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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, the centre of mass of the system (trolley + child) will move with a speed :

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The speed of the CM of the (trolley + child) system remains unchanged when the child gets up and runs about on the trolley in any manner, bocause no external force acts on the system. The forces involved in running on the trolley are internal to this system and hence there is no change in velocity of centre of mass of the system.
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MODERN PUBLICATION-SYSTEMS OF PARTICLES AND ROTATIONAL MOTION-NCERT FILE ( NCERT TEXTBOOK EXERCISES )
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