The Polar

We all have seen a graph like Fig. 1, called a “polar”, that describes the air speed vs. sink rate of a glider. You can find them in the aircraft operator’s manual. This one is for a fictional sailplane that may have a best L/D in the low to mid 40kt range, a minimum sink in the low 30kt range, and a very low stall speed of 23kts.

The problem that I wanted to solve was how to come up with an equation that would allow me to find the minimum sink rate and the best speed to fly. The method I settled on was to use vectors and vector calculus to quantify the polar. If you would like to brush up on vectors read this.

If you define v(t) as a vector that extends from (0,0) to the curve it would be composed of two component vectors, x(t)i and y(t)j. The two factors i and j are unit vectors whose magnitudes are 1. That unit magnitude is multiplied by x(t) and y(t) to get the actual magnitude of the component vector. OK, go back and read the link at the end of the last paragraph.

Got it? Fig. 2 below is Fig. 1 with the vectors drawn in:

In this case x(t) is around 27kts and y(t) 1.7kts – those would be the values you multiply with the unit vectors i and j. The magnitude of v(t) can be calculated using the Pythagorean  Theorm:

With a little imagination you can visualize moving the point of v(t) back and forth along the curve. If you move it to the right far enough you’ll encounter a very well known (to sailplane pilots) situation. With v(t) anchored at (0,0) and moved far enough to the right along the curve, eventually it will be tangent to the curve and x(t)i will point at your best speed to fly.

“This is all good and maybe even marginally interesting.”, I hear you cry. “But what does it have to do with finding the best L/D and minimum sink?” Well, stay tuned for the next exciting chapter where I’ll determine just what x(t) and y(t) are.

Next: The Equation

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