Question

A ball is bouncing down a flight of stairs. The coefficient of restitution is $e$. The height of each step is $d$ and the ball descends one step each bounce. After each bounce it rebounds to a height $h$ above the next lower step. The height is large compared with the width of step so that the impacts are effectively head-on. Find the relationship between $h$ and $d$.

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

Answer 1

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

Since $v^2-u^2=2as$

When falling down, $2ah=v_1^2$

When bouncing back up, $2a(h-d)=v_2^2$

Thus the coefficient of restitution $e=\frac{v_2}{v_1}=\sqrt{\frac{h-d}{h}}$

$⟹h=\frac{d}{1-e^2}$