This paper uses quadratic surface Lyapunov functions to efficiently check if a double integrator in feedback with a saturation nonlinearity has L₂-gain less than gamma. We show that for many of such systems, the L₂-gain are non-conservative in the sense that they are approximately equal to the lower bound found by replacing the saturation with a constant gain of 1. These results allow the use of classical analysis tools like mu-analysis or IQCs to analyze systems with double integrators and saturations, including servo systems like some mechanical systems, satellites, hard-disks, CD players, etc.