MOTION1. 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

Concept of a Point ObjectIn 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

DistanceDistance 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.

DisplacementDisplacement 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.

ScalarScalar quantity is a physical quantity which has magnitude only.

Eg.: Length, Mass, Time, Speed, Energy, etc.

VectorVector quantity is a physical quantity which has both magnitude as well as direction.

Eg.: Displacement, Velocity, Acceleration, Momentum, Force, etc.

Physical QuantityPhysical 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.

SpeedSpeed 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 SpeedA 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 MotionA 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 VelocityA 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) 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.

Uniform VelocityA 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 VelocityWhen a body is moving with variable velocity, the velocity of the body at any instant is called instantaneous velocity.

AccelerationAcceleration 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) in cgs system of units

ii) m/s2 (m s-2) in SI system of units and

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

Dis

tan

ce

O

A

B

The slope of the distance – time graph indicates speed.

Speed = AB

OB

Uniform Motion

Speed – Time Graph (Uniform Speed)

Time

Sp

eed

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

Distance – Time Graph (Uniform Acceleration)

Time

Dis

tan

ce

O

Non-Uniform MotionNon-uniform speed

Distance – Time Graph (Uniform Retardation)

Time

Dis

tan

ce

O

Non-uniform speed

Speed (Velocity) – Time Graph: (Uniform Retardation)

Time

Vel

oci

ty

O B

AThe slope of the velocity – time graph indicates retardation.

Retardation = AO

OB

Speed (Velocity) – Time Graph (Uniform Acceleration)

Time

Vel

oci

ty

O B

AThe slope of the velocity – time 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: (Uniform Acceleration)

Time

Vel

oci

ty

O B

AThe slope of the velocity – time 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 vat

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)

2x t

s =(u + u + at)

2x t

s =(2u + at)

2x 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

Vel

oci

ty (

m/s

)

O C

Bv

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

Ve l

oci

ty (

m/s

)

Time (s)O C

Bv

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

Vel

oci

ty (

m/s

)

Time (s)O C

Bv

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.

rO

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.

1. NCERT Exercise Questions .