Question

A ball drops from the ceiling of the room, and after rebounding twice from the floor, reaches a height equal to half that of ceiling. Find the coefficient of restitution?

• A. $\sqrt{\frac{1}{2}}$
• B. $\frac{1}{2}$
• C. $4\sqrt{\frac{1}{2}}$
• D. $3\sqrt{\frac{1}{2}}$

Answer 1

The correct option is C

$4\sqrt{\frac{1}{2}}$

Let $h$ = height of ceiling.

$\Rightarrow$ speed before first impact = $\sqrt{2gh}$ $\Rightarrow$ speed after first impact = $e\sqrt{2gh}$

The ball comes back for second impact, now its speed just before second impact = speed after first impact.

$\Rightarrow$ before second impact, speed = $e\sqrt{2gh}$ $\Rightarrow$ after second impact, speed = $e^2\sqrt{2gh}$

This speed is the initial speed $u$ for upward motion after second impact.

High attained = $\frac{u^2}{2g}$ $\Rightarrow \frac{h}{2}$ = $\frac{[e^2\sqrt{2gh}{]}^2}{2g}\Rightarrow \frac{h}{2}=\frac{e^2\left(2gh\right)}{2g}\Rightarrow e=\frac{1}{\sqrt{2}}$