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.

$A$ uniform rod $AB$ of length $L$ and mass $M$ is lying on a smooth table. $A$ small particle of mass $m$ strike the rod with a velocity $v_0$ at point $C$ a distance $x$ from the centre $O$. The particle comes to rest after collision. The value of $x$, so that point $A$ of the rod remains stationary just after collision, is :

Ratio of masses and radii of two circular rings are $1 : 2$ and $2 : 1$ respectively then ratio of moment of inertia will be

The mass per unit length of a rod of length $l$ is given by : $\lambda = \frac{M_0x}{l}$ ,where $M_0$ is a constant and $x$ is the distance from one end of the rod. The position of  centre of mass of the rod is

A rod of length $L$ is held vertically on a smooth horizontal surface. The top end of the rod is given a gentle push. At a certain instant of time, when the rod makes an angle $\theta$ with horizontal the velocity of $COM$ of the rod is $v_0$ . The velocity of the end of the rod in contact with the surface at that instant is 

The angular momentum of a projectile projected at an angle $\theta $ with the horizontal with speed $u$ about the point of projection when it is at the highest point of its trajectory is