Ornithologists have found that the power (in joules per second) consumed by a certain pigeon flying at velocity v m/s is described well by the function P(v) = 17v⁻¹ + 10⁻³v³ joules/s. Assume that the pigeon can store 5 × 104 joules of usable energy as body fat.
(a) Show that at velocity v, a pigeon can fly a total distance of D(v) = (5 × 10⁴)v/P(v) if it uses all of its stored energy.
(b) Find the velocity vp that minimizes P.
(c) Migrating birds are smart enough to fly at the velocity that maximizes distance traveled rather than minimizes power consumption. Show that the velocity v_d which maximizes D(v) satisfies P'(vd) = P(vd)/v_d. Show that v_d is obtained graphically as the velocity coordinate of the point where a line through the origin is tangent to the graph of P (Figure
34).

(d) Find v_d and the maximum distance D(v_d).

Transcribed Image Text:

4 Power (joules/s) Minimum power consumption 5 1 Maximum distance traveled 10 Velocity (m/s) 15