Consider a body that moves horizontally through a medium whose resistance is proportional...

Consider a body that moves horizontally through a medium whose resistance is proportional to the square of the velocity v, so that dv/dt = −kv2. Show that

and that

Note that, in contrast with the result of Problem, x(t) → + ∞ as t → + ∞. Which offers less resistance when the body is moving fairly slowly-the medium in this problem or the one in Problem? Does your answer seem consistent with the observed behaviors of x(t) as t → + ∞?

Problem

Suppose that a body moves through a resisting medium with resistance proportional to its velocity v, so that dv/dt = −kv.

(a) Show that its velocity and position at time t are given by

v(t) = v0e−kt

and

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(b) Conclude that the body travels only a finite distance, and find that distance.