A sphere A impinges directly on an identical sphere B at rest. If e is the coefficient of restitution then the ratio of the velocities of A and B after impact is

\frac{1 + e}{1 - e}

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\frac{1 - e}{1 + e}

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\frac{e}{1 - e}

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\frac{e}{1 + e}

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Solution

The correct option is B $\frac{1 - e}{1 + e}$
Let $v_{A}$ & $v_{B}$ be the below their after collection in the same direction by momentum conservation
$m v_{A} + m v_{B} = m v$
$= 1 v_{A} + v_{B} = v$
Newton's law of restitution along common normal
$\frac{v_{B} - v_{A}}{o - v_{A}} = - e \frac{r e l a t i v e v e l o c i t y o f s e p r a t i o n}{r e l a t i v e v e l o c i t y o f a p p r o a c h} = e$
$v_{B} - v_{A} = e v$
$- v_{A} + v_{B} = e v$
$2 v_{B} = (1 + e) v$
$= v_{B} = (\frac{1 + e}{2}) v$
$v_{A} = (\frac{1 - e}{2}) v$
$∴ \frac{v_{A}}{v_{B}} = (\frac{1 - e}{1 + e})$

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