Two balls of masses `m` and `2m` and momenta `4p` and `2p` (in the directions shown) collide as shown in figure. During collsion, the value of linear impulse between them is J. In terms of J and p find coefficient of restitution 'e'. Under what condition collision is elastic. Also find the condition of perfectly inelastic collision.

Answer

A, B, C

Explanation

Directions of linear impulses on the two colliding bodies at the time of collision are shown in fig.

Linear impulse is equal to the change in linear momentum. Hence, momenta of the balls after collision are shown in figure.
Let `v_1` and `v_(2)` are their velocities before collision [in the directions shown in figure(a)] and `v_1^'` and `v_2` are the velocities after collision as shown in figure (c).
v=("linear momentum"/("mass")
e=("relative speed of separation")/("relative speed of approach")`
or `e=(v_1'+v_2')/(v_1+v_2)=((J-4p)/(m)+(J-2p)/(2m))/((4p)/(m)+(2p)/(2m))`
or `e=(3J-10p)/(10p)=(3J)/(10p)-1`
For elastic collision,
`e=1rArr(3J)/(10p)=2` or `J=(20)/(3p)`
For perfectly inelastic collision,
`e=0rArr(3J)/(10p)=1` or `J=10/3p`