## Question 5: Define elastic and inelastic collisions. Give examples in each case. Derive mathematical equations for calculating final velocities of the elastically colliding bodies in one dimension.

There are two types of collisions.

### (1) Elastic Collisions

Collisions in which kinetic energy as well as linear momentum of the system remains conserved are called elastic collisions. Thus in elastic collisions,

Momentum before collision = Momentum after collision

K.E before collision = K.E after collision

Example: Take the example of the collisions atoms and molecules of a gas in a container. These collisions are perfectly elastic and the K.E and momentum remain conserved. Similarly, the collisions of atoms and subatomic particles are also example of elastic collisions.

### (2) Inelastic collisions

Collisions in which the momentum of the system is conserved but the K.E is not conserved are called inelastic collisions. Thus in inelastic collisions,

Momentum before collision = Momentum after collision

K.E before collision ≠ K.E after collision

Example: Inelastic collisions occur frequently in our daily life. Take the example of two colliding cars. Both cars have K.E before the collision and usually no K.E after the collision. The K.E is converted to sound and work done to permanently distort the bodies of the car.

### Equations for the final velocities of the bodies in head-on collision

Let two balls of masses m1 and

If before and after the collision, the motion of the colliding bodies appears on a single line, the collision is said to be one-dimensional or head-on.

m2, respectively, collide head-on as shown in the figure. Let’s further assume m1 is approaching towards m2. Let v1 and v2 are their velocities before collision and the final velocities after collision are v1and v2 respectively. The law of conservation of momentum says that

Similarly, according to the law of conservation of energy,

So, in case of elastic collision, the relative speed of approach (v1-v2) before collision is equal to the relative speed of separation (v2 – v1) after collision.

Final velocities of the bodies can now be calculated from the above equations.

From equation (IV)

So particle ‘m1’ will go with this velocity after collision.

To find the velocity of ‘m2’ after collision, we consider equation (IV) once more.

Put this value in equation (i)

This equation represents the velocity of body m2 after collision.