A child is standing at one end of a long trolley moving with a speed v on a smooth horizontal track. If the child starts running towards the other end of the trolley with a speed u, the centre of mass of the system (trolley $+$ child) will move with a speed.

The correct option is C. v

The child is running towards the other end of the trolley with a speed u. However, the running of the child will produce no effect on the velocity of the centre of mass of the system consisting of the trolley and the child. This is because the force due to the boy's motion is purely internal. Internal forces produce no effect on the motion of the bodies on which they act.

Therefore no external force is involved in the boy—trolley system, the boy's motion will produce no change in the velocity of the centre of mass of the trolley. The centre of mass of the trolley is moving at a speed v and hence the centre of mass of the system will continue to move with velocity v.