Question

A ball is bouncing down a set of stairs. The coefficient of restitution is $e$. The height of each step is $d$ and the ball bounces one step at each bounce. After each bounce the ball rebounds to a height $h$ above the next lower step. Neglect height of each step in comparison to $h$ and assume the impacts to be effectively head on. Which of the following relation is correct? (given that $h>d$)

• A. $\frac{h}{d}=1-e^2$
• B. $\frac{h}{d}=1-e$
• C. $\frac{h}{d}=\frac{1}{1-e^2}$
• D. $\frac{h}{d}=\frac{1}{1-e}$

Answer 1

The correct option is C $\frac{h}{d}=\frac{1}{1-e^2}$

Consider the given condition as shown below.

As we can see in the figure ball jumps from a stair $\left(1\right)$ and achieve a height of $(h-d)$.
Hence, the speed of the ball just after hitting the surface is
$v=\sqrt{2g(h-d)}$ where, $v=\text{Magnitude of velocity just after the collision}$
After that ball hit the another stair with the speed $v_1$ given by applying WPE from topmost point to just before hitting the stairs,
$\Rightarrow 0+mgh=\frac{1}{2}m{v}_1^2+0$
$\Rightarrow v_1=\sqrt{2gh}$ where, $v_1=\text{Magnitude of velocity just before the collision.}$
As we know
Coefficient of restitution $e=\frac{\text{Velocity of seperation}}{\text{Velocity of approach}}$
$\Rightarrow e=\frac{v_1-0}{0-v^{\prime }}=\frac{\sqrt{2g(h-d)}}{0-(-\sqrt{2gh})}$
$\Rightarrow e^2=\frac{(h-d)}{h}$
$\Rightarrow e^2=1-\frac{d}{h}$
$\Rightarrow \frac{d}{h}=1-e^2$
$\Rightarrow \frac{h}{d}=\frac{1}{1-e^2}$