Question

A ball is let fall from a height $h_0$. There are n collisions with the earth. If the velocity of rebound after n collision is $u_n$ and the ball rises to a height $h_n$, then coefficient of restitution e is given by-

• A. $e^n=\sqrt{\frac{h_n}{h_0}}$
• B. $e^n=\sqrt{\frac{h_0}{h_n}}$
• C. $ne=\sqrt{\frac{h_n}{h_0}}$
• D. $\sqrt{ne}=\sqrt{\frac{h_n}{h_0}}$

Answer 1

The correct option is A $e^n=\sqrt{\frac{h_n}{h_0}}$

The velocity of the earth before and after collision may be assumed zero.
Hence $e^n=\frac{v_1}{v_0}\times \frac{v_2}{v_1}\times \frac{v_3}{v_2}\times .....\frac{v_n}{v_{n-1}}$
$v_n=$ velocity after $n^{th}$ rebound
$v_0=$ velocity with which the ball strikes the earth first time
Hence $e^n=\frac{v_n}{v_0}=\frac{\sqrt{2g{h}_n}}{\sqrt{2g{h}_0}}$ $\therefore e^n=\sqrt{\frac{h_n}{h_0}}$