Air Resistance In (14) of Section 1.3 we saw that a differential equation describing...

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

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 40 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 find an explicit expression for s(t) if s(0) = 0.

Reference: DE in Problem 40 in Exercises 2.1.

Terminal Velocity In Section 1.3 we saw that the autonomous differential equation

where k is a positive constant and g is the acceleration due to gravity, is a model for the velocity v of a body of mass m that is falling under the influence of gravity. Because the term −kv represents air resistance, the velocity of a body falling from a great height does not increase without bound as time t increases. Use a phase portrait of the differential equation to find the limiting, or terminal, velocity of the body. Explain your reasoning.