Chapter 3.2, Problem 15E

Air Resistance A differential equation for the velocity v of a falling mass m subjected to air resistance proportional to the square of the instantaneous velocity is

$m\frac{dv}{dt}=mg-k{v}^2,$

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 part (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 41 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.