GOTO 90
IF ( N .EQ. 1 ) RETURN
---------------------------------|*** PERFORM PIVOT OPERATIONS ***|
---------------------------------J = A(L+H)
T = X(J)
X(J) = X(L)
X(L) = T
L = L - 1
IF ( L .GT. 0 ) GOTO 120
RETURN
END
191