Chapter 14.2, Problem 48E

Bird Migration Suppose a migrating bird flies at a velocity v, and suppose the amount of time the bird can fly depends on its velocity according to the function T(v). Source: A Concrete Approach to Mathematical Modelling.

(a)    If E is the bird’s initial energy, then the bird’s effective power is given by kE/T, where k is the fraction of the power that can be converted into mechanical energy. According to principles of aerodynamics,

      $\frac{kE}{T}=aS{v}^3+I,$

 where a is a constant, S is the wind speed, and I is the induced power, or rate of working against gravity. Using this result and the fact that distance is velocity multiplied by time, show that the distance that the bird can fly is given by

      $D\left(v\right)=\frac{kEv}{aS{v}^3+I}.$

(b)    Show that the migrating bird can fly a maximum distance by flying at a velocity

      $v={\left(\frac{I}{2aS}\right)}^{1/3}.$