An object of mass $m$ is sliding down a hill of arbitrary shape and after traveling a certain horizontal path stops because of friction. The friction coefficient is different for different segments of the entire path but it is independent of the velocity and direction of motion. The work done that a force must perform to return the object to its initial position along the same path would be:

$A$ bullet of $10\, \text{g}$, moving with velocity $v$, collides head-on with the stationary bob of a pendulum and recoils with velocity $100\, \text{m/s}$. The length of the pendulum is $0.5\, \text{m}$ and the mass of the bob is $1\, \text{kg}$. The minimum value of $v$ in $\text{m/s}$ so that the pendulum completes a vertical circle is: (Assume the string to be inextensible and $g=10\, \text{m/s}^2$)

From a building, two balls $A$ and $B$ are thrown such that $A$ is thrown upwards and $B$ is thrown downwards (both vertically) with the same speed. If $v_A$ and $v_B$ are their respective velocities on reaching the ground, then:

$A$ body of mass $0.5 \ kg$ is supplied with a power $P$ (in watt) which varies with time $t$ (in second) as $P = 3t^2 + 3$. If the velocity of the body at time $t = 0$ is zero, then the velocity of the body at time $t = 3 \ s$ is (in $ms^{-1}$)

Answer carefully,with reasons:
$(a)$ In an elastic collision of two billiard balls,is the total kinetic energy conserved during the short time of collision of the balls (i.e.,when they are in contact)?
$(b)$ Is the total linear momentum conserved during the short time of an elastic collision of two balls?
$(c)$ What are the answers to $(a)$ and $(b)$ for an inelastic collision?
$(d)$ If the potential energy of two billiard balls depends only on the separation distance between their centres,is the collision elastic or inelastic?
(Note: We are talking here of potential energy corresponding to the force during collision,not gravitational potential energy.)

$A$ car of mass $1000 \ kg$ has a motor of efficiency $20 \%$. If burning of one litre of petrol supplies $6 \times 10^7 \ J$ of energy,the amount of petrol used in accelerating the car from rest to $43.2 \ km \ h^{-1}$ is (in $cc$)