= Pressure-driven cork ejection
{title2=$\ddot x=P_gx/(\rho l^2)$}
A cylindrical plug of radius $a$, length $l$ and density $\rho$ is pushed outward by gauge pressure $P_g$. If its remaining contact length is $l-x$ and its constant radial contact pressure is $P_c$, the resisting <kinetic friction> is $2\pi a(l-x)\mu P_c$. At the marginal initial condition $\pi a^2P_g=2\pi al\mu P_c$, the net outward force becomes $\pi a^2P_gx/l$. With mass $\rho\pi a^2l$,
$$
\ddot x=b^2x,\quad b=\frac1l\sqrt{\frac{P_g}{\rho}},\quad x(t)=\frac{v_0}{b}\sinh(bt),\quad \dot x^2=v_0^2+b^2x^2.
$$
Thus the exit speed at $x=l$ tends to $\sqrt{P_g/\rho}$ as $v_0\to0^+$. Exactly $v_0=0$ leaves the marginal configuration stationary, and the exit time $b^{-1}\operatorname{arsinh}(bl/v_0)$ diverges in that limit. An attached extra mass changes the acceleration but not the pressure-minus-friction work, if the contact geometry is unchanged.
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