Question

Ball 1, with a mass of 140 g and traveling at 13 m/s , collides head on with ball 2, which has a mass of 320 g and is initially at rest.

What is the final velocity of the ball 1 if the collision is perfectly elastic?

Express your answer in meters per second.

What is the final velocity of the ball 2 if the collision is perfectly elastic?

What is the final velocity of the ball 1 if the collision is perfectly inelastic?

What is the final velocity of the ball 2 if the collision is perfectly inelastic?

Answer 1

Given, the mass of ball 1 () = 140 g = 0.14 kg

initial speed of the ball 1 () = 13 m/s

the mass of ball 2 () = 320 g = 0.32 kg

initial speed of the ball 2 () = 0 m/s

The coefficient of restitution (e) is defined as

So,

For perfectly elastic collision :

In this case the momentum and energy both remains conserved , before and after collision.

Let the final velocity of ball 1 is and final velocity of ball 2 is .

From conservation of momentum, we can write

or, ............................(1)

We also know that in the case of perfectly elastic collision

or,

or,  

or,  

If

Then from equation (1),

or,

or,

or,

So,

If

  

or,

or,

or,

So,   

It means nothing has changed after collision. So, this case is not possible.

Hence,

Final velocity of ball 1 = -5.087 m/s (i.e. in backward direction) and

final velocity of ball 2 = 7.91 m/s .

  

For perfectly inelastic collision :

In this case only the momentum remains conserved, before and after collision.

And after collision both body moves with same velocity (or they stick together) because e = 0.

Let the final velocity of both balls is .

Then from conservation of momentum, we can write

   

or,

or,

or,

Hence, the final velocity of both the ball is 3.956 m/s .

For any doubt please comment .