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Partially Inelastic Collision

A small ball ...

Question

A small ball moves toward right with a velocity $v$ . It collides with the wall and returns back and continues to and fro motion. If the average speed for first to and fro motion of the ball is $(2 / 3) v$ , find the coefficient of restitution of impact.

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Solution

The ball moves from $A$ to $B$ with a speed $v_{1} = v$ and returns from $B$ to $A$ with a speed $v_{2}$ say
Since, ${\to v}_{2} = - e {\to v}_{1}$
$\Rightarrow v_{2} (^- i) = - e v_{1} (^i) \Rightarrow v_{2} = e v_{1} = e v (∵ v_{1} = v)$
Therefore
$\to v = \frac{A B + B A}{\frac{A B}{v_{1}} + \frac{B A}{v_{2}}} = \frac{2 v_{1} v_{2}}{v_{1} + v_{2}}$
Substituting respective values we get
$v_{1} = v; v_{2} = e v$
we obtain
$\to v = \frac{2 v (e v)}{v + e v} = \frac{2 e v}{1 + e} \Rightarrow \frac{\to v}{v} = \frac{2 e}{1 + e} . . . (1)$
Since it is given that $\to v = (2 / 3) v$
$\Rightarrow \frac{\to v}{v} = \frac{2}{3} . . . . . (2)$
Equating (1) and (2) we obtain
$\Rightarrow e = 0.5$

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