In the figures $(i), (ii)$ and $(iii)$ shown, the objects $A, B$ and $C$ have the same mass $m$. The string, spring, and pulley are massless. Object $C$ strikes object $B$ with velocity $u$ in each case and sticks to it. Find the ratio of the velocity of $B$ in case $(i)$ to $(ii)$ to $(iii)$.

Shown in the diagram is a system of two bodies,a block of mass $m$ and a disc of mass $4m$,held in equilibrium. If the string $3$ is burnt,find the acceleration of the disc. Neglect the masses of the pulleys $P$ and $Q$. The coefficient of friction between the block and the horizontal surface is $0.5$ and the friction between the disc and the string is zero. (Take $g = 10 \ m/s^2$)

$A$ block of mass $m$ is lying on a rough inclined plane having an inclination $\alpha = \tan^{-1}(\frac{1}{5})$. The inclined plane is moving horizontally with a constant acceleration of $a = 2 \text{ ms}^{-2}$ as shown in the figure. The minimum value of the coefficient of friction,so that the block remains stationary with respect to the inclined plane,is (Take $g = 10 \text{ ms}^{-2}$):

The driver of a car travelling at velocity $v$ suddenly sees a broad wall in front of him at a distance $d$. He should

$A$ $2 \, kg$ mass starts from rest on a smooth inclined surface with an inclination of $30^\circ$ and a length of $2 \, m$. What distance $S$ (in meters) will it travel before coming to rest on a rough horizontal surface with a coefficient of friction of $0.25$?

Give the magnitude and direction of the net force acting on:
$(a)$ a drop of rain falling down with a constant speed,
$(b)$ a cork of mass $10\; g$ floating on water,
$(c)$ a kite skillfully held stationary in the sky,
$(d)$ a car moving with a constant velocity of $30\; km/h$ on a rough road,
$(e)$ a high-speed electron in space far from all material objects,and free of electric and magnetic fields.