In free space, a shell moving with velocity $60\,m/s$ along the positive $x$ -axis. When it  passes through origin, it explodes into two peices of mass ratio $1 : 2.$ After the  explosion, the velocity of the centre of mass is ........ $m/s$.

A circular stage is free to rotate about vertical axis passing through centre. $A$ tortoise is  sitting at corner of stage. Stage is provided angular velocity $\omega_0$. If tortoise start  moving along one chord at constant speed with respect to stage then how the angular  velocity of stage $\omega(t)$ vary with time $t$ :-

When a uniform solid sphere and a disc of the same mass and of the same radius rolls down a rough inclined plane from rest to the same distance, then the ratio of the time taken by them is

The instantaneous angular position of a point on a rotating wheel is given by the equation $\theta (t) = 2t^3 -6t^2$. The torque on the wheel becomes zero at $t$ $=$ ........ $\sec.$

A child is standing at one end of a long trolley moving with a speed $v$ on a smooth horizontal floor. If the child starts running towards the other end of the trolley with a speed $u$, the centre of mass of the system (trolley + child) will move with a speed.

Figure shows a thin metallic triangular sheet $ABC.$ The mass of the sheet is $M.$ The moment of inertia of the sheet about side $AC$ is