A child standing at one end of a long trolley moving with 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 will at a speed of what?

v+((u/length ofthe trolley)+1)

mass of trolley=M,mass of child=m

answer=(Mv-mu)/(M+m)

the child runs opposite to the direction of trolley or at the same direction ?
i

the centre of mass will be going with speed v as no external effect or force is acting on the system and momentum will remain conserved. initial speed of trolley = v and hence final speed will be = v.

the friction due to the foot of the child act in equal and opposite pairs on the system and hence no effect is produced in the momentum. we are considering trolley and child together as a system.