Two identical balls are rolling without slipping on a horizontal plane as shown in figure. They undergo a prefect elastic collision. Just after collision, the velocity of image of the top most point of A with respect to the plane mirror is $x v,$ then $x =_________$

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answer is 4.

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Detailed Solution

By momentum conservation :

$m \times 2 v + (- mv) = {mv}_{1} + {mv}_{2}$

$v = v_{1} + v_{2}$

K.E conservation :

$\frac{1}{2} m (2 v)^{2} + \frac{1}{2} {mv}^{2} = \frac{1}{2} {mv}_{1}^{2} + \frac{1}{2} {mv}_{2}^{2} \Rightarrow 5 v^{2} = v_{1}^{2} + v_{2}^{2}$

$v_{2} = v - v_{1}$

$5 v^{2} = v_{1}^{2} + {(v - v_{1})}^{2}$

$v_{1} = \frac{v \pm 3 v}{2} = 2 v or - v$

$- v is not possible so v_{2} = 2 v$

$v_{top} = v - 2 rω = - v$

$V_{iM} = - (3 v - (- v)) = - 4 v$

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