A ball is bouncing down a flight of stairs. The coefficient of...

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

Options

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

Correct answer

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

Step-by-step solution

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

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

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