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 ?

When the child is running on the floor of the trolley , the child exterts force on the trolley by his feet , this is action.The trolley also exerts same force in opposite direction on the child , this is reaction. These are internal forces . But no net external force is acting on the trolley - child system .Hence acceleration of C.M. of the system is zero and velocity of the system remains unchanged.