Explanation:

When a shell follows a parabolic path and explodes somewhere in its flight, the center of mass of the fragments will continue to move along the same parabolic path. This is because of the conservation of momentum and the fact that external forces acting on the system are negligible.

Conservation of Momentum:
According to the law of conservation of linear momentum, the total momentum of a system remains constant if no external forces act on it. In the case of the exploding shell, the initial momentum of the system is zero as the shell is at rest. When the shell explodes, the fragments move in different directions with different velocities, but the total momentum of the system still remains zero. This means that the momentum of the fragments in one direction is balanced by the momentum of the fragments in the opposite direction.

External Forces:
In the absence of any external forces, the center of mass of the system will continue to move in a straight line with a constant velocity. This is known as Newton's first law of motion. Since no external forces act on the system after the explosion, the center of mass will continue to move in the same parabolic path as before the explosion.

Center of Mass:
The center of mass of a system is the point where the entire mass of the system can be considered to be concentrated. In the case of the exploding shell, the center of mass is initially at rest and located at the highest point of the parabolic path. After the explosion, the fragments move away from the center of mass, but the center of mass itself continues to move along the same parabolic path.

Conclusion:
Therefore, the correct answer is option 'C' - the center of mass of the fragments will continue to move along the same parabolic path. This is because of the conservation of momentum and the absence of any external forces acting on the system.