$A$ child sits stationary at one end of a long trolley moving uniformly with a speed $V$ on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner,what is the speed of the $CM$ of the (trolley + child) system?

(V) The speed of the $CM$ remains $V$.
The child is running arbitrarily on a trolley moving with velocity $V$. The running of the child constitutes internal forces within the (trolley + child) system.
According to the law of conservation of momentum,the velocity of the centre of mass $(CM)$ of a system changes only if there is an external force acting on the system.
Since the floor is smooth (frictionless) and there are no other external horizontal forces acting on the system,the net external force on the (trolley + child) system is zero.
Therefore,the velocity of the $CM$ of the system remains unchanged,regardless of the child's motion on the trolley. The speed of the $CM$ remains $V$.