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(A) Since the system (man + trolley) is on a smooth horizontal surface,there is no external horizontal force acting on the system.Therefore,the positi

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$\frac{mL}{m + M}$

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(A) Since the system (man + trolley) is on a smooth horizontal surface,there is no external horizontal force acting on the system.Therefore,the position of the center of mass of the system remains unchanged.Let $x_m$ be the displacement of the man and $x_t$ be the displacement of the trolley with respect to the ground.Let the trolley move a distance $x$ towards the left,so $x_t = -x$.The man moves a distance $L$ relative to the trolley towards the right.So,the displacement of the man with respect to the ground is $x_m = L - x$.Using the center of mass displacement formula: $m x_m + M x_t = 0$.Substituting the values: $m(L - x) + M(-x) = 0$.$mL - mx - Mx = 0$.$mL = (m + M)x$.$x = \frac{mL}{m + M}$.Thus,the distance moved by the trolley with respect to the ground is $\frac{mL}{m + M}$.

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