Chapter 3.1, Problem 35E

Air Resistance In (14) of Section 1.3 we saw that a differential equation describing the velocity v of a falling mass subject to air resistance proportional to the instantaneous velocity is

$m\frac{dv}{dt}=mg-kv,$

where k > 0 is a constant of proportionality. The positive direction is downward.

1. (a) Solve the equation subject to the initial condition v(0) = v₀.
2. (b) Use the solution in pan (a) to determine the limiting, or terminal, velocity of the mass. We saw how to determine the terminal velocity without solving the DE in Problem 40 in Exercises 2.1.
3. (c) If the distance s, measured from the point where the mass was released above ground. is related to velocity v by ds/dt = v(t), find an explicit expression for s(t) if s(0) = 0.