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Final Velocities

A moving body...

Question

A moving body with a mass $m_{1}$ and velocity u strikes a stationary body of mass $m_{2}$ . The masses $m_{1}$ and $m_{2}$ should be in the ratio $m_{1} / m_{2}$ so as to decrease the velocity of the first body to $2 u / 3$ and giving a velocity of $v$ to $m_{2}$ assuming a perfectly elastic impact. Then the ratio $m_{1} / m_{2}$ is :

A

5

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B

1 / 5

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C

1 / 25

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D

25

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Solution

The correct option is A $5$
$m_{1} u = m_{1} \frac{2 u}{3} + m_{2} v$ (By conservation of linear momentum)
$\Rightarrow \frac{1}{3} m_{1} u = m_{2} v$ .................(i)
Also $e = \frac{| v_{1} - v_{2} |}{| u_{2} - u_{1} |}$
$\Rightarrow v - \frac{2 u}{3} = u \Rightarrow v = \frac{5}{3} u$ ............(ii)
From (i) and (ii), $\frac{1}{3} m_{1} u = \frac{5}{3} m_{2} u \Rightarrow \frac{m_{1}}{m_{2}} = 5$

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