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 get up and runs about on the trolley in any manner, what is the speed of C.M. (Centre of Mass) of the (trolley + child) system

Trolley and the child make the system, speed of the C.M. of this system remains same irrespective of the movement of the child inside the trolley in any manner because C.M. is affected by external forces and not by the internal forces.