Two particles $A$ and $B$ (both initially at rest) start moving towards each other under a mutual force of attraction. At the instant, when the speed of $A$ is $v$ and the speed of $B$ is $2v$, the speed of the centre of mass is

$A$ particle of mass $10 \text{ g}$ is moving towards east with a velocity of $10 \text{ ms}^{-1}$ and another particle of mass $15 \text{ g}$ is moving towards north with a velocity of $5 \text{ ms}^{-1}$. The magnitude of the velocity of the centre of mass of the system of the two particles is (in $\text{ ms}^{-1}$)

$A$ block of mass $M$ is tied to one end of a massless rope. The other end of the rope is in the hands of a man of mass $2M$ as shown in the figure. Initially,the block and the man are resting on a rough plank of mass $2M$ as shown in the figure. The whole system is resting on a smooth horizontal surface. The man pulls the rope. The pulley is massless and frictionless. What is the magnitude of the displacement of the plank when the block meets the pulley? (The man does not leave his position on the plank during the pull).

The centre of mass of two masses $m$ and $m'$ moves by a distance $\frac{x}{5}$ when mass $m$ is moved by a distance $x$ and $m'$ is kept fixed. The ratio $\frac{m'}{m}$ is

Two bodies of mass $4 \,kg$ and $5 \,kg$ are moving along east and north directions with velocities $5 \,m/s$ and $3 \,m/s$ respectively. The magnitude of the velocity of the centre of mass of the system is:

Two particles of masses $m_1$ and $m_2$ $(m_1 > m_2)$,initially at rest,move towards each other under an inverse square law force of attraction. Pick out the correct statement about the centre of mass $(CM)$ of the system.