Question

Why the center of mass of the fragments of the projectile continues along the same parabolic path which it would have followed if there were no explosion.

Answer 1

This Demonstration maps the motion of an object with a mass of 10 kg that splits into two pieces at the peak of its flight. The first piece is stopped in midair by a specially contrived explosion in such a way that its subsequent trajectory is straight down to the ground. You can change the mass of the second piece to see how its path and velocity are affected. Note the motion of the center of mass (, the black dot) and how it relates to the two fragments.