URGENT ! 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 , then the centre of mass of the system ( trolley + child) will move with a speed ???

When the child is standing on the trolley, both child and trolley are moving with the same velocity = v

So, their centre of mass is also moving with the velocity of the vehicle,

v_com = mv + Mv / (m + M) = v

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the velocity of centre of mass changes only if the system is subjected to net external force. But since the only forces involved in the child + trolley system are internal(action-reaction pair) when the child runs on the trolley. Therefore, the velocity of their centre of mass will remain unchanged. It will be v