Question

A ball is dropped from height h on the ground. If the coefficient of restitution is e, the height to which the ball goes up after it rebounds for the nth time is :

• A. $\frac{h}{e^{2n}}$
• B. $\frac{e^{2n}}{h}$
• C. $h{e}^{2n}$
• D. $h{e}^n$

Answer 1

The correct option is C $h{e}^{2n}$

Let $v_0$ be the velocity with which the ball strikes the Earth first time and $v_n$ after the nth rebound.

The coefficient of restitution is

$e=\frac{v_1}{v_0}=\frac{v_2}{v_1}=\frac{v_3}{v_2}=$.....$=\frac{v_n}{v_{n-1}}$

$\therefore e^n=\frac{v_1}{v_0}\times \frac{v_2}{v_1}\times \frac{v_3}{v_2}\times$.....$\times \frac{v_n}{v_{n-1}}=\frac{v_n}{v_0}$

Here, $v_0=\sqrt{2gh}$ and $v_n=\sqrt{2gH}$

$e^n=\frac{\sqrt{2gH}}{\sqrt{2gh}}=\frac{\sqrt{H}}{\sqrt{h}}$

$e^{2n}=\frac{H}{h}\Rightarrow H=h{e}^{2n}$