We showed in Example 1.2 that the distance a ball falls as a function of time, starting from rest and subject to both gravity \(g\) downward and a quadratic drag force upward, is

where \(v_{T}\) is its terminal velocity.

(a) Invert this equation to find how long it takes the ball to reach the ground in terms of its initial height \(h\).

(b) Check your result in the limits of small \(h\) and large \(h\). (For part (b) it is useful to know the infinite series expansions of the functions \(e^{x},(1+x)^{n}\), and \(\ln (1+x)\) for small \(x\).)

Data from Example 1.2

A river of width \(D\) flows uniformly at speed \(V\) relative to the shore. A swimmer swims always at speed \(2 V\) relative to the water.

• If the swimmer dives in from one shore and swims in a direction perpendicular to the shoreline in the reference frame of the flowing river, how long does it take her to reach the opposite shore, and how far downstream has she been swept relative to the shore?

• If instead she wants to swim to a point on the opposite shore directly across from her starting point, at what angle should she swim relative to the direction of the river flow, and how long would it take her to swim across?

Transcribed Image Text:

y=(/g) In(cosh(gt/vr)),