Solved: Air Resistance A differential equation governing the velocity v of a falling
Chapter 2, Problem 17(choose chapter or problem)
A differential equation governing the velocity v of a falling mass m subjected to air resistance proportional to the square of the instantaneous velocity is
\(m \frac{d v}{d t}=m g-k v^{2}\),
where \(k>0\) is the drag coefficient. The positive direction is downward.
(a) Solve this equation subject to the initial condition \(v(0)=v_{0}\).
(b) Use the solution in part (a) to determine the limiting, or terminal, velocity of the mass. We saw how to determine the terminal velocity without solving the DE in Problem 41 in Exercises 2.1.
(c) If distance s, measured from the point where the mass was released above ground, is related to velocity v by \(d s / d t=v(t)\), find an explicit expression for s(t) if \(s(0)=0\).
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