Suppose that a motorboat is moving at 40 ft/s when its motor suddenly quits, and that 10 s...

Suppose that a motorboat is moving at 40 ft/s when its motor suddenly quits, and that 10 s later the boat has slowed to 20 ft/s. Assume, as in Problem, that the resistance it encounters while coasting is proportional to its velocity. How far will the boat coast in all?

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.