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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 $d$.
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The correct answer is:
$h=\frac{d}{1-e^2}$
The ball falls a distance $h$ from its highest (rest) position and rebounds a distance $(h-d)$.
Thus the coefficient of restitution $e=\sqrt{\frac{h-d}{h}}$ $e^2=\frac{h-d}{h} \quad$ or $\quad h=\frac{d}{1-e^2}$
Thus the coefficient of restitution $e=\sqrt{\frac{h-d}{h}}$ $e^2=\frac{h-d}{h} \quad$ or $\quad h=\frac{d}{1-e^2}$
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