Chapter 9.3, Problem 50E

An object of mass m is moving horizontally through a medium which resists the motion with a force that is a function of the velocity; that is,

$m\frac{d^{2}s}{d{t}^2}=m\frac{dv}{dt}=f(v)$

where v = v(t) and s = s(t) represent the velocity and position of the object at time t, respectively. For example, think of a boat moving through the water.

(a) Suppose that the resisting force is proportional to the velocity, that is, f(v) = −kv, k a positive constant. (This model is appropriate for small values of v.) Let v(0) = v₀ and s(0) = s₀ be the initial values of v and s. Determine v and s at any time t. What is the total distance that the object travels from time t = 0?

(b) For larger values of v a better model is obtained by supposing that the resisting force is proportional to the square of the velocity, that is, f(v) = −kv², k > 0. (This model was first proposed by Newton.) Let v₀ and s₀ be the initial values of v and s. Determine v and s at any time 1. What is the total distance that the object travels in this case?