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A body of mass $m$ moving with velocity $v$ collides head on with another body of mass $2\, m$ which is initially at rest. The ratio of $K.E.$ of the colliding body before and after collision will be
$1 : 1$
$2 : 1$
$4 : 1$
$9 : 1$
Solution
Velocity of first body after collision is $:-$
$\mathrm{v}_{1}=\left(\frac{\mathrm{m}_{1}-\mathrm{m}_{2}}{\mathrm{m}_{1}+\mathrm{m}_{2}}\right) \mathrm{u}_{1}+\frac{2 \mathrm{m}_{2} \mathrm{u}_{2}}{\mathrm{m}_{1}+\mathrm{m}_{2}}$
$=\left(\frac{m-2 m}{m+2 m}\right) v+\frac{2 \times(2 m) \times 0}{m+2 m}=\frac{-v}{3}$
$\mathrm{KE}_{\mathrm{i}}=\frac{1}{2} \mathrm{mv}^{2}, \quad \mathrm{KE}_{\mathrm{f}}=\frac{1}{2} \mathrm{m}\left(\frac{-\mathrm{v}}{3}\right)^{2}=\frac{1}{2} \frac{\mathrm{mv}^{2}}{9}$
$\frac{\mathrm{KE}_{i}}{\mathrm{KE}_{\mathrm{f}}}=\frac{\frac{1}{2} \mathrm{mv}^{2}}{\frac{1}{2} \frac{\mathrm{mv}^{2}}{9}}=9$
