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A child is standing at one end of a long...

A child is standing at one end of a long trolley moving with a speed v on a smooth horizontal floor. 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

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The force exerted by the cliild is an internal force acting on the system of child and trolley. Since there is no external force the speed of centre of mass remains the same as v.
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