Which of the following potential energy curves cannot possibly describe the elastic collision of two billiard balls? Here r is the distance between centers of the balls, R is the radius of each ball. 

If collisions are elastic collisions then potential energy is a function of position only.
Potential Energy is zero after the two colliding bodies make contact with each other and are now separate. 
The potential energy of a system of two masses is inversely proportional to the separation between them.
In the given case, with decrease in separation, potential energy increases after two bodies make the contact with each other.
i.e.          ,             for , 
   
Therefore, 
It will become zero (i.e., V(r) = 0) when the two balls touch each other, i.e., at r = 2R, where R is the radius of each billiard ball.
The potential energy curves given in figures (i), (ii), (iii), (iv), and (vi) do not satisfy these two conditions.
Hence, they do not describe the elastic collisions between them.
Possible graph is,