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EVERYTHING MOVES!!!!!
THERE IS NO APSOLUTE REST!!!!
Example:
 A frame of reference – is a perspective from which a
system is observed together with a coordinate system used to
describe motion of that system.
Classical Mechanics: deals with the physical laws describing the
motion of bodies under the action of a system of forces.
It successfully describes the motion for object that are
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1. large compared with the dimensions of atoms (10 m)
2. moving at speeds that are small compared to the speed
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of light (3x10 m/s)
Kinematics – is the branch of classical mechanics that describes
the motion of objects and systems (groups of objects) without
consideration of the forces acting on them. Motion is described
in terms of distance/displacement, speed/velocity, and
acceleration.
Dynamics – is the study of forces explaining why objects change
the velocity. Explains motion and causes of changes using
concepts of force and energy.
The movement of an object through space can be quite
complex.
There can be internal motions, rotations, vibrations, etc…
This is the combination of rotation
(around its center of mass) and the
motion along a line - parabola.
If we treat the hammer as a particle
the only motion is translational
motion (along a line) through space.
A racing car travels round a circular track of radius 100 m.
The car starts at O. When it has travelled to P its displacement as
measured from O is
A
B
C
D
100 m due East
100 m due West
100 √ m South East 
100 √ m South West
 Vectors and Scalars
Each physical quantity will be either a scalar or a vector.
 Scalar is a quantity which is fully described by a magnitude
(or numerical value with appropriate unit) alone.
Temperature, length, mass, time, speed, …
 Vector is a quantity which is fully described by both
magnitude (or numerical value and unit) and direction.
Displacement, velocity, force,…
Scalar
distance - 50 km
speed - 70 km h-1
Vector
displacement: 50 km, E
velocity:
70 km h-1, S-W
scalars obey the rules of ordinary algebra:
2 kg of potato + 2 kg of potato = 4 kg of potato
Kinematics in One Dimension (along a line)
Vectors obey the rules of vectors’ algebra:
The sum of two vectors depends on their directions.
Our objects will be represented as point objects (particles) so
they move through space without rotation.
Simplest motion: motion of a particle along a line – called:
translational motion or one-dimensional (1-D) motion.
 Average and Instantaneous Velocity
 Displacement is the shortest distance in a given direction.
The displacement tells us
how far an object is from
its starting position and in
what direction.
(it tells us how the object
is displaced)
in some time
interval
at one
instant
when we say velocity we
mean instantaneous velocity
 DEF:
Average velocity is the displacement divided by elapsed time.
v avg =
x 2 -x1
Δx
=
t 2 -t1
Δt
SI unit : m/s
(it obviously has direction, the same as displacement)
Example: an ant wanders from P to Q position
distance traveled = 1 m
shortest distance from P to Q = 0.4 m
displacement of the ant = 0.4 m, SE
Example:
1) x1 = 7 m, x2 = 16 m
2) x1 = 7 m, x2 = 2 m
x3 = 12 m ∆x = 5 m
“+” direction & distance
∆x = –5m
“–” direction & distance
 Instantaneous velocity
Instantaneous velocity is the velocity at one instant.
The speedometer of a car reveals information about the
instantaneous speed of your car. It shows your speed at a
particular instant in time.
If direction is included you have instantaneous velocity.
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 Average and Instantaneous Speed
Example:
How fast do your eyelids move when you blink? Displacement is
zero, so vavg = 0. How fast do you drive in one hour if you drive
zigzag and the magnitude of the displacement is different from
distance?
To get the answers to these questions we introduce speed:
A racing car travels round a circular track of
radius 100 m. The car starts at O.
It travels from O to P in 20 s.
Average speed - during some time interval
The car starts at O. It travels from O back to O in 40 s.
 Speed is the distance object covers per unit time.
Its velocity was 0 m/s. Its speed was 2πr/t = 16 m/s.
v avg
distance travelled
=
Δt
it tells us how fast the object is moving
KCR train has travelled a distance of about 6.7 km from
University Station to Tai Po Market Station. But if we measure
their separation by drawing a straight line on the map, we will
find that Tai Po Market Station is only 5.4 km from University
Station, roughly in the North-West direction. This is the
displacement of the train.
Its velocity was 10 m/s.
Its speed was πr/t = 16 m/s.
 Motion with constant velocity – uniform motion
in that case, velocity is the same at all times so v = vavg
at all times, therefore:
v=
x
t
or
x = vt
Object moving at constant velocity covers the same distance in
the same interval of time.
 Acceleration
 DEF: Acceleration is the change in velocity per unit time
m
SI unit:  a  = s = m/s2
s
it has direction, the same as the change of velocity
a = 3 m/s2 means that velocity changes 3 m/s every second!!!
If an object’s initial velocity is 4 m/s then after one second it will
be 7 m/s, after two seconds 10 m/s,…
We take the KCR trip from University Station to Tai Po Market
Station. It takes about 6 minutes to travel a distance of 6.7 km.
Thus,
The displacement of Tai Po Market Station from University
Station is 5.4 km, so
in the
North-West direction. This is smaller than the average speed of
the train.
So why do we care of velocity at all? OK, it gives us direction
what is very important (just imagine airplane controller with
information only on speed of airplanes not on directions). But we
saw that average speed is greater in general than magnitude of
average velocity. So why is concept of velocity so important?
Because acceleration is a vector, all of equations are vector
equations.
Acceleration can be in any direction to the velocity and the
motion will depend on that.
ONLY:
if motion is 1-D without changing direction:
average speed = magnitude of average velocity because
distance traveled = displacement
• instantaneous speed = magnitude of instantaneous velocity
 Uniformly Accelerated Motion
motion with constant acceleration
let:
t = the time interval for which the body accelerates
a = constant acceleration
u = the velocity at time t = 0, the initial velocity
v = the velocity after time t, the final velocity
x = the displacement covered in time interval t
1. from the definition of a:
v = u + at
→
velocity v at any time t = initial velocity u
increased by a, every second
example:
u = 2 m/s
a = 3 m/s
t (s)
0
1
2
3
v (m/s)
2
5
8
11
(
)
speed increases 3 m/s
EVERY second.
← arithmetic sequence, so:
(
)
 In general: for the motion with constant acceleration:
v avg =
u+ v
2
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ALL TOGETHER:
Examples of changing direction only:
 Any Motion
 Motion with constant velocity – uniform motion
v = vavg at all times, therefore:
x = vt
 Uniform Accelerated Motion equations
A stone is rotating around the center of a circle.
The speed is constant, but velocity is not –
direction is changing as the stone travels around,
therefore it must have acceleration.
Velocity is tangential to the circular path at any time.
ACCELERATION IS ASSOCIATED WITH A FORCE!!!
The force (provided by the string) is forcing the stone to move in
a circle giving it acceleration perpendicular to the motion –
toward the CENTER OF THE CIRCLE - along the force. This is the
acceleration that changes velocity by changing it direction only.
When the rope breaks, the stone goes off in the tangential
straight-line path because no force acts on it.
In the case of moon acceleration is caused by gravitational force
between the earth and the moon. So,
acceleration is always toward the earth.
That acceleration is changing velocity
(direction only).
1. weakening gravitational force would
result in the moon getting further and
further away still circling around earth.
In addition to these equations to solve a problem with constant
acceleration you’ll need to introduce your own coordinate
system, because displacement, velocity and acceleration are
vectors (they have directions).
2. no gravitational force all of a sudden: there wouldn’t be
acceleration – therefore no changing the velocity (direction) of
the moon, so moon flies away in the direction of the velocity at
that position ( tangentially to the circle).
3. The moon has no speed – it moves toward the earth –
accelerated motion in the straight line – crash
Acceleration can cause: 1. speeding up 2. slowing down
3. and/or changing direction
So beware: both velocity and acceleration are vectors. Therefore
• 1. if velocity and acceleration (change in velocity) are in the
same direction, speed of the body is increasing.
• 2. if velocity and acceleration (change in velocity) are in the
opposite directions, speed of the body is decreasing.
• 3. If an object changes direction even at constant speed it
is accelerating. Why? Because the direction of the car is
changing and therefore its velocity is changing. If its velocity
is changing then it must have acceleration.
4. High speed – result the same as in the case of weakening
gravitational force
Only the right speed and acceleration (gravitational force) would
result in circular motion!!!!!!!
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What can we find from graphs of motion ?
From graph velocity vs. time
From graph displacement vs. time
First 5 seconds:
Slope of the graph =
First 5 seconds of motion:
Slope of the graph =
recognizing that as definition of velocity,
come to conclusion:
slope of the displacement – time graph is velocity
Therefore for the first 5 s: a = 50/5 =10 ms²
Therefore for the first 5 s: v = 25/5 =10 m/s
slope =
Recognizing this as definition of acceleration:
Come to conclusion:
slope of the velocity – time graph is acceleration
Δx
= v avg
Δt
vavg gives us no details of the motion between initial and final
points.
Determinedisplacement: a is constant, so
Average velocity of a particle during the time interval Δt is
equal to the slope of the straight line joining the initial (P) and
final (T) position on the position-time graph.
Recognizing this as area of triangle, conclusion is:
Area under a velocity-time graph is the displacement.
During first 5 s the object has travelled: ½ x 50 x 5 = 125m
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From displacement – time graph we can find velocity
(instantaneous and average) by calculating the slope
The slope represent acceleration.
The area under graph represent its displacement.
 Free Fall
Free fall is vertical (up and/or down) motion of a body
where gravitational force is the only or dominant force
acting upon it
Gravitational force gives all bodies regardless of mass or shape,
when air resistance can be ignored, the same acceleration.
The slope is the rate of change of acceleration (jerk).
Area under the graph v – t is the change in velocity.
This acceleration is called free fall or gravitational acceleration
(symbol g – due to gravity).
Free fall acceleration at Earth’s surface is about 9.8 m/s2 toward
the center of the Earth.
Let’s throw an apple equipped with a speedometer
upward with some initial speed u.
That means that apple has velocity u as it leaves our hand.
The speed would decrease by 9.8 m/s every second on the way
up, at the top it would reach zero, and increase by 9.8 m/s for
each successive second on the way down.
g depends on how far an object is from the center of the Earth.
The farther the object is, the weaker the attractive gravitational
force is, and therefore the gravitational acceleration is smaller.
At the bottom of the valley you accelerate faster (slightly)
then on the top of the Himalayas.
Gravitational acceleration at the distance 330 km from the
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surface of the Earth (where the space station is) is 7.8 m/s .
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In reality – good vacuum (a container with the air pumped out)
can mimic ideal free fall.
August 2, 1971 experiment was conducted on the Moon – David
Scott simultaneously released geologist’s hammer and falcon’s
feather. Falcon’s feather dropped like the hammer. They
touched the surface at the same time.
 Equations are the ones for uniform accelerated
motion with a = g
How much time will the students have to save themselves? What
is the velocity/speed of the pen when it reaches the ground?
Givens:
u = 0 m/s (dropped)
2
g = 10 m/s
Unknowns:
t=?
v=?
y
v = u + gt
𝑎
𝑢𝑡
𝑡
2
200 = 0 + 5 t
t
6.3 s
at the top: v = 0 (object is changing direction)
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Very often we will use: g = 10 m/s (simplicity)
To describe the motion we need coordinate system.
EXAMPLES:
1. Dr. Huff, a very strong lady, throws a ball upward with initial
speed of 20 m/s.
How high will it go? How long will it take for the ball to come
back?
Givens:
Unknowns:
u = 20 m/s
t=?
2
g = - 10 m/s
y=?
at the top v = 0
v = u + gt
v = 0 + 10 t
v = 63 m/s (speed)
velocity is 63 m/s downward
3. Mrs. Radja descending in a balloon at the speed of 5 m/s
above our school drops her car keys from a height of 100 m.
How much time will the students have to save themselves?
What is the velocity of the keys when they reach the ground?
t=?
v=?
y
𝑢𝑡
𝑎
𝑡
Equations, substitution, solution:
100
1. at the top:
v = u + gt = 0
20 – 10t = 0
t=2s
free fall is symmetrical up and down
t = 4 s for the whole motion
2. max height
time can be only positive
t=4s
still enough time to save yourself – start running
v = u + gt
v = 50 m/s, downwards
it is dangerous speed; run away as fast as you can
20 m
You would get the same result if you chose y axis to be positive
downwards. In that case
g = 10 m/s/s, u = – 20 m/s.
t = 2 s to the top (–20 + 10t = 0)
=
negative sign in the coordinate system in which downward is
positive means that
is upward.
2. Mr. Rutzen, hovering in a helicopter 200 m above our school
suddenly drops his pen.
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4. Dr. Huff, our very strong lady, goes to the roof and throws a
ball upward. The ball leaves her hand with speed 20 m/s.
Ignoring air resistance calculate
a. the time taken by the stone to reach its maximum height
b. the maximum height reached by the ball.
c. the height of the building is 60 m. How long does it take for
the ball to reach the ground?
d. what is the speed of the ball as it reaches the ground?
Graphs of free fall motion
v =gt
v = 10 t
Time
(s)
0
1
2
3
4
a. at the top: v = u + gt = 0
20 – 10t = 0
we ignore the height of any human being
Velocity
(m/s)
0
10
20
30
40
t=2s
c. y = - 60 m
2
– 60 = 20 t – 5 t
2
t – 4 t – 12 = 0
t=2±4
as the can not e negative t = 6 s
u=0
y = - 80 m
changing slope –
changing speed → acceleration
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– 80 = – 5 t
t=4s
t=4+2=6s
as you can see equation
takes care of direction of initial velocity
d.
v = u + gt
v = 20 – 10 x 6 = – 40 m/s
(direction : velocity)
speed at the bottom is 40 m/s
y = 5 t2
constant slope –
constant acceleration
or
from the top
g 2
t
2
Distance
(m)
0
5
20
45
60
b. maximum height:
or
y=
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Air Drag and Terminal Velocity
If air resistance can not be neglected, there is additional force
(drag force) acting on the body in the direction opposite to
velocity.
Acceleration is getting smaller due to air resistance and
eventually becomes zero.
When the force of the air resistance equals gravity, the object
will stop accelerating and maintain the same speed. This is called
terminal velocity/speed. It is different for different bodies.
If a raindrops start in a cloud at a height h = 1200m above the
surface of the earth they hit us at 340mi/h; serious damage
would result if they did. Luckily: there is an air resistance
preventing the raindrops from accelerating beyond certain
speed called terminal speed….
How fast is a raindrop traveling when it hits the ground?
It travels at 7m/s (17 mi/h) after falling approximately only 6 m.
This is a much “kinder and gentler” speed and is far less
damaging than the 340mi/h calculated without drag.
The terminal speed for a skydiver is about 60 m/s
(pretty terminal if you hit the deck)