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

Air Resistance A differential equation governing 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 the drag coefficient. The positive direction is downward.

(a) Solve this 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 39 in Exercises 2.1.

(c) If 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 39 in Exercises 2.1.:

Terminal Velocity The autonomous differential equation

where k is a positive constant of proportionality called the drag coefficient 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 or drag, the velocity of a body falling from a great height does not increase without bound as time t increases.

(a) Use a phase portrait of the differential equation to find the limiting, or terminal, velocity of the body. Explain

your reasoning.

(b) Find the terminal velocity of the body if air resistance is proportional to v ². See pages 22 and 26.