First slide

Coefficient of restitution

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 enough compared with the width of step so that the impacts are effectively head on. Find the relationship between h and d

Moderate

A

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

$\large h = \frac{d}{{1+ {e^2}}}\$
C

$\large h = \frac{d}{{1+ {e}}}\$
D

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

Solution

Let u = velocity of ball just before impact. v = velocity of ball just after impact.
$\large \therefore u^2=2gh\;and\;v^2=2g(h-d)$
$\large Now\;e=v/u\Rightarrow e^2=\frac {h-d}{h}\Rightarrow h=\frac {d}{(1-e^2)}$

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