(a) Show that y = tanh t satisfies the differential equation dy/dt = 1 − y² with initial condition y(0) = 0.
(b) Show that for arbitrary constants A, B, the function

y = A tanh(Bt)

satisfies

(c) Let v(t) be the velocity of a falling object of mass m. For large velocities, air resistance is proportional to the square of velocity v(t)². If we choose coordinates so that v(t) > 0 for a falling object, then by Newton’s Law of Motion, there is a constant k > 0 such that

Solve for v(t) by applying the result of (b) with A = √gm/k and B = √gk/m.

(d) Calculate the terminal velocity

(e) Find k if m = 150 pounds and the terminal velocity is 100 miles per hour.

Transcribed Image Text:

dy dt AB - B y(0) = 0