$A$ ball of mass $m$ is thrown upward and another ball of same mass is thrown downward so as to move freely under gravity. The acceleration of the centre of mass is .........

$A$ man of $50\, kg$ mass is standing in a gravity-free space at a height of $10\, m$ above the floor. He throws a stone of $0.5\, kg$ mass downwards with a speed of $2\, m/s$. When the stone reaches the floor,the distance of the man above the floor will be ...... $m$.

$A$ trolley of length $L$ has a pistol fixed on its wall as shown in the figure. The mass of the (trolley + pistol) is $M$. $A$ bullet of mass $m$ is fired from the pistol with a velocity $v_0$ and hits the opposite wall. What is the distance moved by the trolley during the time the bullet travels to hit the opposite wall?

On a smooth horizontal plane,a uniform string of mass $M$ and length $L$ is lying in a state of rest. $A$ man of the same mass $M$ is standing next to one end of the string. Now,the man starts collecting the string. Finally,the man collects all the string and puts it in his pocket. What is the displacement of the man with respect to the Earth in the process of collection?

The figure shows a man of mass $m$ standing at the end $A$ of a trolley of mass $M$ placed at rest on a smooth horizontal surface. The man starts moving towards the end $B$ with a velocity $u_{rel}$ with respect to the trolley. The length of the trolley is $L$. The distance moved by the man with respect to ground is

$A$ child sits stationary at one end of a long trolley moving uniformly with a speed $V$ on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner,what is the speed of the $CM$ of the (trolley + child) system?