$A$ 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.

State whether the following statements are true or false:
$(a)$ If the magnitude of force and length are increased by $4$ times,the magnitude of energy increases by $16$ times.
$(b)$ In an inelastic collision,both momentum and energy are conserved.
$(c)$ If work is done on a system by non-conservative forces,the potential energy increases.

$A$ particle of mass $m$ moving horizontally with velocity $v_0$ strikes a smooth wedge of mass $M$,as shown in the figure. After the collision,the ball starts moving up the inclined face of the wedge and rises to a height $h$. Choose the correct statement related to the wedge $M$.

Statement $(I)$: The slope of the kinetic energy-displacement curve of a body in motion is directly proportional to its acceleration.
Statement $(II)$: From a height of $15 \ m$, a ball is projected vertically upwards with a velocity of $30 \ m/s$. If the ball rises to the same height after hitting the ground, the loss of its energy on hitting the ground is $30 \%$.
Statement $(III)$: The velocity acquired by a body of mass '$m$' after travelling a fixed distance from rest under the action of a constant force is directly proportional to mass '$m$'.
Which of the following is correct?

$A$ block of mass $m(=0.1 \ kg)$ is hanging over a frictionless light fixed pulley by an inextensible string of negligible mass. The other end of the string is pulled by a constant force $F$ in the vertically downward direction. The linear momentum of the block increases by $2 \ kg \ m/s$ in $1 \ s$ after the block starts from rest. Then, (given $g=10 \ m/s^2$):

$A$ particle of unit mass is moving along the $x$-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column $I$ ($a$ and $U_0$ are constants). Match the potential energies in column $I$ to the corresponding statement$(s)$ in column $II$.

Column $I$	Column $II$
$(A) U_1(x) = \frac{U_0}{2} \left[1 - \left(\frac{x}{a}\right)^2\right]^2$	$(P)$ The force acting on the particle is zero at $x = a$.
$(B) U_2(x) = \frac{U_0}{2} \left(\frac{x}{a}\right)^2$	$(Q)$ The force acting on the particle is zero at $x = 0$.
$(C) U_3(x) = \frac{U_0}{2} \left(\frac{x}{a}\right)^2 \exp \left[-\left(\frac{x}{a}\right)^2\right]$	$(R)$ The force acting on the particle is zero at $x = -a$.
$(D) U_4(x) = \frac{U_0}{2} \left[\frac{x}{a} - \frac{1}{3}\left(\frac{x}{a}\right)^3\right]$	$(S)$ The particle experiences an attractive force towards $x = 0$ in the region $|x| < a$.
	$(T)$ The particle with total energy $\frac{U_0}{4}$ can oscillate about the point $x = -a$.