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Ballistic Pendulum and Projectile Motion
The initial velocity of a ball shot from a spring gun is determined by the equations
for projectile motion and by the equations for a ballistic pendulum.
Projectile motion: For a projectile whose initial velocity is horizontal. The
initial velocity (vi) can be calculated from the measurement of the range x and the vertical
distance y as shown in Fig. 1.
Fig. 1
In such a case it must be remembered that the horizontal component of the
velocity always remains constant and is not affected by the constant downward pull of
gravity. Furthermore the vertical component of the motion is unaffected by its horizontal
flight and hence the body falls vertically, as does a freely falling object having no
horizontal motion. The actual motion of such a projectile is the combination of its
horizontal constant velocity and its downward uniformly accelerated velocity. During the
time interval t required for the projectile to reach the floor, it will have moved
horizontally through a distance
vix =
(l)
x
t
and during the same interval, because of the acceleration due to gravity g, it will have
fallen a distance
y = 1/2 gt2.
(2)
The time of the motion can be found from equation 2. Then, the initial velocity
can be found.
Ballistic Pendulum: For a system where there is no external force, momentum of
the system is conserved. That is, the momentum before the collision must be equal to the
momentum after the collision. In the present case, an inelastic collision occurs between
the ball from the spring gun and the pendulum.
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The momentum of the ball just before impact shall be equal to the combined
momenta of the ball and pendulum an instant after impact. In the ballistic pendulum used
in this experiment the velocity of the pendulum before impact is zero, and hence its
momentum before impact is zero. The momentum of the ball before impact is the
product of its mass m and its initial velocity vi just before impact. Since the projectile
becomes imbedded in the pendulum after impact, the ball and pendulum an instant after
impact have a common velocity vf and the combined momenta is (mb + mp)vf. From the
law of the conservation of momentum
Total momentum before impact=Total momentum after impact
mbvi = (mb + mp)vf
(3)
from which the initial velocity is given by
vf =
(4)
mb vi
(mb + m p )
As a result of the impact, the pendulum containing the projectile swings about its
point of support, and thus the center of gravity rises through a vertical distance h.
Knowing this velocity after impact, it is possible to determine the height. The kinetic
energy of the system an instant after impact must, by conservation of energy, equal the
increase in potential energy gained by the pendulum when it reaches its highest point. By
equating the kinetic energy to the potential energy:
(5)
KE = PEg
1
(mb + m p )v 2f = (mb + m p )gh
2
from which
v2
h=
2g
(6)
If the maximum height the center of mass the pendulum swings through is h and
the length of pendulum from the center of mass to the point of rotation is L, then the
angle θ is given by
(7)
" h%
θ = cos $1− '
# L&
−1
Compare this angle to the average angle obtained from your experiemtal trials and
calculate a percent error.
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Apparatus:
1. Ballistic pendulum
2. Scale
3. Tape measure
4. Steel Marble
5. Plumb bob
6. Carbon paper
PROCEDURE
1. In the first part of the experiment, the initial velocity of the projectile is obtained
from measurements of the range and fall. The apparatus should be set near one
edge of a level table. In this part the pendulum is not used and should be swung up
onto the rack so that it will not interfere with the free flight of the ball.
2.
Get the gun ready for firing by placing the ball inside the spring gun and using the
plunger, push the ball in the gun until the trigger is engaged at the highest setting.
Make sure the ball is resting against the spring and has not rolled toward the front
of the gun. The ball is fired horizontally so that it strikes a target placed on the
floor. Fire the ball and determine approximately where it strikes the floor. Place a
sheet of white paper on the floor so that the ball will hit it near its center and cover
it with carbon paper; tape the corners of the paper to keep it from moving around.
In this way a record can be obtained of the exact spot where the ball strikes the
floor. Fire the ball a total of six times.
3.
Measure the horizontal distance from the point of projection (There is a circle with
an x inside on the side of the spring gun.) to the point in the middle of the cluster of
shots with the floor. Use a plumb bob to locate the point on the floor directly below
the ball as it leaves the gun.
4.
Measure the vertical fall of the ball, that is, the vertical distance of the point of
projection above the floor.
5.
Calculate the initial velocity of the ball by using equations (1) and (2).
6.
Calculate the final velocity of the ball and the pendulum using conservation of
momentum (4). Show work on the data sheet and record the results in the table.
7.
Calculate the height of the pendulum using equation (6) and the predicted angle of
the pendulum using equation (7).
8.
Loosen the thumbscrew at the top of the pendulum and carefully remove the
pendulum from its support. Weigh the pendulum and the ball separately and record
the values obtained. Measure the length of the pendulum from the axis of rotation
to the center of mass of the pendulum.
9. Get the gun ready for firing. Release the pendulum and allow it to hang freely. Reset
the angle indicator to zero. When the pendulum is at rest, pull the trigger, thereby
firing the ball into the pendulum. This will cause the pendulum with the ball inside
it to swing up and drag the angle indicator with it. Record the maximum angle on
the data sheet. Repeat five times.
10.
Compute the average angle of the pendulum. And compare to the calculated angle.
Find a percent error.
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Data
Projectile Motion:
Horizontal distance of the Projectile (x) = _______________
Vertical distance (h) = ______________
Show work for initial velocity calculation below:
Initial velocity of the ball by projectile motion (vi)= _______________
Mass of ball (mb) = _____________
Mass of Pendulum (mp) = _______________
Show work for final velocity calculation below:
Final velocity of the ball and pendulum (vf)= _______________
Show work for height of pendulum below:
Height of pendulum (h) = _______________
Length of Pendulum (L) = _____________
Show work for angle of pendulum below:
Calculated angle of pendulum (θ) = _______________
Experimental Angle (θ) = 1.______ 2.______ 3.______ 4.______ 5.______
Average experimental angle (θ) = _________
Percent difference between calculated and experimental θ = ___________
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QUESTIONS
1.
For the ballistic pendulum, calculate the kinetic energy of the ball before the
collision and the kinetic energy of the ball and pendulum just after the collision.
2.
Using the result of question 1, find the fraction of the kinetic energy lost during the
collision with the initial kinetic energy. Compare this fraction to the ratio of the
mass of the pendulum to the total mass of the pendulum and the ball.
Fraction Lost =
KE f − KEi
−m p
=
KEi
mb + m p
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