Two particles which are initially at rest move towards each other under the action of their mutual attraction. If their speeds are $v$ and $2v$ at any instant,then the speed of the center of mass of the system is,

Obtain an expression for the velocity of the centre of mass for a system of $n$ particles.

Three particles $A, B$ and $C$ of masses $m, 2m$ and $3m$ are moving towards north, south and east respectively. If the velocities of the particles $A, B$ and $C$ are $6 \,ms^{-1}, 12 \,ms^{-1}$ and $8 \,ms^{-1}$ respectively, then the velocity of the centre of mass of the system of particles is (in $\,ms^{-1}$)

$A$ 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:

$A$ man of mass $M$ stands at one end of a plank of length $\ell$,which lies at rest on a frictionless surface. The man walks to the other end of the plank. If the mass of the plank is $3M$,what is the distance moved by the man relative to the ground?

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 ..... $m$.