= Critical-damping overshoot from initial velocity
{title2=$v_0>\gamma z_{\mathrm{eq}}$}
A mass-spring-damper subject to a constant load obeys $\ddot z+2\gamma\dot z+\gamma^2z=\gamma^2z_{\mathrm{eq}}$ at <critical damping>. With $z(0)=0$ and $\dot z(0)=v_0$,
$$
z(t)=z_{\mathrm{eq}}+[-z_{\mathrm{eq}}+(v_0-\gamma z_{\mathrm{eq}})t]e^{-\gamma t}.
$$
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 $v_0>g\sqrt{m/k}$.
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