Motion

MOTION
1. Motion and Rest
2. Distance and Displacement
3. Uniform Motion
4. Non-uniform Motion
5. Speed
6. Velocity
7. Acceleration
8. Equations of Uniformly Accelerated Motion
9. Graphical Representation of Motion
10.Distance-Time Graph
11.Speed-Time Graph
12.Derivation of Equations of Motion by Graphical Method
13.Uniform Circular Motion
14.Calculation of Speed of a Body in Uniform Circular Motion
Created by C. Mani, Education Officer, KVS Silchar Region

Concept of a Point Object
In mechanics, a particle is a geometrical mass point or a material body of
negligible dimensions. It is only a mathematical idealization.
Examples:
Earth
In practice, the nearest approach to a particle is a body, whose size is much
smaller than the distance or the length measurements involved.

Motion and Rest
A ball is at rest w.r.t. a stationary man.
A car is at rest w.r.t. a stationary man.
A ball is moving w.r.t. a stationary man.
A car is moving w.r.t. a stationary man.

Motion:
An object is said to be in motion if it changes its position with respect to its
surroundings and with time.
Examples:
1. Moving cars, buses, trains, cricket ball, etc.
2. All the planets revolving around the Sun.
3. Molecules of a gas in motion above 0 K.
Rest:
An object is said to be at rest if it does not change its position with respect to
its surroundings and with time.
Examples: Mountains, Buildings, etc.
Rest and Motion are relative terms
An object which is at rest can also be in motion simultaneously.
Eg. The passengers sitting in a moving train are at rest w.r.t. each other
but they are also in motion at the same time w.r.t. the objects like trees,
buildings, etc.

Motion and Rest are Relative Terms
Car is moving w.r.t. stationary man.
Car is moving w.r.t. stationary man.

Motion and Rest are Relative Terms
Both the cars are at rest w.r.t. stationary man.
Both the cars are moving w.r.t. a stationary man.
Both the cars are at rest w.r.t. each other.

In the examples of motion of ball and car, man is considered to be
at rest (stationary).
But, the man is standing on the Earth and the Earth itself moves
around the Sun as well as rotates about its own axis.
Therefore, man is at rest w.r.t. the Earth but is rotating and
revolving around the Sun.
That is why motion and rest are relative terms !
Motion and Rest are Relative Terms – How?

A ship is sailing in the ocean. Man-A in the ship is running on the board in
the direction opposite to the direction of motion of the ship. Man-B in
the ship is standing and watching the Man-A.
Analyse the following cases to understand motion and rest !
1. Man-A w.r.t. Man-B
2. Man-A w.r.t. ship
3. Man-B w.r.t. ship
4. Ship w.r.t. still water
5. Man-A w.r.t. still water
6. Man-B w.r.t. still water
7. Ocean w.r.t. the Earth
8. Ocean w.r.t. the Sun
9. Earth w.r.t. the Sun
10.Ship w.r.t. the Sun
11.The Sun w.r.t. Milky Way Galaxy
12. Milky Way Galaxy w.r.t. other galaxies
Your imagination should not ever stop !

Reference Point or Origin
While describing motion, we use reference point or origin
w.r.t. which the motion of other bodies are observed.
In the previous examples, a man at rest is used as
reference point or origin.
We can use any object as reference point. For example, a
car at rest or in motion can be used as reference point.
When you travel in a bus or train you can see the trees,
buildings and the poles moving back.
To a tree, you are moving forward and to you, the trees are
moving back.
Both, you and the trees, can serve as reference point but
motion can not be described without reference point.
What effect do you get when you play video game involving
car racing?

Motion in a straight line
The motion of a body may take place in one dimension, i.e. in a straight line.
This can be represented graphically by plotting a graph between the position
of the body and the time taken by it. This is called position-time graph.
Origin, unit and direction of position measurement of an object
1. The distance measured to the right of the origin of the position axis is taken
positive and the distance measured to the left of the origin is taken negative.
2. The origin for position can be shifted to any point on the position axis.
3. The distance between two points on position-axis is not affected due to the
shift in the origin of position-axis.
-x -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
+x
(in km)

-t -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
+t
(in hours)
Origin, unit and sense of passage of time
1. The time measured to the right of the origin of the time-axis is taken
positive and the time measured to the left of the origin is taken negative.
2. The origin of the time-axis can be shifted to any point on the time-axis.
3. The negative time co-ordinate of a point on time-axis means that object
reached that point a time that much before the origin of the time-axis
i.e. t = 0.
4. The time interval between two points on time-axis is not affected due to
the shift in the origin of time-axis.

When the same point is chosen as origins for position and time
O
x = 0 km
t = 0 h
When the different points are chosen as origins for position and time
Origin for time
x = 40 km
t = 8 h
x = 30 km
t = 6 h
x = 55 km
t = 11 h
x = -40 km
t = -6 h
x = 0 km
t = 2 h
x = -10 km
t = 0 h
x = 15 km
t = 5 h
Origin for position and time
BA C
Origin for position
O BA C

Distance
Distance travelled by a body is the actual length of the path covered by it
irrespective of the direction in which the body travels.
N
5 km
2 km
5 km
Distance travelled is 7 km.
Distance travelled is 10 km.

Displacement
Displacement of a body is the shortest (straight line) distance between its
initial position and final position along with direction.
N
5 km
2 km
5 km
Displacement is 6.57 km in the direction shown by the arrow
mark.
Displacement is 0 km.

A C
B
Distance
Displacement
Conclusions about displacement:
1. The displacement is a vector quantity.
2. The displacement has units of length.
3. The displacement of an object in a given time interval can be
positive, zero or negative.
4. The actual distance travelled by an object in a given time interval
can be equal to or greater than the magnitude of the
displacement.
5. The displacement of an object between two points does not tell
exactly how the object actually moved between those points.
6. The displacement of a particle between two points is a unique
path, which can take the particle from its initial to final position.
7. The displacement of an object is not affected due to the shift in
the origin of the position-axis.

Scalar
Scalar quantity is a physical quantity which has magnitude only.
Eg.: Length, Mass, Time, Speed, Energy, etc.
Vector
Vector quantity is a physical quantity which has both magnitude as well as
direction.
Eg.: Displacement, Velocity, Acceleration, Momentum, Force, etc.
Physical Quantity
Physical quantity is a quantity which can be measured and expressed in
magnitude (value with or without unit).
Eg.: 1. Length can be measured and expressed as 5 m.
2. Relative Density can be measured and expressed as 0.8
S. No. Distance Displacement
1
2
Distance is a scalar quantity. Displacement is a vector quantity.
Distance travelled by a moving
body cannot be zero.
Final displacement of a moving
body can be zero.

Speed
Speed is defined as the time rate of change of distance of a body.
Note:
1. Speed is a scalar quantity.
2. Speed is either positive or zero but never negative.
3. Speed of a running car is measured by ‘speedometer’.
4. Speed is measured in
i) cm/s (cm s-1
) in cgs system of units
ii) m/s (m s-1
) in SI system of units and
iii) km/h (km.p.h., km h-1
) in practical life when distance and time
involved are large.
or
Speed is defined as the distance travelled by a body in unit time.
Speed =
Time taken
Distance travelled
If a body travels a distance ‘s’ in time ‘t’, then its speed ‘v’ is given by:
v =
s
t

Uniform Speed
A body has a uniform speed if it travels equal distances in equal intervals
of time, no matter how small these time intervals may be.
A body has a uniform motion if it travels equal distances in equal intervals
of time, no matter how small these time intervals may be.
Non-uniform Motion
A body has a non-uniform motion if it travels unequal distances in equal
intervals of time.
Uniform Motion
Variable Speed
A body is said to be moving with variable speed, if it covers unequal
distances in equal intervals of time, howsoever small these intervals
may be.

Average Speed
Average speed of a body is the ratio of total distance
travelled to the total time taken to cover this distance.
Eg.:
Let a car covers first 25 km in 1 h, next 35 km in ½ h
and last 30 km in 1 h, then the average speed is =
(25 + 35 + 30) / (1 + ½ +1) = 36 km/h.
Average Speed =
Total Time taken
Total Distance travelled
Instantaneous Speed
When a body is moving with variable speed, the speed
of the body at any instant is called instantaneous
speed.
vav =
stot
ttot

Velocity
Velocity is defined as the time rate of change of displacement of a body.
or
Velocity is defined as the distance travelled by a body in unit
time in a given direction.
If a body travels a distance ‘s’ in time ‘t’ in a given direction, then its
speed ‘v’ is given by:
v =
s
t
Velocity =
Time taken
Distance travelled in a given direction
=
Time taken
Displacement
Note:
1. Velocity is a vector quantity.
2. Direction of velocity is the same as the direction of displacement of the
body.
3. Velocity can be either positive, zero or negative.
4. Velocity can be changed in two ways:
i) by changing the speed of the body or
ii) by keeping the speed constant but by changing the direction.

Variable Velocity
A body is said to be moving with variable velocity, if its speed or its direction
or both change(s) with time.
Average Velocity
When a body moves with variable velocity, the average velocity of the body is
the ratio of the total displacement covered by it to the total time taken.
Note: No effort or force is required to move the body with uniform velocity.
Velocity is measured in
i) cm/s (cm s-1
ii) m/s (m s-1
iii) km/h (km.p.h., km h-1
) in practical life when distance and time involved
are large.
Uniform Velocity
A body is said to be moving with uniform velocity, if it travels in a specified
direction in a straight line and moves over equal distances in equal intervals
time, no matter how small these time intervals may be.
Average velocity =
2
Initial velocity + Final velocity
=
u + v
2

Difference between Speed and Velocity
Speed Velocity
1. Speed is the time rate of change of
distance of a body.
1. Velocity is the time rate of
change of displacement of a body.
2. Speed tells nothing about the
direction of motion of the body.
2. Velocity tells the direction of
motion of the body.
4. Speed of the body can be
positive or zero.
4. Velocity of the body can be
positive, zero or negative.
3. Speed is a scalar quantity. 3. Velocity is a vector quantity.
5. Average speed of amoving
body can never be zero.
5. Average velocity of a moving
body can be zero.
Instantaneous Velocity
When a body is moving with variable velocity, the velocity of the body at any
instant is called instantaneous velocity.

Acceleration
Acceleration is defined as the time rate of change of its velocity.
Acceleration =
Time taken for change
Change in velocity
Acceleration =
Time taken
Final velocity - Initial velocity
or
Suppose a body moving with initial velocity ‘u’ changes to final velocity
‘v’ in time ‘t’, then
a =
v - u
t
Note:
1. Acceleration is a vector quantity.
2. Direction of acceleration is the same as the direction of velocity of the
body.
3. Acceleration can be either positive, zero or negative.
4. Acceleration of a body is zero when it moves with uniform velocity.

Acceleration is measured in
i) cm/s2
(cm s-2
ii) m/s2
(m s-2
iii) km/h2
(km h-2
) in practical life when
distance and time involved are large.
Uniform Acceleration
A body is said to be moving with uniform acceleration, if it travels in a
straight line and its velocity increases by equal amounts in equal
intervals of time.
A body has uniform acceleration if its velocity changes at a uniform rate.
or

Eg.:
The motion of a freely falling body is
uniformly accelerated motion.
The motion of a sliding block on a
smooth inclined plane is uniformly
accelerated motion.

Non-uniform Acceleration
A body is said to be moving with non-uniform acceleration, if its velocity
increases by unequal amounts in equal intervals of time.
A body has non-uniform acceleration if its velocity changes at a non-
uniform rate.
or
Eg.:
The motion of a car on a crowded city road. Its speed (velocity) changes
continuously.
Retardation or Deceleration of Negative Acceleration
A body is said to be retarded if its velocity decreases w.r.t. time.
A car is decelerating to come to a halt.

GRAPHICAL REPRESENTATION OF MOTION
Distance – Time Graph
(Uniform Speed)
Time
Distance
O
A
B
The slope of the distance – time
graph indicates speed.
Speed =
AB
OB
Uniform Motion
Speed – Time Graph
(Uniform Speed)
Time
Speed
O
A
B
The area of the speed – time graph
indicates distance travelled.
D
C
t1 t2
The area of ABCD gives the
distance travelled between the time
t1 and t2 seconds.
C

(Uniform Acceleration)
Time
Distance
O
Non-Uniform Motion
Non-uniform speed
(Uniform Retardation)
Time
Distance
O
Non-uniform speed

Speed (Velocity) – Time Graph:
(Uniform Retardation)
Time
Velocity
O B
A
The slope of the velocity – time
graph indicates retardation.
Retardation =
AO
OB
Speed (Velocity) – Time Graph
Time
Velocity
O B
A
graph indicates acceleration.
Acceleration =
AB
OB
The area of AOB gives the distance
travelled.
The area of AOB gives the distance
travelled.

Speed (Velocity) – Time Graph:
Time
Velocity
O B
A
graph indicates acceleration.
Acceleration =
AB
OB
When the initial speed is not zero
The area of OCAB gives the
distance travelled.
C

EQUATIONS OF UNIFORMLY ACCELERATED MOTION
Consider a body moving with initial velocity ‘u’ accelerates at uniform rate
‘a’. Let ‘v’ be the final velocity after time ‘t’ and ‘s’ be the displacement.
u va
t
We know that:
a =
v - u
t
Cross multiplying, v – u = at
or v = u + at
The equation v = u + at is known as the first equation of motion.
First equation of motion
Acceleration =
Time taken
Final velocity - Initial velocity
s

we get
vav =
u + v
2
or
The equation s = ut + ½ at2
is known as the second equation of motion.
Second equation of motion
Average velocity =
2
Initial velocity + Final velocity
Distance travelled = Average velocity x Time
From the first equation of motion we have, v = u + at
Substituting for v in equation (1),
(1)s =
(u + v)
2
x t
s =
(u + u + at)
2
x t
s =
(2u + at)
2
x t
or s =
2ut + at2
2
or s = ut + ½ at2

Third equation of motion
From the first equation of motion we have,
v - u = at
We know that: vav =
u + v
2
or u + v = 2vav
(1)
or v + u = 2vav (2)
Multiplying eqns. (1) and (2), we get
v2
- u2
= 2atvav
v2
- u2
= 2asor vav x t = s
or v2
= u2
+ 2as
The equation v2
= u2
+ 2as is known as the third equation of motion.

EQUATIONS OF UNIFORMLY ACCELERATED MOTION
BY GRAPHICAL METHOD
First equation of motion
Acceleration =
Time taken for change
Change in velocity
Velocity(m/s)
O C
B
v
u
E
A D
Time (s)t
a =
BD
AD
a =
AE
OC
a =
OE - OA
OC
a =
v - u
t
v – u = at
or v = u + at

Second equation of motion
Velocity(m/s)
Time (s)O C
B
v
u
E
A D
t
The area of trapezium OABC gives
the distance travelled.
s = ½ x OC x (OA + CB)
s = ½ x t x (u + v)
s = ½ x t x (u + u + at)
s = ½ x (2ut + at2
)
s = ut + ½ at2

Third equation of motion
The area of trapezium OABC gives
the distance travelled.
s = ½ x OC x (OA + CB)
s = ½ x t x (u + v)
(v + u) =
2s
t
From the first equation of motion we have,
(v – u) = at (2)
(1)
Multiplying eqns. (1) and (2), we get
v2
- u2
= 2as
or v2
= u2
+ 2as
Velocity(m/s)
Time (s)O C
B
v
u
E
A D
t

UNIFORM CIRCULAR MOTION
The motion of a body in a circular or angular
or curved path is called circular or angular or
curvilinear motion.
When a body moves in a circular path with
uniform speed, its motion is called uniform
circular motion.
Note that the velocity changes at each and
every instant and hence the body experiences
acceleration.
r
O
Velocity of a body in circular path =
2πr
t

Examples of Uniform Circular Motion
1. Artificial satellites move under uniform circular
motion around the earth.
2. The moon moves in uniform circular motion around
the earth.
3. The tip of a second’s hand of a clock exhibits
uniform circular motion.

Acknowledgemen
tThe objects such as body of the red car, aeroplane,
ship, moving man, earth, rotating earth, sun and
galaxy are taken from various web-sites.

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