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).

$(a)$ No : In an elastic collision, the total initial kinetic energy of the balls will be equal to the total final kinetic energy of the balls. This kinetic energy is not conserved at the instant the two balls are in contact with each other. In fact, at the time of collision, the kinetic energy of the balls will get converted into potential energy.

$(b)$ Yes : In an elastic collision, the total linear momentum of the system always remains conserved.

$(c)$ No; Yes : In an inelastic collision, there is always a loss of kinetic energy, i.e., the total kinetic energy of the billiard balls before collision will always be greater than that after collision.The total linear momentum of the system of billiards balls will remain conserved even in the case of an inelastic collision.

$(d)$ Elastic : In the given case, the forces involved are conservation. This is because they depend on the separation between the centres of the billiard balls. Hence, the collision is elastic.