Fire cracker exploding at rest- Consider the explosion of a fire cracker at rest into number of fragments. The explosion takes place not because of any external force but due to chemical reactions occurring in the cracker. Through the fragments fly off in different directions with different velocities of the center of mass remains zero even after the collision. Thus the center of mass of a fire cracker at rest remains at rest even after the explosion as shown in following figure.
A fire cracker exploding in mid air
When a cracker is fired at the angle with the horizontal and when it explodes in air, then different fragments of the cracker follow different parabolic paths with different velocities. But the center of mass of all the fragments of the cracker continues to move in initial parabolic path as shown by line in following figure. The reason is that the resultant of the external forces (which are gravitational in nature) on the cracker act at the center of mass where whole mass of all the fragments can be supposed to be concentrated.