$A$ 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 .................. $km$.

Consider a two-particle system with particles having masses $m_1$ and $m_2$. If the first particle is pushed towards the centre of mass through a distance $d$,by what distance should the second particle be moved,so as to keep the centre of mass at the same position?

$A$ 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)?

$A$ block of mass $M$ is tied to one end of a massless rope. The other end of the rope is in the hands of a man of mass $2M$ as shown in the figure. Initially,the block and the man are resting on a rough plank of mass $2M$ as shown in the figure. The whole system is resting on a smooth horizontal surface. The man pulls the rope. The pulley is massless and frictionless. What is the magnitude of the displacement of the plank when the block meets the pulley? (The man does not leave his position on the plank during the pull).

Two particles of equal masses have velocities $\overrightarrow{v_1} = 2\hat{i} \ m/s$ and $\overrightarrow{v_2} = 2\hat{j} \ m/s$. The first particle has an acceleration $\overrightarrow{a_1} = (3\hat{i} + 3\hat{j}) \ m/s^2$,while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a

$A$ ball of mass $m$ is thrown upward and another ball of same mass is thrown downward so as to move freely under gravity. The acceleration of the centre of mass is .........