Which of the following potential energy curves in the figure cannot possibly describe the elastic collision of two billiard balls? Here $r$ is the distance between the centres of the balls.

(A) For an elastic collision between two hard billiard balls,the potential energy $V(r)$ must satisfy two conditions:
$1$. When the distance $r$ between the centers is greater than $2R$ (where $R$ is the radius of each ball),the balls do not interact,so the potential energy $V(r) = 0$.
$2$. When the distance $r$ is less than $2R$,the balls are in contact and undergo deformation,leading to a rapid increase in potential energy. At the point of contact $r = 2R$,the potential energy should be zero,and for $r < 2R$,it should be positive and increasing.
Looking at the given curves:
- Curve $(v)$ shows $V(r) = 0$ for $r \ge 2R$ and $V(r) > 0$ for $r < 2R$,which correctly describes the interaction.
- Curves $(i), (ii), (iii), (iv),$ and $(vi)$ do not satisfy these physical requirements. For example,$(ii)$ shows potential energy increasing with distance,which is incorrect,and $(i)$ and $(vi)$ show non-zero potential energy for $r > 2R$.
Therefore,the curves that cannot describe the elastic collision are $(i), (ii), (iii), (iv),$ and $(vi)$.