A sphere of mass $m$ moving with constant velocity hits another sphere of same mass at rest. If $e$ is the coefficient of restitution. The ratio of their velocities after collision is :

1 + e

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\frac{1 + e}{2}

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\frac{1 + 2 e}{1 - 2 e}

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

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Solution

The correct option is C $\frac{1 - e}{1 + e}$
Using momentum conservation,
$m u = m v_{1} + m v_{2}$ .....................(1)
Using coefficient of restitution equation,

$e = \frac{v_{2} - v_{1}}{u}$

$\Rightarrow v_{2} - v_{1} = u e$ ...................(2)
Solving equations (1) and (2),

$v_{2} = \frac{1 + e}{2} u$ and $v_{1} = \frac{1 - e}{2} u$ .
Therefore ratio of this given as,
$\frac{v_{1}}{v_{2}} = \frac{1 - e}{1 + e}$

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