A mon of mass 'm' moves on a plank of mass 'M' with a constant velocity 'u' with respect to the plank (as shown in the figure). If the plank rests on a smooth horizontal surface, then determine the final velocity of the plank.

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Solution

The correct option is A

As no external forces are acting on the system in horizontal direction , so its linear momentum remain constant in that direction.Also there will be no shift in postion of center of mass. i.e $△$ $X_{c m}$ = 0 .Let $v_{1}$ and $v_{2}$ are the velocties of the man and the plank with respect to ground. Then we have.
$[{\to V}_{m a n}]_{p l a n k}$ = $[{\to V}_{m a n}]_{g r o u n d}$ - $[{\to V}_{m a n}]_{g r o u n d}$
Then $[{\to V}_{m a n}]_{g r o u n d}$ = $[{\to V}_{m a n}]_{p l a n k}$ + $[{\to V}_{m a n}]_{g r o u n d}$
or $v_{1}$ = u - $v_{2}$
Initially the system is at rest , therefore 0 = m $v_{1}$ - m $v_{2}$
or 0 - m(u - $v_{2}$ ) - M $v_{2}$ or $v_{2}$ = $\frac{m u}{m + M}$

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