First slide

Coefficient of restitution

Question

A ball is dropped from a height I on the ground. If the coefficient of restitution is e, the height to which the ball goes up after it rebounds for the nth time is

Moderate

A

${he}^{2 n}$
B

${he}^{n}$
C

$\frac{e^{2 n}}{h}$
D

$\frac{h}{e^{2 n}}$

Solution

Let v _o be the velocity with which the ball strikes the earth first time and v _n be the velocity after n^threbound. According to the definition of coefficient of restitution, we have
$e = \frac{V_{1}}{V_{0}} = \frac{V_{2}}{V_{1}} = \frac{V_{3}}{V_{2}} = \dots = \frac{V_{n}}{V_{n - 1}} F u r t h e r e^{n} = \frac{v_{1}}{v_{0}} \times \frac{v_{2}}{v_{1}} \times \frac{v_{3}}{v_{2}} \times \dots \times \frac{v_{n}}{v_{n - 1}} = \frac{v_{n}}{v_{0}} H e r e v_{0} = \sqrt{2 g h} and v_{n} = \sqrt{(2 g H)} ∴ e^{n} = \frac{\sqrt{2} g H}{\sqrt{2 g h}} = \frac{\sqrt{H}}{\sqrt{h}} or e^{2 n} = \frac{H}{h} or H = h e^{2 n}$

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