Air Resistance A differential equation for the velocity v of a falling mass m subjec...

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

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

(a) Solve the equation subject to the initial condition v(0) = v0.

(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.

(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.

(reference problem 41 in exercise 2.1)

Suppose the model in Problem 40 is modified so that air resistance is proportional to v², that is, See Problem 17 in Exercises 1.3. Use a phase portrait to find the terminal velocity of the body. Explain your reasoning.

(reference problem 17 in exercise 1.3)

For high-speed motion through the air—such as the skydiver shown in Figure 1.3.16, falling before the parachute is opened —air resistance is closer to a power of the instantaneous velocity v(t). Determine a differential equation for the velocity v(t) of a falling body of mass m if air resistance is proportional to the square of the instantaneous velocity.