Two identical smooth  balls are rolling without slipping on a horizontal plane as shown in figure. They undergo a perfect elastic collision. Just after collision, the velocity of image of the bottom point of A with respect to the plane mirroris xV, then x = ____ 

 

For identical smooth  balls, after elastic collision velocities are exchanged

 $\therefore$after collision ${\mathrm{V}}_{\mathrm{A}}=\mathrm{V}(-\mathrm{i})\text{ and }{\mathrm{V}}_{\mathrm{B}}=2\mathrm{Vi}$
Velocity of bottom point of A is 
$\begin{array}{l}\mathrm{V}{\text{'}}_{\mathrm{A}}=-\mathrm{V}\left(\mathrm{i}\right)+\left(2\mathrm{r}\omega \right)(-\widehat{\mathrm{i}})\\ \mathrm{V}{\text{'}}_{\mathrm{A}}=3\mathrm{V}(-\widehat{\mathrm{i}})[\because \mathrm{V}=\mathrm{r}\omega ]\text{ since }\left|\right|\overrightarrow{\mathrm{V}{\text{'}}_{\mathrm{A}}}|=|\overrightarrow{{\mathrm{V}}_{\mathrm{M}}}\mid \end{array}$
Velocity of bottom of A with respect to mirror is zero.

So the velocity of image of bottom of A will be zero.