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Instantaneous Velocity

A sphere of m...

Question

A sphere of mass $m_{1}$ in motion hits directly another sphere of mass $m_{2}$ at rest and sticks to it, the total kinetic energy after collision is 2/3 of their total K.E before the collision. Find the ratio $m_{1} : m_{2}$ .

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Solution

Velocity after collision is $v_{1}$ .
$m_{1} v = (m_{1} + m_{2}) v_{1} v_{1} = \frac{m_{1}}{(m_{1} + m_{2})} v$
K.E before collision= $\frac{3}{2}$ k.e after collision.
$\frac{1}{2} m_{1} v_{2} = \frac{3}{2} \times \frac{1}{2} (m_{1} + m_{2}) {v_{1}}^{2} m_{1} v_{2} = \frac{3}{2} \times \frac{{m_{1}}^{2}}{(m_{1} + m_{2})} v^{2} 2 (m_{1} + m_{2}) = 3 m_{1} 2 m_{2} = m_{1} \frac{m_{1}}{m_{2}} = 2$

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