 Pressure-driven cork ejection

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 $\pi a^2{P}_g=2\pi al\mu P_c$, the net outward force becomes $\pi a^2{P}_{g}x/l$. With mass $\rho \pi a^{2}l$,Thus the exit speed at $x=l$ tends to $\sqrt{P_g/\rho }$ as $v_0\to 0^{+}$. Exactly $v_0=0$ leaves the marginal configuration stationary, and the exit time $b^{-1}\mathrm{arsinh}(bl/v_0)$ diverges in that limit. An attached extra mass changes the acceleration but not the pressure-minus-friction work, if the contact geometry is unchanged.