Chapter D1.3, Problem 45E

Free fall An object in free fall may be modeled by assuming that the only forces at work are the gravitational force and air resistance. By Newton’s Second Law of Motion (mass × acceleration = the sum of the external forces), the velocity of the object satisfies the differential equation

             $\underset{\text{mass}}{\underset{︸}{m}}\cdot \underset{\text{acceleration}}{\underset{︸}{v^{\prime }(t)}}=\underset{\text{external forces}}{\underset{︸}{mg+f(v)}},$

where f is a function that models the air resistance (assuming the positive direction is downward). One common assumption (often used for motion in air) is that f(v) = – kv² , where k > 0 is a drag coefficient.

  a.    Show that the equation can be written in the form v'(t) = g – av², where a = k/m.

b.    For what (positive) value of v is v′(t) = 0? (This equilibrium solution is called the terminal velocity.)

  c.    Find the solution of this separable equation assuming v(0) = 0 and 0 < v² < g/a.

d.    Graph the solution found in part (c) with g = 9.8 m/s², m = 1, and k = 0.1, and verify that the terminal velocity agrees with the value found in part (b).