$A$ uniform rod of mass $M$ and length $L$ is placed on a smooth horizontal surface. $A$ horizontal force $F$ is applied at one end of the rod at an angle $\theta$ with the rod. The acceleration of the centre of mass of the rod is:

$30 \,kg$ boy stands at the far edge of a floating plank, whose near edge is against the shore of a river. The plank is $10 \,m$ long and weighs $10 \,kg$. If the boy walks to the near edge of the plank, how far from the shore does the plank move (in $\,m$)?

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 $L$. When the man reaches the end $B$,the distance moved by the trolley with respect to ground is

Three blocks $A$,$B$ and $C$ are arranged as shown in the figure such that the distance between two successive blocks is $10 \ m$. Block $A$ has a mass of $10 \ kg$,block $B$ has a mass of $25 \ kg$,and block $C$ has a mass of $15 \ kg$. Block $A$ is displaced towards block $B$ by $2 \ m$ and block $C$ is displaced towards block $B$ by $3 \ m$. The distance through which the block $B$ should be moved so that the centre of mass of the system does not change is

$A$ 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?

Two blocks of masses $5 \, kg$ and $2 \, kg$ are connected by a spring of negligible mass and placed on a frictionless horizontal surface. An impulse provides a velocity of $7 \, m/s$ to the heavier block in the direction of the lighter block. The velocity of the centre of mass is ............ $m/s$.