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  4. A solid sphere rolling on a rough horizontal surface with a linear speed v collides elastically with a fixed, smooth, vertical wall. The speed of the sphere after it has started pure rolling in the backward direction is

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Q. A solid sphere rolling on a rough horizontal surface with a linear speed $v$ collides elastically with a fixed, smooth, vertical wall. The speed of the sphere after it has started pure rolling in the backward direction is

Solution:

Before collision
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$mvR - \left(\frac{2}{5} mR ^{2}\right) \times \omega= mv ^{\prime} R +\left(\frac{2}{5} mR ^{2}\right) \times \omega^{\prime}$
$\Rightarrow mvR -\frac{2}{5} mR ^{2} \times \frac{ v }{ R }= mv ^{\prime} R +$
$\left(\frac{2}{5} mR ^{2}\right) \times \frac{ v ^{\prime}}{ R }$
on solving $\frac{3 v }{5}=\frac{7 v ^{\prime}}{5} \Rightarrow v ^{\prime}=\frac{3 v }{7}$