Physics is cool Part-1

This is a very nice problem. First, we have to analyse the motion of the blocks.

• When the small block covers the first half of semi-circle for the first time, the big block is at rest.

• When the small block moves to the second half of the semi-circle, the big block moves to the right.

• Subsequently, the big block continuously move to the right.

Now, velocity of small block at the bottom most part of the semi-circle is $v_{i}\sqrt{2gr}$ . This can be obtained by conserving energy.

When the block moves to the second half of the semi-circle, the big block attains velocity. Since, there is no friction anywhere, we can conserve momentum and energy using basic equations.

Conservation of Momentum:

$m{v}_{i}=m{v}_{1}+M{v}_{2}$

Conservation of Energy:

$mgr=\frac{m{{v}_{1}}^{2}}{2}+\frac{M{{v}_{2}}^{2}}{2}$

Solving these solutions, we get 2 pairs of solutions.

$(1)$ ${v}_{1}=\sqrt{2gr}$ and ${v}_{2}=0$

$(2)$ ${v}_{1}=\frac { m-M }{ m+M } \sqrt { 2gr }$ and ${v}_{2}=\frac { 2m }{ m+M } \sqrt { 2gr }$

Hence, ${v}_{2max}=\frac { 2m }{ m+M } \sqrt { 2gr }$

On inserting the values the answer comes out to be 18.