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

$A$ man inside a freely falling box throws a heavy ball towards a side wall. The ball keeps on bouncing between the opposite walls of the box. We neglect air resistance and friction. Which of the following figures depicts the motion of the centre of mass of the entire system (man,the ball and the box)?

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

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 $m/s$):

Explain translational motion by the given illustration.

$A$ projectile of mass $M$ is fired so that the horizontal range is $4\, km$. At the highest point,the projectile explodes into two parts of masses $M/4$ and $3M/4$ respectively. The heavier part starts falling down vertically with zero initial speed. The horizontal range (distance from the point of firing) of the lighter part is .................. $km$.