A block of mass $M$ with a semicircular track of radius$R$ rests on a horizontal frictionless surface. A uniform cylinder of radius$r$and mass $m$ is released from rest from the top point $A.$ The cylinder slips on the semicircular frictionless track.

  

The velocity of the block when the cylinder reaches point $\left(B\right)$ is:

In a vertical plane inside a smooth hollow thin tube a block of same mass as that of tube is released as shown in the figure. When it is slightly disturbed it moves towards right. By the time the block reaches the right end of the tube, displacement of the tube will be (where $R$is mean radius of tube). Assume that the tube remains in vertical plane.

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 $S$.