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 collisions is $v_n$ 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 $e^n=\sqrt{\frac{h_n}{h_0}}$

velocity with which the ball rebounds after $n$ collision is given by $v_n=e^n{v}_{initial}$

$\therefore$ $v_n=e^n{v}_o$ ..............(1)

Given : Initial height from which the ball is dropped $=h_o$

Height upto which the ball rebounds after n collision $=h_n$

$⟹v_o=\sqrt{2g{h}_o}$ and $v_n=\sqrt{2g{h}_n}$

Using (1), $\sqrt{2g{h}_n}=e^n\times \sqrt{2g{h}_o}$ $⟹e^n=\sqrt{\frac{h_n}{h_o}}$