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 width 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 $\frac{h}{d}=\frac{1}{1-e^2}$

For a stair step, ball free falls from $h$ height, so velocity when it touches the step is $\sqrt{2gh}$

So the velocity of rebound is $e\sqrt{2gh}$

The height it reaches is $e^2\times 2gh/2g=e^{2}h$

As $e^{2}h+d=h$, where $d$ is the height of step.

So, $\frac{h}{d}=\frac{1}{1-e^2}$