Let origin of X-Y coordinate system coincide with center of mass of big block that has mass M.
A small ball of mass M/2 is rolling on the surface of semi-spherical ( or it can be semi-hollow cylindrical surface )
made in the big block as shown in figure.
When the ball is at a distance -x from y axis, x-coordinate xcm of center of mass of combined system of ( big block + ball )
is given by, xcm = (M/2) (-x ) / [ M + (M/2) ] = -x/3 ......................(1)
X-component velocity of centre of mass of combined system , ( dxcm / dt ) = [ (M/2) V1 + M V2 ] / [ M + (M/2) ] .............(2)
where V1 is the speed of ball and V2 is the speed of block
But , if there is no external force acting on the system, centre of mass will not move,
hence x-component velocity of centre of mass of combined system is zero.
Hence from eqn.(2), we have, (V1/2) + V2 = 0 or V2 = - ( V1 /2 )
when the ball reaches the point A, velocity V1 = (2gR)1/2
( Potential energy difference is converted to kinetic energy )
velocity V2 = - [ (gR) / 2 ]1/2