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

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Solution

Let $v_{A}$ and $v_{B}$ be the velocity after impact.
By momentum conservation
$\Rightarrow m v = m v_{A} + m v_{B}$
$\Rightarrow v = v_{A} + v_{B} ⟶ (1)$
$a) =$ before collision
$b) =$ after collision
$\Rightarrow e = \frac{r e l a t i v e v e l o c i t y o f s e p e r a t i o n}{r e l a t i v e v e l o c i t y o f i m p a c t}$
$\Rightarrow e = \frac{v_{B} - v_{A}}{v}$
$v_{B} - v_{A} = e v \to (I I)$
$v_{A} + v_{B} = v - ----------- - \to (I)$
$\Rightarrow v_{B} = \frac{v (1 + e)}{2}$
$\Rightarrow v_{A} = v - v_{B} = v (1 - \frac{1 + e}{2}) = (\frac{1 - e}{2}) v$
$\Rightarrow$ Ratio $= \frac{v_{A}}{v_{B}} = (\frac{1 - e}{1 + e}) .$
Hence, the answer is $(\frac{1 - e}{1 + e}) .$

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

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