01

NOTE
P H Y S I C S

KINEMATICS
INTRODUCTION TO KINEMATICS

What you already know What you will learn

• Differentiation • Rest and Motion

• Vectors • Frame of reference

• Integration • Motion Parameters

Mechanics

Mechanics is the study of cause and effect of motion of bodies.

Kinematics Dynamics

Branch of mechanics which analyses the Branch of mechanics which analyses

effects i.e., parameters and properties the cause of motion of bodies i.e., the
of motion. force(s).

Rest and Motion - Motion is a combined property of the object and the observer. There is no
meaning of rest or motion without the observer. Nothing is in absolute rest or in absolute motion.
An object is said to be in motion with respect to an observer, if its position changes with respect to
that observer. For motion, it may happen that the observer moves with respect to the object or the
object moves with respect to the observer.

Motion is always relative.

Let us take one example to understand this

Case 1 Case 2
Boy is moving. The ball and girl are stationary. Boy and ball are moving with same speed and
in the same direction. The girl is stationary.

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v v

Observations recorded:

Boy: feels the ball is moving because he is in Boy: feel the ball is at rest because the speed
relative motion with the ball of the ball is the same
Girl: feels the ball is at rest Girl: ball will appear in motion

Thus, if we do not mention an observer, we cannot predict the motion of the ball as it will appear
different for different observers. This example of understanding motion is meaningless unless the
observer is defined.
So, we will now define motion by defining the observer, the frame (frame of reference) in which the
observer sits and other related parameters.
Frame of reference : The frame attached with the observer.
Reference point : The point from where measurements are taken.

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Dimension

The dimension of a mathematical space is the minimum number of independent information

needed to specify any point within it.

Ram: 3iˆ
Motion in 1D x axis
2ˆj − 4kˆ
In 1D motion, particles are constrained to move in a straight y axis
ˆi + 5 ˆj − 3kˆ
line and can only change their direction of motion opposite
to their original direction of motion. 3iˆ
2ˆj − 4kˆ
Consider three friends Ram, Shyam, and Sita moving along -z axis
the paths, which can be expressed in vector form as follows: ˆi + 5 ˆj − 3kˆ

3i ˆ
Ram: 3iˆˆ
3i
ˆ
2ˆˆˆjjj −
Shyam: 2
2 − 4k
− 4kˆˆ
4k
ˆˆˆii + ˆ − 3k ˆ
Sita: i++55 ˆˆjjj −
5 3kˆˆ
− 3k y axis
As the vectors represent the straight directions, motion of x axis
3iˆ
these three friends is actually in 1D. One should never think
ˆi, ˆj and kˆ makes the problem of one, two or three dimensions. 2ˆj − 4kˆ
Ram is moving along the x-axis in a straight line, Shyam is -z axis ˆi + 5 ˆj − 3kˆ
moving in a straight line in a plane, and Sita is moving in a
straight line in a space as shown in the figure.

Motion in 2D
If motion of a particle is constrained to move in any direction in a plane, then it is known as two-
dimensional motion.
Example: An ant moving on the floor.
Motion in 3D
Motion of particles in any direction in 3D space is three-dimensional motion.
Example: A housefly flying in a room.

Following three points define the motion:

• Where? y
Space coordinates define the body with respect to the defined
reference point. The position is expressed in x, y, and z coordinates.
• When?
x
Time coordinate defines the location of the body at a specific time.
O
• Who reported?
The one who reports the event (space time coordinates) is the z
observer.
These three questions explain any event completely.

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Motion Parameters

Position Acceleration Velocity Distance

Displacement Time Speed

• Distance travelled by an object depends on the spatial reference frame. Similarly, displacement,
speed, velocity, acceleration, momentum, and kinetic energy are also dependent on the
reference frame.
• Time of the event is independent of the spatial reference frame because time will remain
constant irrespective of which frame you are observing it from.
• Distance between two points is independent of the frame of reference.
Position
The position of a point is its location in the space at a certain moment of time with respect to a
frame of reference. It is specified by x, y, and z coordinates in spatial coordinates as shown in the
figure.
y
y
P (x, y, z)
P(x)
0 x P (x, y)
0 x
0 x

z
• Position vector
y
It specifies the position of point with respect to origin.
P (x, y, z)
For a point with coordinates (x, y, z) as shown in the figure, position
vector is as follows:
 ˆ ˆ
r = xi + yj + zk̂
x

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• Displacement vector
y B
It specifies the position of a particle with respect to another A (x2, y2, z2)
particle.
(x1, y1, z1)
Displacement vector for point B from A is as follows: x

s =( x 2 − x 1) ˆi + ( y 2 − y 1) ˆj + ( z2 − z1) k̂

Distance
Distance covered by a body is the total length of actual path covered in travelling from initial to final
positions.
• Distance is scalar quantity
• Metre is the SI unit of distance

Displacement
Displacement is the shortest distance from initial to final position of a body undergoing motion
• Displacement is vector quantity.
Distance vs Displacement
When a particle moves in a straight line, distance and displacement will be equal in magnitude.

Example 1
A ball moves in a straight line from point A to B and then travels in the reverse direction to point C.
• Distance travelled by ball = 10 units + 8 units +
8 units = 26 units.
• Displacement of ball = 10 units + 8 units - 8 units
= 10 units, as distance travelled from B to C is in O
negative direction. C
A B

10 units 8 units

Example 2
A boy and girl were initially at origin, O. Now, consider the boy travels 2 m in negative x-direction
and the girl travels 2 m in positive x-direction.

O x = -2 m O x=2m

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• Distance travelled by boy = 2m

• Distance travelled by girl = 2m
• Displacement of boy = -2m
So, in vector form, displacement of boy = -2 î m, where î is a unit vector in x-direction.

• Displacement of girl = +2m

So, in vector form, displacement of boy = +2 î m, where î is a unit vector in x-direction.

Distance ≥ |Displacement|

BOARDS NEET
Displacement along circular path

B
Distance covered = ADB (arc length) = r𝜽
Linear Displacement = AB
r C D
𝜽/2
0 𝜽/2 AC = 0A sin 𝜽
2
r A 𝜽
AB = 2AC = 2 r sin
2

A body moves over one-fourth of a circular arc from A to B in a circle of radius r. The
magnitudes of distance travelled and displacement will be, respectively

(A) πr , r √2 (B) πr , r (C) πr, r (D) πr, r

2 2 √2

Solution y
B

Distance travelled = l
I = r𝜽 r
x
πr O A
I=
2

Displacement = √ r 2 + r 2 = r √ 2
Option A is correct.

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If a particle is moving on a square track of the side 𝑎. What

will be the distance and displacement of the particle when it
reaches his initial position.

Solution

As the particle moves along the perimeter of square,

Distance travelled = 4a units
As it gets back to original position in the end,
Displacement = 0 unit
Note: Displacement for the body moved in any closed loop is zero.

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