Chapter 8.2, Problem 26P

An object of mass $m$ falls from rest under gravity subject to an air resistance proportional to its speed. Taking the $y$ axis as positive down, show that the differential equation of motion is $m(dv/dt)=mg-kv,$ where $k$ is a positive constant. Find $v$ as a function oft, and find the limiting value of $v$ as $t$ tends to infinity; this limit is called the terminal speed. Can you find the terminal speed directly from the differential equation without solving it? Hint: What is $dv/dt$ after $v$ has reached an essentially constant value?

Consider the following specific examples of this problem.

(a) A person drops from an airplane with a parachute. Find a reasonable value of $k$

(b) In the Millikan oil drop experiment to measure the charge of an electron, tiny electrically charged drops of oil fall through air under gravity or rise under the combination of gravity and an electric field. Measurements can be made only after they have reached terminal speed. Find a formula for the time required for a drop starting at rest to reach $99\%$ of its terminal speed.