Critical-damping overshoot from initial velocity ($v_0>\gamma z_{\mathrm{eq}}$)

A mass-spring-damper subject to a constant load obeys $\ddot{z}+2\gamma \dot{z}+{\gamma }^{2}z={\gamma }^2{z}_{\mathrm{eq}}$ at critical damping. With $z(0)=0$ and $\dot{z}(0)=v_0$,For $z_{\mathrm{eq}}>0$, the displacement exceeds its final value at a positive time exactly when $v_0>\gamma z_{\mathrm{eq}}$. Thus a critically damped step response released from rest is non-overshooting, but critical damping alone does not forbid overshoot from arbitrary initial momentum. For a vertical scale with downward coordinate, $z_{\mathrm{eq}}=mg/k$ and $\gamma =\sqrt{k/m}$, making the threshold $v_0>g\sqrt{m/k}$.