  The changeover from normal to super diffusion in time dependent billiards is
explained analytically. The unlimited energy growth for an ensemble of bouncing
particles in time dependent billiards is obtained by means of a two dimensional
mapping of the first and second moments of the velocity distribution function.
We prove that for low initial velocities the mean velocity of the ensemble
grows with exponent ~1/2 of the number of collisions with the border, therefore
exhibiting normal diffusion. Eventually, this regime changes to a faster growth
characterized by an exponent ~1 corresponding to super diffusion. For larger
initial velocities, the temporary symmetry in the diffusion of velocities
explains an initial plateau of the average velocity.
