A particle moving with kinetic energy E makes an head on elastic collision with an identical particle at rest. During the collision

Momentum of the system is always conserved

Potential energy of the system is maximum when both the particles move with same velocity i.e., mass becomes double while momentum is constant.

$K=\frac{P^2}{2m}$

or 

$K\propto \frac{1}{m}$

i.e., at this moment KE becomes half of $\frac{E}{2}.$ The rest is in the form of elastic potential energy. Therefore, minimum kinetic energy of system is E/2 and at the same moment elastic potential energy of the system is maximum viz, E/2.