# Class – 11 Physics, Chapter – 10, Work, Power and Energy

Definition Of Work : When a force is applied on an object to displace it from its initial position then it is said to be work is done. It is the Dot Product of Force and Displacement.   Types Of Work :

1 ) Positive Work : When the applied force and displacement of the objects take place in the same direction or at acute angle , then it is said to be positive work is done. That is the applied force and displacement are parallel to each other and angle between them is zero degree and the value of cos0 is 1 i.e

W = FS

OR

W = FS cosθ

Examples :

1. Kicking of a football
2. Moving a chair
3. pushing and moving a table
4. A car moving forward
5. throwing a stone

2 ) Negative Work :  When the applied force and displacement of the objects take place in the opposite direction, then it is said to be negative work is done. That is the applied force and displacement are antiparallel to each other and angle between them is 180 degree and the value of cos180 is ( – 1 ) i.e

W = -FS

Examples :

1. Two boys push each other
2. In Tug of War when opponent team pulls hard.
3. Trying to catch a heavy object, hands move down

3 ) Zero Work :  When the applied force and displacement of the objects take place perpendicularly, then it is said to be zero work is done. That is the applied force and displacement are normal to each other and angle between them is 90 degree and the value of cos90 is 0 . when we apply a force on the object but it does not displace from its position, then there is zero work done .

W = 0

Examples :

1. Pushing a wall
2. Walking on the road
3. Motion of planet on its orbit

#### Summary of types of work Unit Of Work :

1 ) SI – Unit —- Joule

2 ) CGS – Unit ——— erg

Dimension Of Work :

[ W ] = [ F ] [ S ] = [ MLT-2 ] [ L ] = [ ML2T-2 ]

Workdone By Variable Force :

In everyday life, most of the force are variable in nature.

Whenever we apply variable force on the object to displace it from its previous position, the workdone in this case may be termed as workdone by variable force.  Definition Of Power : The rate of doing work is called as Power.

Power = Work / time

P = dw / dt

where dw = small workdone

dt = time interval

It is the scalar product of Force and Velocity.

P = F . v

Unit Of Power :

1 ) SI – Unit : ( 1) Watt ( 2 ) J / Sec ( 3 ) N-m / sec

2 ) CGS unit : ( 1 ) gm – cm / sec ( 2 ) dyne – cm / sec

Dimension Of Power :

[ Power ] = [ Work ] / [ Time ] = [ ML2T-2 ] / [ T ] = [ ML2T-3 ]

Definition of energy :

The capacity of doing work is called energy.

Types Of Energy :

There are several types of energy which are given below:

1. Mechanical Energy
2. Chemical energy
3. Atomic energy
4. Light energy
5. Sonic energy
6. Wind energy
7. Thermal energy
8. Nuclear energy
9. Electromagnetic energy
10. Gravitational energy
11. Ionization energy
12. Electrical energy
13. Heat energy
15. Elastic energy
16. Internal energy

These are not enough but in the syllabus of class – 11 physics, we will discuss only mechanical energy in detail.

##### Mechanical Energy :

The energy possessed by an object according to its structure, configuration, position and motion, is called as Mechanical energy .

There are two types of mechanical energy

1. Kinetic energy
2. potential energy

Kinetic energy :

The energy possessed by an object with respect to its motion, is termed as kinetic energy.

Examples :

1 ) A moving car 2 ) Moving fan

3 ) The arrow flying through air when the string is released on the bow. There are basically several types of kinetic energy

Rotational kinetic energy, Translational kinetic energy, Vibrational kinetic energy, Translational – rotational kinetic energy and etc.

Expression for kinetic energy :

Let

u = initial velocity = 0    ( ∵ the object is initially at rest )

v = final velocity

a = acceleration

m = mass of the object

t = time

F = force applied on the object to displace it from its position

s = displacement

From Newton’s 2nd law of motion we have

F = ma —————————————— ( 1 )

From equation of motion we have

v2 = u2 + 2as

⇒ v2 = 2as

⇒ s = v2 / 2a

Now, workdone = force * displacement = ma * v2 / 2a = mv2 / 2 Potential energy :

The energy possessed by an object due to its configuration, structure and height is called as potential energy. This energy is associated with the objects at rest.

Examples :

1 ) A stretched string

2 ) A coiled spring

3 ) A snow pack

4 ) A stretched rubber band

5 ) A rock sitting at the edge of cliff

6 ) A raised weight

7 ) stored water in a tank or dam

Different types of potential energy:

1 ) Gravitational Potential Energy :

The energy required to bring a body of definite mass from infinity to a point in the gravitational field is called as Gravitational potential energy. In case of Earth, this is the amount of work done to raise a body to a particular height.

2 )Electrostatic potential energy :

The energy of a system of two or more charged bodies is called as Electrostatic potential energy.

3) Elastic potential energy :

When a mass attached to a spring is displaced from its equilibrium position, a restoring force is come into play. The work done against the restoring force is stored in the form of potential energy. This type of potential energy is termed as Elastic potential energy.

Expression Of The  Gravitational Potential Energy : Let an object of mass ‘ m ‘ is raised to a height of ‘ h ‘. The work done in raising the object is stored in the form of potential energy.

∴ Potential energy = Force by which the object is raised * height = Weight * height = mgh

where g = acceleration due to gravity.

##### P.E = mgh

Expression for the elastic potential energy

or

potential energy of a spring : The potential energy of spring

Let us assume that a very light and perfectly elastic spring is fixed at one end from a rigid support. Other end of the support is attached with a body of mass ‘m’. ( In the above figure, red circular part is assumed as the body. Let the body is displaced from it’s equilibrium position. Due to elastic property of spring, the restoring force come into play and the whole system is seemed to be in to and fro motion.

Let

displacement be = x

restoring force = F

According to Hook’s Law

∝ – x —————-( 1 )

F = – kx

where k = spring constant or force constant

Work done against the  restoring force is given by

w = ∫ Fdx

where dx = small displacement

w = ∫ ( – kx ) dx

w = – k ∫ xdx

w = – k x2 / 2 + c

where ‘c’ is integration constant

when x = 0 then w = 0

0 = – k ( 0 )2 / 2 + c

c = 0

w = – kx2 / 2

this work done is stored in the form of elastic potential energy given by

#### ∴  U = kx2 / 2 Principle of conservation of energy :

It states that

” Energy can not be created nor be destroyed but can be converted from one form to another ”

OR

” Energy of an isolated system remain constant ”  Conservative force : When the work done by a force in moving a body from one point to another is independent of the followed between the two points then the force is said to be conservative force.

Example : Gravitational force, elastic force and electrostatic force etc.

Properties of Conservative force :

• The work done is independent of path followed between the initial and final position of a body.
• The work done by this force depends only on the initial and final position of the body.
• The work done by this force in moving a particle along a closed path is zero.
• The work done by this force is always reversible.
• Mechanical energy of a body is always conserved under the action of conservative force.

Non conservative force :  When the work done by a force in moving a body from one point to another is dependent of the followed between the two points then the force is said to be conservative force. It is also termed as dissipative force.

Example : Frictional force, force due to air resistance.

Properties of Non Conservative force :

• The work done is dependent of path followed between the initial and final position of a body.
• The work done by this force does not depend only on the initial and final position of the body.
• The work done by this force in moving a particle along a closed path is non zero.
• The work done by this force is always irreversible.
• Mechanical energy of a body is not conserved under the action of conservative force.
##### Collision :

an accident that happen when two bodies hit each other with force , is called as collision and these vodies are termed as colliding objects or colliding bodies. Types of collision :

1 ) Head – on – collision : when two bodies moving in opposite direction crash into each other. The collision occurred is called as head – on – collision. This one of the most dangerous accident because the impact is doubled due to the travelling speed f each body. It is also termed as fave – to – face collision.  What 2 ) Elastic collision : The collision in which there is no loss of kinetic energy in the colliding system, is termed as elastic collision.

Properties of Elastic collision are discussed in the following way.

• Principle of conservation of linear momentum is obeyed.
• Total kinetic energy of the system is conserved.
• The total energy of the system is conserved.
• The mechanical energy is not converted into other form of energy like sound energy, light energy and heat energy etc.
• Forces involved during the interaction are conservative nature.

3 ) Inelastic collision: The collision in which the kinetic energy of the system is not conserved due to action of internal friction, is called as inelastic collision. This type of collision obeys the principle of conservation of linear momentum.

4 ) Perfectly inelastic collision : The collision in which the colliding bodies are stuck together after collision and then move with a common velocity , is called as perfectly inelastic collision.

Properties of Perfectly Inelastic collision are discussed in the following way.

• Total energy is conserved
• The linear momentum of the system is conserved.
• Kinetic energy is not conserved.
• A part of mechanical energy may be converted into other form of energy like sound energy, heat energy and light energy.
• Forces involved during the interaction are non – conservative nature.
##### Collision in one dimension

a ) Inelastic collision in one dimension

1 ) Expression for final velocity Let

m1 = mass of first colliding object

m2 = mass of second colliding object

u1 = initial velocity of first colliding object

u2 = initial velocity of second colliding object

v = final velocity of the combined mass

applying principle of conservation of linear momentum

m1u1 + m2u2 = ( m1 + m2 )v

v = ( m1u1 + m2u2 ) / ( m1 + m2 )

2 ) Expression for loss in kinetic energy

Δk = m1u21 / 2 + m2u22 / 2 – ( m1 + m2 )v2 / 2

Δk = [ m21u21 + m1m2u22 + m1m2u22 + m22u22 – m21u21 – m22u22 -2m1mu1u2 ] / 2 ( m1 + m2 )

Δk = [ m1m2u21 + m1m2u22 – 2m1m2u1u2 ] / 2( m1 + m2 )

Δk = [ m1m2 ( u12 + u22 – 2u1u2 ) ] / 2 ( m1 + m2 )

Δk = [ m1m2 ( u1 – u2 )2 ] /  2 ( m1 + m2 )

# Special cases :

1 ) when u2 = 0 i.e the second colliding body is at rest

v = ( m1u1 + m2u2 ) / ( m1 + m2 )

v = m1u1( m1 + m2

and

Δk = [ m1m2 ( u1 – u2 )2 ] /  2 ( m1 + m2 )

Δk = m1m2u12 /  2 ( m1 + m2 )

2 ) when m1 = m2 = m ( say ) i.e both the colliding bodies have equal masses

v = ( m1u1 + m2u2 ) / ( m1 + m2 )

v = m ( u1 + u2 ) / 2m

v = ( u1 + u2 ) / 2

and

Δk = [ m1m2 ( u1 – u2 )2 ] /  2 ( m1 + m2 )

Δk =  m2 ( u1 – u2 )2  /  4m

Δk =   m( u1 – u2 )2  / 4

3 ) when m1 = m2 = m ( say ) and  u2 = 0

v = ( m1u1 + m2u2 ) / ( m1 + m2 )

v = mu1 / 2m

v = u1 / 2

Δk = [ m1m2 ( u1 – u2 )2 ] /  2 ( m1 + m2 )

Δk = m2u12 / 4m

Δk = mu2 / 4

b ) Elastic collision in one dimension

1 ) Coefficient of restitution  ( e ) Elastic Collision in one dimension

Let

m1 = mass of first colliding object

m2 = mass of second colliding object

u1 = initial velocity of first colliding object

u2 = initial velocity of second colliding object

v1 = final velocity of the first colliding object

v = final velocity of the second colliding object

applying principle of conservation of linear momentum

m1u1 + m2u2 = m1v1 + m2v2

m1u1 – m1v1 =  m2v2 – m2u2

m1 ( u1 – v1 ) = m2 ( v2 – u2 ) ———————————- ( 1 )

applying principle of conservation of kinetic energy

m1u21 / 2 + m2u22 / 2 = m1v21 / 2 + m2v22 / 2

m1u21 – m1v21 =  m2v22 – m2u22

m1 ( u21 – v21 ) = m2 ( v22 – u22 )

m1 ( u1 – v1 )( u1 + v1 ) = m2 ( v2 – u2 )( v2 + u2 )

by using equation ( 1 ) we get

m2 ( v2 – u2 )( u1 + v1 ) =  m2 ( v2 – u2 )( v2 + u2 )

( u1 + v1 ) = ( v2 + u2 ) ——————————— ( 2 )

( v2 – v1 ) = ( u1 – u2 )

( v2 – v1 ) / ( u1 – u2 ) = 1

e = 1

where e = ( v2 – v1 ) / ( u1 – u2 ) = coefficient of restitution

2 ) Expression for final velocity of first colliding body

from equation ( 2 ) we get

( u1 + v1 ) = ( v2 + u2 )

v2 = u1 + v1 – u2

putting this value in equation ( 1 ) we get

m1 ( u1 – v1 ) = m2 ( v2 – u2 )

m1 ( u1 – v1 ) = m2 ( u1 + v1 – u2  – u2 )

( m1 u1 – m1v1 )= ( m2u1 + m2v1 – 2m2u2 )

( m2v1 + m1v1 ) =  m1 u1 – m2u1 +  2m2u2

v1( m1 + m2 ) = u1( m1 – m2 ) +  2m2u2

v=  u1( m1 – m2 ) / ( m1 + m2 ) +  2m2u2 / ( m1 + m2 )

3 ) Expression for final velocity of second colliding body

from equation ( 2 ) we get

( u1 + v1 ) = ( v2 + u2 )

v1 = ( v2 + u2 – u1 ) ——————————— ( 3 )

putting this value in equation ( 1 ) we get

m1 ( u1 – v1 ) = m2 ( v2 – u2 )

putting this value in equation ( 1 ) we get

m1 ( u1 – v2 – u2 + u1 ) = m2 ( v2 – u2 )

( m1u1 – m1v2 – m1u2 + m1u1 ) = ( m2v2 – m2u2 )

( m2v2 + m1v2 ) = ( m1u1– m1u2 + m1u1 +  m2u2 )

v2( m2 + m1) = ( 2m1u– m1u2  +  m2u2 )

v2=  2m1u / ( m2 + m1)  – u2( m1  –  m2 ) / ( m2 + m1

# special cases :

1 ) when u2 = 0 i.e the second colliding body is at rest

v=  u1( m1 – m2 ) / ( m1 + m2 ) +  2m2u2 / ( m1 + m2 )

v=   u1( m1 – m2 ) / ( m1 + m2 )

and

v=  2m1u / ( m2 + m1)  – u2( m1  –  m2 ) / ( m2 + m1)

v=  2m1u / ( m2 + m1)

2 ) when m1 = m2 = m ( say ) i.e both the colliding bodies have equal masses

v=  u1( m1 – m2 ) / ( m1 + m2 ) +  2m2u2 / ( m1 + m2 )

v= 0 + 2mu2 / ( 2m )

v= u2

and

v=  2m1u / ( m2 + m1)  – u2( m1  –  m2 ) / ( m2 + m1)

v=  2mu / ( m + m)

v= u1

3 ) when m1 = m2 = m ( say ) and  u2 = 0

v=  u1( m1 – m2 ) / ( m1 + m2 ) +  2m2u2 / ( m1 + m2 )

v= 0 + 0

v= 0

and

v=  2m1u / ( m2 + m1)  – u2( m1  –  m2 ) / ( m2 + m1)

v=   2mu / ( m + m)

v= u1

### Practice questions

1 ) Define work.

Ans : When a force is applied on an object to displace it from its initial position then it is said to be work is done. It is the Dot Product of Force and Displacement.

2 ) what do you mean by positive work ?

Ans :

Positive Work : When the applied force and displacement of the objects take place in the same direction or at acute angle , then it is said to be positive work is done. That is the applied force and displacement are parallel to each other and angle between them is zero degree and the value of cos0 is 1 i.e

W = FS

OR

W = FS cosθ

Examples :

1. Kicking of a football
2. Moving a chair
3. pushing and moving a table
4. A car moving forward
5. throwing a stone

3 ) what do you mean by negative work ?

Ans :

Negative Work :  When the applied force and displacement of the objects take place in the opposite direction, then it is said to be negative work is done. That is the applied force and displacement are antiparallel to each other and angle between them is 180 degree and the value of cos180 is ( – 1 ) i.e

W = -FS

Examples :

1. Two boys push each other
2. In Tug of War when opponent team pulls hard.
3. Trying to catch a heavy object, hands move down

4 ) what is meant by zero work ?

Ans :

Zero Work :  When the applied force and displacement of the objects take place perpendicularly, then it is said to be zero work is done. That is the applied force and displacement are normal to each other and angle between them is 90 degree and the value of cos90 is 0 . when we apply a force on the object but it does not displace from its position, then there is zero work done .

W = 0

Examples :

1. Pushing a wall
2. Walking on the road
3. Motion of planet on its orbit

5 ) Which physical quantity has [ ML2T-2 ] as dimension ?

Ans : Work

6 ) What is the relation between work and displacement ?

Ans : Work ∝ displacement ( when force remains constant )

i.e greater is the displacement greater is the amount of work and vice versa.

7 ) What is the relation between work and velocity ?

Ans : Let

w = work

f = force

v = change in velocity

a = acceleration

m = mass of the object

d = displacement

w = fd = mad = mvd / t = mv ( d / t ) = mv2

w = mv2

∝ v2

8 ) Write down the dimension of power.

Ans :  [ ML2T-3 ]

9 ) Derive the expression for kinetic energy of a body of mass ‘ m ‘  and velocity ‘ v ‘ .

Ans :

Let

u = initial velocity = 0    ( ∵ the object is initially at rest )

v = final velocity

a = acceleration

m = mass of the object

t = time

F = force applied on the object to displace it from its position

s = displacement

From Newton’s 2nd law of motion we have

F = ma —————————————— ( 1 )

From equation of motion we have

v2 = u2 + 2as

⇒ v2 = 2as

⇒ s = v2 / 2a

Now, workdone = force * displacement = ma * v2 / 2a = mv2 / 2

10 ) Calculate the potential energy in a string having a body of mass at one end ( this massive body is fixed at this end ) .

Ans : Let us assume that a very light and perfectly elastic spring is fixed at one end from a rigid support. Other end of the support is attached with a body of mass ‘m’. ( In the above figure, red circular part is assumed as the body. Let the body is displaced from it’s equilibrium position. Due to elastic property of spring, the restoring force come into play and the whole system is seemed to be in to and fro motion.

Let

displacement be = x

restoring force = F

According to Hook’s Law

∝ – x —————-( 1 )

F = – kx

where k = spring constant or force constant

Work done against the  restoring force is given by

w = ∫ Fdx

where dx = small displacement

w = ∫ ( – kx ) dx

w = – k ∫ xdx

w = – k x2 / 2 + c

where ‘c’ is integration constant

when x = 0 then w = 0

0 = – k ( 0 )2 / 2 + c

c = 0

w = – kx2 / 2

this work done is stored in the form of elastic potential energy given by

U = kx2 / 2

11 ) Which principle of conservation is common in all types of collision ?

Ans : Principle of conservation of linear momentum

12 ) A body is at rest. Another body hits the stationary body with the velocity u. If both the bodies have equal masses then what will be the velocity of the stationary body after elastic collision ?

Ans : Elastic Collision in one dimension

Let

m1 = mass of first colliding object = m

m2 = mass of second colliding object = m

u1 = initial velocity of first colliding object = u

u2 = initial velocity of second colliding object = 0

v1 = final velocity of the first colliding object = v

v = final velocity of the second colliding object

applying principle of conservation of linear momentum

m1u1 + m2u2 = m1v1 + m2v2

m1u1 – m1v1 =  m2v2 – m2u2

m1 ( u1 – v1 ) = m2 ( v2 – u2 ) ———————————- ( 1 )

putting the value of u1 , v1, m1, m2 and u2

m ( u – v ) = m ( v2 – 0 )

m ( u – v ) = v2  ————————————- ( 2 )

applying principle of conservation of kinetic energy

m1u21 / 2 + m2u22 / 2 = m1v21 / 2 + m2v22 / 2

m1u21 – m1v21 =  m2v22 – m2u22

m1 ( u21 – v21 ) = m2 ( v22 – u22 )

m1 ( u1 – v1 )( u1 + v1 ) = m2 ( v2 – u2 )( v2 + u2 )

by using equation ( 1 ) we get

m2 ( v2 – u2 )( u1 + v1 ) =  m2 ( v2 – u2 )( v2 + u2 )

( u1 + v1 ) = ( v2 + u2 ) ——————————— ( 3 )

putting the value of u1 , v1 and u2

u + v = v2 + 0

u + v = v2 ——————————- ( 4 )

On adding equation ( 2 ) and equation ( 4 ) we get

2 u = 2 v2

u =  v2

hence the stationary body will move with the initial velocity of moving body i.e u after elastic collision.

13 ) What is the difference between elastic and inelastic collision ?

Ans : The difference between elastic and inelastic collision are here under.

##### Inelastic collision
1 ) Original shape of the colliding

bodies regain after elastic collision.

1 ) Original shape of the collidingbodies do not  regain after inelastic collision.
2 ) There is no loss of kinetic energy

that is the kinetic energy is conserved.

2 ) There is loss of kinetic energy that is the

kinetic energy is conserved.

3 ) The value of coefficient of restitution

is unity.

3 ) The value of coefficient of restitution

is not unity.

4 ) Total energy of the colliding system

is constant.

4 ) Total energy of the colliding system

is not constant.

5 ) Mechanical energy of the system is not

converted into other form of energy like

light energy, heat energy and sound energy.

5 ) Mechanical energy of the system is not

converted into other form of energy like

light energy, heat energy and sound energy.

14 ) Explain conservative force and write down the it various properties ?

Ans : Conservative force : When the work done by a force in moving a body from one point to another is independent of the followed between the two points then the force is said to be conservative force.

Example : Gravitational force, elastic force and electrostatic force etc.

Properties of Conservative force :

• The work done is independent of path followed between the initial and final position of a body.
• The work done by this force depends only on the initial and final position of the body.
• The work done by this force in moving a particle along a closed path is zero.
• The work done by this force is always reversible.
• Mechanical energy of a body is always conserved under the action of conservative force.

15 ) What is non – conservative force ?

Ans : When the work done by a force in moving a body from one point to another is dependent of the followed between the two points then the force is said to be conservative force. It is also termed as dissipative force.

Example : Frictional force, force due to air resistance.

16 ) What is the difference between conservative and non conservative forces ?

##### Non – conservative force
1 ) The work done is independent

of path followed between the initial

and final position of a body.

1 ) The work done is dependent

of path followed between the initial

and final position of a body.

2 ) The work done by this force

depends only on the initial and

final position of the body.

2 ) The work done by this force does not

depend only on the initial and

final position of the body.

3 ) The work done by this force in

moving a particle along a closed

path is zero.

3 ) The work done by this force in

moving a particle along a closed

path is non zero.

4 ) The work done by this force is

always reversible.

4 ) The work done by this force is

always irreversible.

5 ) Mechanical energy of a body is

always conserved under the action

of conservative force.

5 ) Mechanical energy of a body is

always conserved under the action

of conservative force.

17 ) What is power ? What is SI unit of power ? Prove that the power is scalar product of force and velocity.

Ans :

# Power : The rate of doing work is called as Power.

Power = Work / time

P = dw / dt

#SI – Unit  of power is Watt.

# Let

F = force

s = displacement

w = work

v = velocity = s / t

t = time

P = power

Now Work = force . Displacement

w = F . s

here, the bold letters represent the vector notation.

P = w / t

P = F . / t

P =  F . v

Hence it is proved that power is the scalar product of force and velocity.

18 ) Derive the expression for gravitational potential energy of a body lying at a height ‘ h ‘ above the surface of the earth.

Ans : Let an object of mass ‘ m ‘ is raised to a height of ‘ h ‘. The work done in raising the object is stored in the form of potential energy.

∴ Potential energy = Force by which the object is raised * height = Weight * height = mgh

where g = acceleration due to gravity.

##### P.E = mgh

19 ) Find the final velocities of the two bodies of different masses after the elastic collision in one dimension.

Ans : Elastic Collision in one dimension

Let

m1 = mass of first colliding object

m2 = mass of second colliding object

u1 = initial velocity of first colliding object

u2 = initial velocity of second colliding object

v1 = final velocity of the first colliding object

v = final velocity of the second colliding object

applying principle of conservation of linear momentum

m1u1 + m2u2 = m1v1 + m2v2

m1u1 – m1v1 =  m2v2 – m2u2

m1 ( u1 – v1 ) = m2 ( v2 – u2 ) ———————————- ( 1 )

applying principle of conservation of kinetic energy

m1u21 / 2 + m2u22 / 2 = m1v21 / 2 + m2v22 / 2

m1u21 – m1v21 =  m2v22 – m2u22

m1 ( u21 – v21 ) = m2 ( v22 – u22 )

m1 ( u1 – v1 )( u1 + v1 ) = m2 ( v2 – u2 )( v2 + u2 )

by using equation ( 1 ) we get

m2 ( v2 – u2 )( u1 + v1 ) =  m2 ( v2 – u2 )( v2 + u2 )

( u1 + v1 ) = ( v2 + u2 ) ——————————— ( 2 )

v2 = u1 + v1 – u2

putting this value in equation ( 1 ) we get

m1 ( u1 – v1 ) = m2 ( v2 – u2 )

m1 ( u1 – v1 ) = m2 ( u1 + v1 – u2  – u2 )

( m1 u1 – m1v1 )= ( m2u1 + m2v1 – 2m2u2 )

( m2v1 + m1v1 ) =  m1 u1 – m2u1 +  2m2u2

v1( m1 + m2 ) = u1( m1 – m2 ) +  2m2u2

v=  u1( m1 – m2 ) / ( m1 + m2 ) +  2m2u2 / ( m1 + m2 )

from equation ( 2 ) we get

( u1 + v1 ) = ( v2 + u2 )

v1 = ( v2 + u2 – u1 ) ——————————— ( 3 )

putting this value in equation ( 1 ) we get

m1 ( u1 – v1 ) = m2 ( v2 – u2 )

putting this value in equation ( 1 ) we get

m1 ( u1 – v2 – u2 + u1 ) = m2 ( v2 – u2 )

( m1u1 – m1v2 – m1u2 + m1u1 ) = ( m2v2 – m2u2 )

( m2v2 + m1v2 ) = ( m1u1– m1u2 + m1u1 +  m2u2 )

v2( m2 + m1) = ( 2m1u– m1u2  +  m2u2 )

v2=  2m1u / ( m2 + m1)  – u2( m1  –  m2 ) / ( m2 + m1

20 ) What is the difference between kinetic energy and potential energy ?

21 ) Show that the mechanical energy of a body is conserved, when it falls freely under the action of gravitational field.

22 ) What is coefficient of restitution ? Discuss its value in different types of collision.

23 ) Discuss the elastic collision of two bodies of equal masses in two dimension.

24 ) Discuss the inelastic collision of two bodies of equal masses in two dimension.

25 ) when a body is at rest and other moving body hits it with w some initial velocity. The collision is perfectly inelastic collision in two dimension. Find the final velocity of the stationary body after the collision.