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Bouncing Ball Example

A moving part...

Question

A moving particle of mass $m_{1}$ has a perfectly elastic direct collision with a stationary body of mass $m_{2}$ . If the two bodies now move in opposite directions with same speed, then what will be the ratio of their masses?

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Solution

As shown in figure the $u_{1}$ and $u_{2}$ is related as
$\Rightarrow u_{1} = \frac{(m_{2} - m_{1})}{m_{1} + m_{2}} v$

$\Rightarrow u_{2} = \frac{2 m_{1}}{m_{1} + m_{2}} v$ $∴ u_{1} = u_{2}$

$\Rightarrow \frac{u_{1}}{u_{2}} = \frac{m_{2} - m_{1}}{{2 m}_{1}} = 1$

$\Rightarrow m_{2} - m_{1} = 2 m_{1}$

$\Rightarrow \frac{m_{1}}{m_{2}} = \frac{1}{3} .$
Hence, the answer is $m_{1} = 1, m_{2} = 3.$

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