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