# Elastic Collision

## Introduction:

A collision occurs when two objects come in direct contact with each other. It is the event in which two or more bodies exert forces on each other in about a relatively short time. There are two types of collisions namely :

• ### Elastic Collision

An elastic collision is one where there is no net loss in kinetic energy in the system as the result of the collision.

• ### Inelastic Collision

An inelastic collision is a type of collision where this is a loss of kinetic energy. The lost kinetic energy is transformed into thermal energy, sound energy and material deformation.

## What is an Elastic Collision?

When two bodies collide but there is no loss in the overall kinetic energy, it is called a perfectly elastic collision.  It can be defined as:

An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies remains the same.

Basically, in the case of collision, the kinetic energy before the collision and after the collision remains the same and is not converted to any other form of energy.

It can be either one-dimensional or two-dimensional. In the real world, perfectly elastic collision is not possible because there is bound to be some conversion of energy, however small.

However, though there is no change in the linear momentum of the whole system, there is a change in the individual momenta of the involved components, which are equal and opposite in magnitude and cancel each other out and the initial energy is conserved.

### Examples of Elastic Collision

• When a ball at a billiard table hits another ball, it is an example of elastic collision.
• When you throw a ball on the ground and it bounces back to your hand, there is no net change in the kinetic energy and hence, it is an elastic collision.

## Elastic Collision Formula

The Elastic Collision formula of momentum is given by:

$m_{1}u_{1}\,&space;+\,&space;m_{2}u_{2}\,&space;=\,&space;m_{1}v_{1}\,&space;+\,&space;m_{2}v_{2}$

Where,

• m1 = Mass of 1st body
• m2 = Mass of 2nd body
• u1 =Initial velocity of 1st body
• u2 = Initial velocity of the second body
• v1 = Final velocity of the first body
• v2 = Final velocity of the second body

The Elastic Collision formula of kinetic energy is given by:

$\frac{1}{2}m_{1}u_{1}^{2}\,&space;+\,&space;\frac{1}{2}m_{2}u_{2}^{2}\,&space;=\,&space;\frac{1}{2}m_{1}v_{1}^{2}\,&space;+&space;\frac{1}{2}m_{2}v^{2}_{2}$

### Question

Two billiard balls collide. Ball 1 moves with a velocity of 6 m/s, and ball 2 is at rest. After the collision, ball 1 comes to a complete stop. What is the velocity of ball 2 after the collision? Is this collision elastic or inelastic? The mass of each ball is 0.20 kg.

Solution:
To find the velocity of ball 2, use a momentum table as follows.

 Objects Momentum Before Momentum After Ball 1 0.20 kg × 6 m/s = 1.2 0 Ball 2 0 0.20 kg × v2 Total 1.2 kg × m/s 0.20 kg × v2

1.2 kg × m/s = 0.20 kg × v2

v2 =1.2 / 0.20 = 6 m/s

To determine whether the collision is elastic or inelastic, calculate the total kinetic energy of the system both before and after the collision.

 Objects KE Before (J) KE After (J) Ball 1 0.50 × 0.20 × 62 = 3.6 0 Ball 2 0 0.50 × 0.20 × 62 = 3.6 Total 3.6 3.6

Since the kinetic energy before the collision is equal to the kinetic energy after the collision (kinetic energy is conserved), this is an elastic collision.

## Difference between Elastic and Inelastic Collision

Some key differences between inelastic and elastic collision are given below in tabular format.

### Inelastic Collision

The total kinetic energy is conserved. The total kinetic energy of the bodies at the beginning and the end of the collision is different.
Momentum does not change. Momentum changes.
No conversion of energy takes place. Kinetic energy is changed into other energy such as sound or heat energy.
Highly unlikely in the real world as there is almost always a change in energy. This is the normal form of collision in the real world.
An example of this can be swinging balls or a spacecraft flying near a planet but not getting affected by its gravity in the end. Example of inelastic collision can be the collision of two cars.

## Applications of Elastic Collision

• The collision time affects the amount of force that an object experiences during a collision. The greater the time over which the collision occurs, the smaller the force acting upon the object. Thus, to maximise the force experienced by an object during a collision, the collision time must be decreased.
• Likewise, to minimise the force, the collision time must be increased. There are several real-world application of this phenomena. The airbags in automobiles increase the collapse time and minimize the effect of force on objects during a collision. Airbag accomplishes this by extending the time required to stop the momentum of the passenger and the driver.