Two particles of the same mass are moving towards each other with velocities $2v$ and $v$ respectively. The velocity of the centre of mass of the system is:

In a system of two particles of masses $m_{1}$ and $m_{2}$,the first particle is moved by a distance $d$ towards the centre of mass. To keep the centre of mass unchanged,the second particle will have to be moved by a distance:

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 $m/s$):

Two objects of masses $200 \ g$ and $500 \ g$ have velocities $10\hat{i} \ m/s$ and $3\hat{i} + 5\hat{j} \ m/s$ respectively. The velocity of their center of mass (in $m/s$) is:

$A$ piece of wood of mass $0.03\, kg$ is dropped from the top of a $100\, m$ height building. At the same time,a bullet of mass $0.02\, kg$ is fired vertically upward,with a velocity $100\, ms^{-1}$,from the ground. The bullet gets embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is ........ $m$. $(g = 10\, ms^{-2})$

Two blocks of mass $m$ each,connected to each other by a massless string,are placed on two inclined planes as shown in the figure. After releasing from rest,find the magnitude of the acceleration of the centre of mass of both the blocks. (Given $g = 10 \, m/s^2$)