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,

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

where $v=v\left(t\right)$ and $s=s\left(t\right)$ 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, $f\left(v\right)=-kv$, k a positive constant. (This model is appropriate for small values of v.) Let $v\left(0\right)=v_0$ and $s\left(0\right)=s_0$ 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, $f\left(v\right)=-k{v}^2,k>0$. (This model was first proposed by Newton.) Let $v_0$ and $s_0$ 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 in this case?