As shown in the figure, a small block of mass m is set in motion...

Since all surfaces are frictionless, there is no external force acting on the system (smaller block and bigger block) in the horizontal direction.

Therefore, the horizontal component of the total linear momentum of the system is conserved.

At point A, the tangent to the semicircular surface is vertical. This means that the velocity of the smaller block relative to the bigger block is purely vertical.

Thus, at the instant the smaller block reaches point A, its horizontal velocity is exactly equal to the horizontal velocity of the bigger block.

Let $V$ be the common horizontal speed of both blocks at this instant.

Applying conservation of linear momentum in the horizontal direction:

$mv = MV + mV$

$mv = (M + m)V$

$V = \dfrac{mv}{M + m}$