**Power**

**Power** is defined as the rate of which the work is done.

Mathematically, Power = Work Done/ time

Thus, power of an agent measures how fast it can do the work.

For constant force,

Power, p = W/t = F.s / t = F. v

Where v = s/t, is linear velocity

If θ be the angle between F and V, then

P = F. v cosθ

**Collisions**

Collison is the mutual interaction between two particles for a short interval of time so that their momentum and kinetic energy may change. In general, collision is an isolated event in which the colliding bodies exert relatively strong forces to one another for relatively short time.

**There are two types of collisions.**

**i. Elastic collision**

Elastic collision is the mutual interaction between two bodies where their momentum and kinetic energy is conserved. It occurs when conservative force is applied to a body.

**Characteristics of an Elastic collision;**

- The momentum is conserved
- Kinetic energy is conserved
- Total energy is conserved
- Forces involved during the interaction is conservative in nature.
- Mechanical energy is not transformed into any other form of energy.

**Elastic collision in one dimension**

If the colliding bodies move in the same path even after collision then it is said to be collision in one direction.

Let us consider two bodies A and B with masses m_{1} and m_{2} moving in a straight line with velocity u_{1} and u_{2} such that u_{1}>u_{2}. After some time, they collide with each other.

Let v_{1} and v_{2 }be the velocities of the bodies A and B respectively after collision such that v_{1}<v_{2}.

From the principle of conservation of momentum,

m_{1}u_{1} + m_{2}u_{2} = m_{1}v_{1} + m_{2}v_{2 }…………. (i)

m_{1}(u_{1}-v_{1}) = m_{2}(v_{2}-u_{2}) ………………. (ii)

In elastic collision,

K.E before collision = K.E. after collision

½ m_{1}u_{1}^{2} + ½ m_{2}u_{2}^{2} = ½ m_{1}v_{1}^{2} + ½ m_{2}v_{2}^{2}

Or, m_{1}(u_{1} – v_{1})(u_{1}+v_{1}) = m_{2}(u_{2} – v_{2})(u_{2}+v_{2}) ……(iii)

Dividing (iii) by (ii), we get

u_{1} -u_{2}=v_{2} -v_{1} ………. (iv)

This shows that in an elastic collision between two particles, the relative velocity of separation after collision is equal to the relative velocity of an approach before the collision.

**ii. Inelastic collision**

The collision in which the momentum is conserved but kinetic energy is not conserved is called Inelastic collision.

**Characteristics of inelastic collision**

- The momentum is conserved
- Total energy is conserved.
- Kinetic energy is not conserved.
- Forces involved during the interaction re non-conservative forces.
- Mechanical energy is transformed into any other form of energy.

**Inelastic collision in one dimension**

Let us consider two perfectly inelastic bodies of mass m_{1} and m_{2}. Body A is moving with velocity u_{1} and B is at rest. After some time, they collide and move together with same velocity v. So, initial momentum before collision = m_{1}u_{1}.

Final momentum before collision = m_{1}u_{1}

Final momentum after collision = (m_{1} + m_{2}) v

Since momentum is conserved, i.e. (m_{1} + m_{2}) v =m_{1}u_{1}

V = m_{1}u_{1}/(m_{1} + m_{2}) ……. (i)

(K.E before collision) / (K.E. after collision) = (½ m_{1}u_{1}^{2}) / ½ (m_{1} + m_{2}) v^{2}

= m_{1}u_{1}^{2} / (m_{1} + m_{2}) [m_{1}u_{1}/ (m_{1} + m_{2})]^{2}

= (m_{1}+m_{2})/m_{1}> 1

Therefore, K.E. before collision = K.E. after collision

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