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 ......... $m/s$.

$A$ trolley of length $L$ has a pistol fixed on its wall as shown in the figure. The mass of the (trolley + pistol) is $M$. $A$ bullet of mass $m$ is fired from the pistol with a velocity $v_0$ and hits the opposite wall. What is the distance moved by the trolley during the time the bullet travels to hit the opposite wall?

Two particles of mass $m_1$ and $m_2$ are separated by a distance $d$. If the first particle is moved towards the centre of mass by a distance $x$,then the distance by which the second particle must be moved towards the centre of mass to keep the centre of mass at the same position is:

Two particles $A$ and $B$,initially at rest,move 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 center of mass of the system is:

$A$ firecracker is thrown with a velocity of $30 \, ms^{-1}$ in a direction that makes an angle of $75^{\circ}$ with the vertical axis. At some point on its trajectory,the firecracker splits into two identical pieces in such a way that one piece falls $27 \, m$ far from the shooting point. Assuming that all trajectories are contained in the same plane,how far will the other piece fall from the shooting point? (Take $g = 10 \, ms^{-2}$ and neglect air resistance)

If a bomb is thrown at a certain angle with the horizontal and after exploding on the way the different fragments move in different directions,then the centre of mass