That is the best explanation I have seen so far. There would be then a similar formula to that of half life?
or charge/discharge of a capacitive circuit?
Yup. The velocity approaches terminal velocity over time hyperbolically.
The formalism is almost the same as a capacitor charging example. Here, suppose we measure the amount of charge
Q that flows through a circuit with applied ideal voltage
V, resistance
R, and capacitor value
C. The equation for charge is as follows:
0 = Q / C + Q' R - V
This is a simple, first-order ODE in one variable, since V is a constant. Rewrite as:
Q' = (V / R) - (Q / R C)
And the familiar solution, where
R C is the characteristic time:
Q = C V (1 - e-t/R C)
This is similar to (but not quite) the same as our situation. For drag, we have a slightly different form -- now
v is velocity:
v' = (m g) - (1/2 Cd ρ v2)
The square term changes the equation quite a bit. Instead of a simple exponential, the solution now takes the form of a hyperbolic tangent function, and solving the equation is more painful. But qualitatively the behavior is the same -- the tanh function can be thought of as a "more square" decaying exponential, one with a more pronounced knee around the characteristic time.
Several times the characteristic unit is familiar to me. In electronic, and in the charge or discharge of a capacitive cct one designs cct using the idea of 5 or 10 times. Its been 20 years though since I was expected to do any such calculations. Now, in the real world, if it don't work, we do not go about re-engineering it, we just fix the &^%$ thing.
In that spirit, you can also just drop things and measure their effective velocity. Find yourself a nice, tall, open stairwell, a stopwatch, and a bagfull of pinecones or ping-pong balls, something draggy and safe. The transition from acceleration to reaching terminal velocity is pretty sharp, and should be directly observable with little effort.