A ball of mass M and a pan of mass m (m lt M) are attached to the ends of a light inextensible string, passing over a smooth light pulley. The ball rests on a horizontal surface, with the pan hanging. A small particle of mass m collides elastically with the pan, falling from above, with a velocity v. Determine the instantaneous velocity of the ball, as it leaves contact with the surface. Find also the velocity with which the particle rebounds.

A ball of mass M and a pan of mass m (m lt M) are attached to the ends

1.	A ball of mass M and a pan of mass m (m lt M) are attached to the ends of a light inextensible string, passing over a smooth light pulley. The ball rests on a horizontal surface, with the pan hanging. A small particle of mass m collides elastically with the pan, falling from above, with a velocity v. Determine the instantaneous velocity of the ball, as it leaves contact with the surface. Find also the velocity with which the particle rebounds.
Answer» Solution :Let `v _(1)` be the VELOCITY ofthe pan (DOWNWARDS) as well as the ball (upwards)and `v _(2)` be the velocity (UPWARD) of the particle just after the collision. If J be the impulse transmitted through the string, then, CONSERVING the momentum during the IMPACT, `mv = mv _(1) - mv _(2) + J ...(1) and J = Mv _(1) ...(2) mv (M +m) v _(1) - mv _(2) ...(3)` Since the collision is elastic, e =1 i.e., velocity of separation = velocity of approach. `implies v _(1) + v _(2) =v ....(4)` Solving (3) and (4), `v _(1) = (2 mv )/( M + 2m) and v _(2) = (Mv)/( M + 2m)`