Chapter D1.1, Problem 54E

Free fall One possible model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation v′(t) = g – bv, where v(t) is the velocity of the object for t ≥ 0, g = 9.8 m/s² is the acceleration due to gravity, and b > 0 is a constant that involves the mass of the object and the air resistance.

a.    Verify by substitution that a solution of the equation, subject to the initial condition v(0) = 0, is $v\left(t\right)=\frac{g}{b}\left(1-e^{-bt}\right)$.

b.    Graph the solution with b = 0.1 s^–1.

c.    Using the graph in part (c), estimate the terminal velocity $\underset{t\to \infty }{\lim }v\left(t\right)$.