Assertion $(A)$: In an elastic collision of two billiard balls,the total kinetic energy $(KE)$ is conserved during the short time of collision of the balls (i.e.,when they are in contact).
Reason $(R)$: Energy spent against friction does not follow the law of conservation of energy.

$A$ force exerts an impulse $I$ on a particle changing its speed from $u$ to $2u$. The applied force and the initial velocity are oppositely directed along the same line. The work done by the force is

The displacement $x$ of a particle moving in one dimension under the action of a constant force is related to the time $t$ by the equation $t = \sqrt{x} + 3$,where $x$ is in meters and $t$ is in seconds. The work done by the force in the first $6$ seconds is.....$J$

$A$ bullet loses $\left( \frac{1}{n} \right)^{th}$ of its velocity while passing through one plank. The number of such planks required to stop the bullet is:

$A$ student skates up a ramp that makes an angle $30^{\circ}$ with the horizontal. He/she starts (as shown in the figure) at the bottom of the ramp with speed $v_0$ and wants to turn around over a semicircular path $xyz$ of radius $R$ during which he/she reaches a maximum height $h$ (at point $y$) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then ($g$ is the acceleration due to gravity):
$(A)$ $v_0^2 - 2gh = \frac{1}{2} gR$
$(B)$ $v_0^2 - 2gh = \frac{\sqrt{3}}{2} gR$
$(C)$ The centripetal force required at points $x$ and $z$ is zero.
$(D)$ The centripetal force required is maximum at points $x$ and $z$.

$A$ locomotive of mass $m$ starts moving so that its velocity varies according to the law $v = k \sqrt{S}$,where $k$ is a constant and $S$ is the distance covered. Find the total work performed by all the forces acting on the locomotive during the first $t$ seconds after the beginning of motion.