Show that in the absence of any external force, the velocity of the centre of mass remains constant.

To show that the velocity of the center of mass is constant when there are no external forces, we start with the definition of the center of mass in terms of masses and positions of particles in a system. By taking the derivative of this position, we see that the velocity of the center of mass is a weighted average of the individual particle velocities.
According to Newton’s second law, the change in momentum of the system is equal to the net external force. In the absence of external forces, the total momentum remains constant, implying that the velocity of the center of mass also remains constant.