From Newton’s Second Law to a Force Balance

Doing Some Modeling: Why Inertia Is Irrelevant found that a cell’s velocity is $F/\gamma$ except for a few microseconds after each change in force, because the inertial stopping time $\tau_i = m/\gamma$ is 10^-9 of the time the cell takes to move its own length. This part turns that observation into the force balance $\gamma\,\dot x = F$. Measuring time in units of the behavior we want to predict puts that ratio, $\ve = \tau_i/T$, in front of the inertial term, and setting $\ve = 0$ gives the force balance. The limit lowers the order of the equation and discards an initial condition, and the point of this part is to see what the reduction throws away and to compute its error in the quantity we actually measure.