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→Adjustments to the Analytical Polar: Headwind and Sinking Air
The problem is much simpler if the geometric method is used: to use the geometric method, imagine setting up a ground speed zero which is different from the airspeed zero. The polar is plotted with respect to the airspeed zero but the tangent ray needs to start from the ground speed zero. Because the ground zero is located in \( V>0\), the tangent is steeper and intersects the polar at a larger \( V \).
=== Effects of Sinking Air ===
If the glider is flying in some air that is sinking with a uniform downward speed \( V_{SA} \), then the polar equation should be adapted into the following form:
\[ V_S = \frac{\rho C_{D0}}{2 \omega} V^3 + \frac{2k \omega}{\pi A \rho} \frac{1}{V} + V_{SA} \]
Using the differentiation method to find the optimum airspeed for covering ground (notice that, because there is no headwind or tailwind, the indicated airspeed is equivalent to ground speed. This is not to say we approximate TAS with IAS, but there is a monotonic relationship between the two which is dictated by the altitude, which is a free variable in our problem.)
\[ \frac{d}{dV}(\frac{V_S}{V}) = 0 \]
\[ \frac{\rho C_{D0}}{\omega} V^4 - V_{SA} V - \frac{4k \omega}{\pi A \rho} = 0 \]
This equation has the solution of \( V=V_{BG} \) if \( V_{SA} = 0 \) as expected, but if \( V_{SA} > 0\), then the solution is \( V>V_{BG} \). The proof of this is left as an exercise for the reader.
=== Effects on the Minimum Sink ===
It should be obvious by now that the above adjustments to the polar have no effect on the minimum sink speed: the difference only arises when \( V \) is divided over to the left side, i.e. a glide ratio is sought after. Physically this makes sense: the minimum sink speed is purely an interaction between the glider and the surrounding air, and if we disregard all relativity to the ground, then the air in which the glider flies can move in whichever possible way (so long as it is not accelerating) and the glider can perform the same macroscopic motion with it without altering the detailed aerodynamics.
