Another factor that increases the gas’s kinetic energy is the quadratic term 2mu^2 in a molecule’s kinetic energy after impact with the piston that remains positive regardless of the oscillation phase. If we denote the pre-impact molecular velocity component perpendicular to the piston’s surface as w_x, then the squared x-component after impact is (w_x\pm 2u)^2=w_x^2\pm 4w_xu+4u^2.
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