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A moving sphere P collides another sphere Q at rest. If the collision takes place along the line joining their centers of mass such that their total kinetic energy is conserved and the fraction of K.E transferred by the colliding particle is $8 / 9$ , the ratio of mass of P and the mass of Q is

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9 : 8

\sqrt{8} : 3

1 : 2

2 : 3

answer is D.

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Detailed Solution

v_{1} = \frac{(m_{1} - m_{2}) u_{1}}{m_{1} + m_{2}} + \frac{2 m_{2} u_{2}}{(m_{1} + m_{2})}

$(u_{2} = 0)$

$= \frac{m_{1} - m_{2}}{m_{1} + m_{2}} \times u_{1}$

$\frac{{4m}_{1} m_{2}}{{(m_{1} + m_{2})}^{2}} = \frac{8}{9}$

$\frac{{4m}_{1} m_{2}}{{(- m_{1} + m_{2})}^{2}} = \frac{{8k}^{2}}{{9k}^{2}}$

${4m}_{1} m_{2} = 8 k^{2}$

${(m_{1} + m_{2})}^{2} = {9k}^{2}$

$m_{1} + m_{2} = 3 k$

${4m}_{1} m_{2} = 8 k^{2}$

$m_{1} m_{2} = 2 k^{2}$

${(m_{1} - m_{2})}^{2} = {(m_{1} + m_{2})}^{2} - {4m}_{1} m_{2}$

$= {9k}^{2} - {8k}^{2}$

$= k^{2}$

$m_{1} - m_{2} = k$

$m_{1} + m_{2} = 3 k$

$\underline{m_{1} - m_{2} = 3 k}$

${2m}_{1} = 4 k$

$m_{1} = 2 k$

$m_{2} = k$

$\frac{m_{2}}{m_{1}} = \frac{1}{2}$

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