$A$ ball is released from a certain height. It loses $50\%$ of its kinetic energy on striking the ground. It will attain a height again equal to

$A$ man is slipping on a frictionless inclined plane and a bag falls down from the same height. Then the velocity of both is related as

Water falls from a $40\,m$ high dam at the rate of $9 \times 10^{4}\,kg$ per hour. Fifty percent of the gravitational potential energy can be converted into electrical energy. Using this hydroelectric energy, the number of $100\,W$ lamps that can be lit is (Take $g = 10\,m/s^2$)

Statement $-1$: Two particles moving in the same direction do not lose all their energy in a completely inelastic collision.
Statement $-2$: Principle of conservation of momentum holds true for all kinds of collisions.

$A$ mass of $2.9 \ kg$ is suspended from a string of length $50 \ cm$ and is at rest. Another body of mass $100 \ g$,which is moving horizontally with a velocity of $150 \ m/s$,strikes and sticks to it. Subsequently,when the string makes an angle of $60^{\circ}$ with the vertical,the tension in the string is $(g = 10 \ m/s^2)$. (in $N$)

$A$ 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 $x = -a$.