Two bodies of masses $m_1$ and $m_2$ are connected by a massless string passing over a massless,frictionless pulley as shown in the figure. Given $m_2 < m_1$. The acceleration of the center of mass of the system is:

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

Two blocks of masses $5 \,kg$ and $2 \,kg$ are connected by a spring of negligible mass and placed on a frictionless horizontal surface. An impulse gives a velocity of $7 \,m/s$ to the heavier block in the direction of the lighter block. The velocity of the centre of mass is ....... $m/s$.

$A$ 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 $A$ and $B$,initially at rest,move towards each other under a mutual force of attraction. At an instance when the speed of $A$ is $v$ and the speed of $B$ is $2v$,the speed of the centre of mass $(CM)$ is:

$A$ small ball $B$ of mass $m$ is suspended with a light inelastic string of length $L$ from a block $A$ of the same mass $m$,which can move on a smooth horizontal surface as shown in the figure. The ball is displaced by an angle $\theta$ from the equilibrium position and then released. The displacement of the block when the ball reaches the equilibrium position is