Question

In the figure shown, a ball of mass m collides perpendicularly on a smooth stationary wedge of mass M. kept on a smooth horizontal plane. If the coefficient of restitution of collision is e, then determine the velocity of the wedge after collision.

Answer 1

The ball of mass $m$ will reflect normally from the wedge.

Let the velocity of the mass $m$ upon reflection be $v_1$.

Apply the conservation of momentum along and perpendicular to the inclined plane of the wedge.

For conservation of momentum along the plane.

$MV=m{v}_{0}sin\theta -m{v}_{1}sin\theta ...\left(1\right)$

For the collision perpendicular to the plane, apply coefficient of restitution.

$e=\frac{relativevelocityafterimpact}{relativevelocitybeforeimpact}=\frac{Vsin\theta +v_1}{-v_0}$

Thus,

$Vsin\theta +v_1=-e{v}_0...\left(2\right)$

From (1) :

$v_1=\frac{m{v}_{0}sin\theta -MV}{msin\theta }$

Substituting in (2),

$Vsin\theta +\frac{m{v}_{0}sin\theta -MV}{msin\theta }=-e{v}_0\frac{V(m{sin}^2\theta -M)+m{v}_{0}sin\theta }{msin\theta }=-e{v}_{0}V=\frac{v_0\cdot msin\theta (1+e)}{(M-m{sin}^2\theta )}$