$A$ particle of mass $m$ is thrown horizontally from the top of a tower and another particle of mass $2m$ is thrown vertically upward. The acceleration of the centre of mass is .............

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

$A$ system consists of two particles of masses $m_1$ and $m_2$ separated by a distance $r$. If the particles move towards each other with velocities $v_1$ and $v_2$ respectively,the velocity of the centre of mass of the system is:

An object $A$ of mass $M$ is falling vertically downwards under the influence of gravity. It breaks into two parts: part $B$ of mass $\frac{1}{3}M$ and part $C$ of mass $\frac{2}{3}M$. The center of mass of the combined parts $B$ and $C$ will be displaced relative to object $A$ in which direction?

$A$ piece of wood of mass $0.03\, kg$ is dropped from the top of a $100\, m$ height building. At the same time,a bullet of mass $0.02\, kg$ is fired vertically upward,with a velocity $100\, ms^{-1}$,from the ground. The bullet gets embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is ........ $m$. $(g = 10\, ms^{-2})$

Two men of masses $55 \ kg$ and $65 \ kg$ are standing at the opposite ends of a boat. The length of the boat is $3.0 \ m$ and its mass is $100 \ kg$. The $55 \ kg$ man walks up to the $65 \ kg$ man and sits beside him. If the boat is in still water,the center of mass of the system will shift by ..... $m$.