Two men $A$ and $B$ are standing on a plank. $B$ is at the middle of the plank and $A$ is at the left end of the plank. The bottom surface of the plank is smooth. The system is initially at rest and masses are as shown in the figure. $A$ and $B$ start moving such that the position of $B$ remains fixed with respect to ground and $A$ meets $B$. Then the point where $A$ meets $B$ is located at

A small sphere of radius$\mathrm{R}$ is held against the inner surface of a larger sphere of radius $6\mathrm{R}$. The masses of large and small spheres are $4\mathrm{M} \mathrm{and} \mathrm{M}$respectively. This arrangement is placed on a horizontal table as shown. There is no friction between any surfaces of contact. The small sphere is now released. The coordinates of the centre of the large sphere when the smaller sphere reaches the other extreme position is :

The diagram shows the velocity-time graph for two masses $R$ and $S$ that collided head on elastically. Which of the following statements is true?

$\mathrm{I}.$ $R$ and $S$ moved in the same direction after the collision.
$\mathrm{II}.$ The velocities of $R$ and $S$were equal at the mid-time of the collision.
$\mathrm{III}.$The mass of $R$ was greater than the mass of $S$.

A system of two blocks $A$ and $B$ are connected by an inextensible massless string as shown. The pulley is massless and frictionless. Initially, the system is at rest when a bullet of mass $m$moving with a velocity $u$as shown hits the block$B$and gets embedded into it. The impulse imparted by tension force to the block of mass $3m$ is :