Critical gravity-tunnel settling time ($b_c=2m{\omega }_0,y/y_s=(1+{\omega }_{0}t)e^{-{\omega }_{0}t}$)

Adding a velocity-proportional force $-b\dot{y}$ produces a damped harmonic oscillator. The fastest non-oscillatory release from rest is critical damping, with the repeated-root solution in the title. Reaching fraction $\epsilon$ of the initial displacement requires $(1+s)e^{-s}=\epsilon$, $s={\omega }_{0}t$. At one percent, $s=6.638352\dots$, rather than $\log 100$. Exact arrival at equilibrium takes infinite time; the tolerance reference must be stated.