State if each of the following statements is true or false. Give reasons for your answer.

$(a)$ In an elastic collision of two bodies, the momentum and energy of each body is conserved.

$(b)$ Total energy of a system is always conserved, no matter what internal and external forces on the body are present.

$(c)$ Work done in the motion of a body over a closed loop is zero for every force in nature.

$(d)$ In an inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system.

When a ball is freely fallen from a given height it bounces to $80\%$ of its original height. What fraction of its mechanical energy is lost in each bounce ?

A 3.628 kg freight car moving along a horizontal rail road spur track at $7.2\; km/hour$ strikes a bumper whose coil springs experiences a maximum compression of $30 \;cm$ in stopping the car. The elastic potential energy of the springs at the instant when they are compressed $15\; cm$ is [2013]

(a) $12.1 \times 10^4\;J$ (b) $121  \times 10^4\;J$ (c) $1.21 \times 10^4\;J$ (d) $1.21  \times 10^4\;J$

If the potential energy of a gas molecule is $U = \frac{M}{{{r^6}}} – \frac{N}{{{r^{12}}}},M$ and $N$ being positive constants, then the potential energy at equilibrium must be