James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

344 Chapter 4 Applications of Differentiation
© Targn Pleiades / Shutterstock.com
APPLIED Project
Planes and Birds: Minimizing Energy
Small birds like finches alternate between flapping their wings and keeping them folded while
gliding (see Figure 1). In this project we analyze this phenomenon and try to determine how
frequently a bird should flap its wings. Some of the principles are the same as for fixed-wing
aircraft and so we begin by considering how required power and energy depend on the speed of
airplanes. 1
FIGURE 1
1. The power needed to propel an airplane forward at velocity v is
P − Av 3 1 BL2
v
where A and B are positive constants specific to the particular aircraft and L is the lift, the
upward force supporting the weight of the plane. Find the speed that minimizes the required
power.
2. The speed found in Problem 1 minimizes power but a faster speed might use less fuel. The
energy needed to propel the airplane a unit distance is E − Pyv. At what speed is energy
minimized?
3. Hows much faster is the speed for minimum energy than the speed for minimum power?
4. In applying the equation of Problem 1 to bird flight we split the term Av 3 into two parts: A bv 3
for the bird’s body and A wv 3 for its wings. Let x be the fraction of flying time spent in flapping
mode. If m is the bird’s mass and all the lift occurs during flapping, then the lift is mtyx and
so the power needed during flapping is
P flap − sA b 1 A wdv 3 1 Bsmtyxd2
v
The power while wings are folded is P fold − A bv 3 . Show that the average power over an entire
flight cycle is
P − xP flap 1 s1 2 xdP fold − A bv 3 1 xA wv 3 1 Bm2 t 2
xv
5. For what value of x is the average power a minimum? What can you conclude if the bird flies
slowly? What can you conclude if the bird flies faster and faster?
6. The average energy over a cycle is E − Pyv. What value of x minimizes E?
1. Adapted from R. McNeill Alexander, Optima for Animals (Princeton, NJ: Princeton University
Press, 1996.)
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.