Suppose that there is a wooden block on a smooth horizontal plane. It is hit by some force and it starts moving along a straight line with a constant velocity. Now, cut the block into two slices horizontally and place them one on another. Assume that the surfaces that touch each other are very smooth, so there is no friction. Then again hit it with the same force. Let's imagine that the force hit the bottom part. So that part only starts moving. As there is no friction the upper part stands still and when the it lose the contact with the bottom slice it will fall down. We can think that the centre of the mass(of two blocks) moves with the same speed which it had, before cut into two slices. But after the upper part fell down the COM also descends. Thus why the path of the COM at the second situation differs from the first situation though we have provided same conditions?
Thus why the path of the COM at the second situation differs from the first situation though we have provided same conditions?
In short, because there was no net external force acting in either the horizontal or vertical direction on the single block, thus per Newton's first law the COM does not accelerate.
In the second example, when the top block leaves the bottom block there is now a net external force acting downward on the top block (its weight). So while the COM of the bottom block does not accelerate, the COM of the top block accelerates downward because it is subjected to a net external force, per Newton's second law. Consequently the COM of the combination accelerates downward.
Hope this helps.