Two identical particles are moving with velocities $2v$ and $v$ respectively. Find the velocity of the center of mass.

Two blocks $A$ and $B$ are joined together with a compressed spring. When the system is released,the two blocks move with unequal speeds in opposite directions as shown in the figure. Select the correct statement.

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

Two bodies of masses $m_1$ and $m_2$ $(m_1 > m_2)$ are tied to the ends of a string which passes over a light frictionless pulley. The masses are initially at rest and then released. The acceleration of the centre of mass is

Two blocks of masses $10 \; kg$ and $30 \; kg$ are placed on the same straight line with coordinates $(0,0) \; cm$ and $(x, 0) \; cm$ respectively. The block of $10 \; kg$ is moved on the same line through a distance of $6 \; cm$ towards the other block. The distance through which the block of $30 \; kg$ must be moved to keep the position of the centre of mass of the system unchanged is

Three bodies having masses $5 \, kg$,$4 \, kg$,and $2 \, kg$ are moving at speeds of $5 \, m/s$,$4 \, m/s$,and $2 \, m/s$ respectively along the $X$-axis. The magnitude of the velocity of the centre of mass is (in $m/s$):