In this talk I present some differential equation whose vector field is perturbed by the fast motion of a particle in a cylinder corresponding to the Z-periodic Lorentz gas, a model introduced by H.A Lorentz in 1905. The solutions of such perturbed equations and their asymptotic behavior when the dynamic (fast motion) accelerates have been broadly studied by Y.Kifer and Khasminskii for probability preserving dynamical systems. Thus it is a known fact that such solution converges at some specific speed to some kind of averaged limit.However, different settings with different asymptotic behavior can be stated for the solution of the perturbed differential equation when the fast motion occurs in the infinite measure preserving Lorentz gas. I will present the case where an averaging result occurs and express it as a limit theorem providing the rate of convergence of the solution of the perturbed equation to a solution of an averaged ordinary differential equation. After explaining the main features of the limit process, I will give some idea of the proof based on the statement of a limit theorem for a non-stationary ergodic sum on the Z-periodic Lorentz gas.