Energy and Distance

Power is rate of energy, $E$, expenditure per time, and so in a flight requiring $T$ time, $P = E/T$. The distance traveled for this energy expenditure at some speed $v$ is $D= v T$. The maximum distance at a given speed, $v$, for a given energy expenditure, $E_{given}$, will be

$$D = v T = E_{given} \frac{v}{P(v)}.$$

To maximize distance as a function of velocity, we must maximize the fraction $v/P$ or minimize the fraction $P(v)/v$. Taking derivatives in $v$,

$$\frac{d}{dv} \left( \frac{P(v)}{v} \right) = \frac{P^\prime (v)}{v} - \frac{P(v)}{v^2} \stackrel{\mbox{\tiny set}}{=} 0.$$

Thus, the speed at which maximum distance is covered satisfies the equation

$$P^\prime (v) = \frac{P(v)}{v}.$$

Graphically, this should be the point on the power curve at which the graph has the same slope as a line passing through the origin.

Of course, a bird has only a fixed amount of energy available for long-distance flight ($E_{avail}$). We can estimate a bird's total available energy using its `empty,' or lean, mass ($m_e$), fat fraction ($F$), energy density of fat ($e$) and the efficiency of converting fat to energy ($\eta$):

$$E_{avail} = F \ m_e \ e \ \eta.$$