A small rubber ball of mass m is placed directly on top of a

A small rubber ball of mass m is placed directly on top of a basketball of mass M >> m and the two are dropped to the floor from a height h. How high does the small rubber ball fly after the collision? Assume all collisions are elastic, and ignore the sizes of the balls compared to h. To help visualize the problem, assume the two balls are slightly separated when the basketball hits the floor.

Solution

Before the collision ,by conservation of energy both balls has same initial velocity

(1/2)mv²=mgh

=>v=V_basketball=V_rubberball=sqrt[2gh]

since the relative velocity has to be equal before and after collision. Before the collision, the basketball bounces up with v and the rubber ball is coming down with v, so their relative velocity is 2v. After the collision, it therefore has to be 2v.After collision the velocity of smaller ball is +3v.

By Conservation of energy

(1/2)mV_rubberball² = mgh_after

(1/2)m*(3v)² =mgh_after

(1/2)*9*2gh =gh_after

h_after=9h