MacCready Speed to Fly Derivation

2024-12-28

Brief MacCready Solution Derivation

Here’s a brief derivation of MacCready speed to fly theory. See The Insight of MacCready Speed to Fly for the bigger picture.

$Minimize: t_{total}(V_{Horizontal}) = t_{Flying} + t_{Climbing}$

$t_{Flying}=D_{Horizontal}/V_{horizontal}$

$t_{Climbing}=\frac{Alt_{Lost}}{V_{Climb}}$

$D_{horizontal}$ : distance between climbs

$V_{horizontal}$ : horizontal velocity

$V_{climb}$ : expected vertical velocity of the next climb

$Alt_{Lost}$ is the altitude lost flying between the thermals, based on our horizontal speed:

$Alt_{Lost}=t_{Flying}*SinkRate(V_{horizontal})$

Substituting everything back into the objective and doing some simplification:

$Min: t_{total} = t_{Flying} + \frac{t_{Flying}*SinkRate(V_{horizontal)}}{V_{Climb}}$

$Min: t_{total} = t_{Flying} (1 + \frac{ SinkRate(V_{horizontal})}{ V_{Climb}})$

$Min: t_{total} = \frac{D_{Horizontal}}{V_{Climb}} * ( \frac{V_{Climb} + SinkRate(V_{horizontal})}{V_{horizontal}})$

$D_{horizontal}$ and $V_{Climb}$ are constants, so they can be factored out, yielding:

$Min: t_{total} = \frac{V_{Climb} + SinkRate(V_{horizontal})}{V_{horizontal}}$

We can solve this by differentiating with respect to $V_{horizontal}$ and finding the intercept:

$0 = (\frac{V_{Climb} + SinkRate(V_{horizontal})}{V_{horizontal}})\frac{d}{dV_{Horizontal}}$

$0 = \frac{SinkRate’(V_{Horizontal})}{V_{Horizontal}} - \frac{V_{Climb} + SinkRate(V_{horizontal})}{V_{horizontal}^2}$

$\frac{SinkRate’(V_{Horizontal})}{V_{Horizontal}} = \frac{V_{Climb} + SinkRate(V_{horizontal})}{V_{horizontal}^2}$

$V_{Climb} + SinkRate(V_{horizontal}) = V_{Horizontal} * SinkRate’(V_{Horizontal})$

This is equivalent to MacCready’s equation, first published in Soaring Magazine, Mar-Apr 1954 issue.

This is likely best interpreted visually on the polar graph: