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?

Two bodies of masses $M$ and $4M$ initially at rest,start moving towards each other due to their mutual attraction. The velocity of their centre of mass when the first body attains a velocity $v_0$ 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 ...... $m$.

$A$ firecracker is thrown with a velocity of $30 \, ms^{-1}$ in a direction that makes an angle of $75^{\circ}$ with the vertical axis. At some point on its trajectory,the firecracker splits into two identical pieces in such a way that one piece falls $27 \, m$ far from the shooting point. Assuming that all trajectories are contained in the same plane,how far will the other piece fall from the shooting point? (Take $g = 10 \, ms^{-2}$ and neglect air resistance)

$A$ small ball $B$ of mass $m$ is suspended with a light inelastic string of length $L$ from a block $A$ of the same mass $m$,which can move on a smooth horizontal surface as shown in the figure. The ball is displaced by an angle $\theta$ from the equilibrium position and then released. The displacement of the centre of mass of the $A+B$ system until the string becomes vertical is

Two particles of equal masses have velocities $\overrightarrow{v_1} = 2\hat{i} \ m/s$ and $\overrightarrow{v_2} = 2\hat{j} \ m/s$. The first particle has an acceleration $\overrightarrow{a_1} = (3\hat{i} + 3\hat{j}) \ m/s^2$,while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a