$A$ body $A$ of mass $M$ while falling vertically downwards under gravity breaks into two parts; a body $B$ of mass $\frac{2}{3} M$ and a body $C$ of mass $\frac{1}{3} M$. The centre of mass of bodies $B$ and $C$ taken together shifts compared to that of body $A$ towards

Two particles of masses $2 \ kg$ and $1 \ kg$ are moving along the same line with speeds of $2 \ m/s$ and $5 \ m/s$ respectively. If both particles are moving in the same direction,the speed of the centre of mass is $v_1$. If both particles are moving in opposite directions,the speed of the centre of mass is $v_2$. Find $v_1$ and $v_2$.

Two bodies of masses $2\ kg$ and $4\ kg$ are moving with velocities $2\ m/s$ and $10\ m/s$ respectively along the same direction. The velocity of their centre of mass will be ......... $m/s$.

$A$ system of two particles has masses $m_1$ and $m_2$. If the particle of mass $m_1$ is pushed towards the centre of mass of the particles through a distance $d$,by what distance should the particle of mass $m_2$ be moved to keep the centre of mass of the particles at the original position?

$A$ body of mass $m_1 = 4 \text{ kg}$ moves at $5 \hat{i} \text{ m/s}$ and another body of mass $m_2 = 2 \text{ kg}$ moves at $10 \hat{i} \text{ m/s}$. The kinetic energy of the centre of mass is:

$A$ mass $m$ is at rest on an inclined plane of mass $M$,which is further resting on a smooth horizontal plane. Now,if the mass $m$ starts moving under gravity,the position of the centre of mass of the system will