Center Of Mass Shifted?

The center of mass of the ball will not move horizontally at all however, it will move vertically. This is because gravity is acting on the ball to the center of mass will obviously move downward at $g$ . But there are no horizontal forces and since the center of mass was at rest horizontally initially it will continue to stay at rest horizontally. The forces that act on the two pieces of the ball are equal and opposite in direction on each other so essentially they are internal forces and not external. This concept for the horizontal motion can be shown mathematically like so,

Let $F_1 =$ Force on little piece and $F_2 =$ Force on bigger piece,

$X_{cm} = (mx_1 + Mx_2) / (m + M)$

Now we will take the derivate of the position of the center of mass with respect to time,

$dX_{cm} / dt = V_{cm} = (mv_1 + Mv_2) / (m + M)$

We will take another derivative but this time of the velocity of the center of mass with respect to time,

$dV_{cm} / dt = a_{cm} = (ma_1 + Ma_2) / (m + M)$

From Newton's second law, we know that $a = F / m$ ,

$a_{cm} = [m( - F / m) + M (F / M)] / (m + M)$

The $m's$ and the $M's$ cancel out in the numerator and we get $-F + F$ left. This cancels out and we get,

$a_{cm} = 0$

Therefore, the velocity of the center of mass does not change because the rate at which it changes is zero. And since the center of mass was initially at rest on the horizontal plane, it will continue to stay at rest throughout the whole trip and remain unshifted horizontally !