Question

A ball drops from a ceiling of a room, and after rebounding twice from the floor reaches a height equal to half that of the ceiling. Show that coefficient of restitution is $\sqrt[4]{4\frac{1}{2}}$

Answer 1

Let $h=$ height of ceiling
$⟹$ Speed before the first impact $=\sqrt{2gh}$
$⟹$ Speed after the first impacts $=e\sqrt{2gh}$
The ball comes back for the second impact.
Before the second impact, speed $=e\sqrt{2gh}$
After the second impact, speed $=e^2\sqrt{2gh}$
Height attained $=\frac{u^2}{2g}⟹\frac{h}{2}=\frac{[e^2\sqrt{2gh}{]}^2}{2g}⟹\frac{h}{2}=\frac{e^4\left(2gh\right)}{2g}$
$⟹e=\frac{1}{\sqrt[4]{42}}$