The last of the aircraft relationships are done, and a mystery about where 0.866 comes from is finally resolved.

Aircraft relationships completed

The last of the aircraft relationships have been dealt with today, which are cross-checks to help ensure that no mistakes have occurred in other equations.

I’ve also found out that WordPress has LaTeX support, so all sorts of LaTeX math symbols can be used, so get ready for some math.

Relation 14

(L/D)_{max} = 101.6 \frac{V_{minS}}{R_{S,min}}

Relation 15

C_{L,minS} = 3.07 \sqrt{eAR\cdot C_{D,0}}

(L/D)_{max} = 0.886 \sqrt{\frac{eAR}{C_{D,0}}}

Relation 16

(L/D)_{max} = \frac{W}{D_{min}}

\theta_{g_{min}} = \frac{180}{\pi}\frac{1}{(L/D)_{max}}

Solving a mystery

I’ve been puzzled for several days though about where the value 0.866 or 0.886 comes from. There have been several potential sources, such as:

But none of these were satisfactory, because they didn’t explain the origin of the figure.

Start with what we know

We are given a couple of ratios, but aren’t told much more than that.

(L/D)_{minS} = \frac{5280}{60}\frac{V_{minS}}{R_{S,min}}

(L/D)_{minS} = 0.866 (L/D)_{max}
Where the formulas for these different parts are:

V_{minS} = \frac{\sqrt{391}}{(3\pi)^{^1/_4}}\frac{\sqrt{^W/_{b_e}}}{\sqrt{\sigma}{A_D}^{^1/_4}} \text{mph}
R_{S,min} = \frac{5280}{60}\sqrt{391}\frac{4}{(3\pi)^{^3/_4}}\sqrt{\frac{W}{\sigma}}\frac{{A_D}^{^1/_4}}{{b_e}^{3/_2}} \text{feet/min}
(L/D)_{max} = \frac{\sqrt{\pi}}{2}\frac{b_e}{\sqrt{A_D}}

But why it is 0.866? Let’s figure out:

Combine and simplify

\text{Solving for} (L/D)_{minS}:
(L/D)_{minS} = \frac{5280}{60}\frac{\sqrt{391}}{(3\pi)^{^1/_4}}\frac{\sqrt{^W}}{\sqrt{b_e}\sqrt{\sigma}{A_D}^{^1/_4}} \times \frac{60}{5280}\frac{1}{\sqrt{391}}\frac{(3\pi)^{^3/_4}}{4}\frac{\sqrt{\sigma}}{\sqrt{W}}\frac{{b_e}^{3/_2}}{{A_D}^{^1/_4}}
bring together common terms:
(L/D)_{minS} = \frac{5280}{5280}\frac{60}{60}\frac{1}{4}\frac{\sqrt{391}}{\sqrt{391}}\frac{(3\pi)^{^3/_4}}{(3\pi)^{^1/_4}}\frac{\sqrt{^W}\sqrt{\sigma}}{\sqrt{W}\sqrt{\sigma}} \times \frac{b_e \sqrt{b_e}}{\sqrt{b_e}}\frac{1}{{A_D}^{^1/_4}{A_D}^{^1/_4}}
and simplify:
(L/D)_{minS} = \frac{\sqrt{3\pi}}{4}\frac{b_e}{\sqrt{{A_D}}}

Now that we have that, we can solve for the difference between them.

x = \frac{(L/D)_{minS}}{(L/D)_{max}}

x = \frac{\sqrt{3\pi}}{4}\frac{b_e}{\sqrt{{A_D}}} \times \frac{2}{\sqrt{\pi}}\frac{\sqrt{A_D}}{b_e}

bring together common terms:
x = \frac{2\sqrt{3}}{4}\frac{\sqrt{\pi}}{\sqrt{\pi}}\frac{b_e}{b_e}\frac{\sqrt{A_D}}{\sqrt{{A_D}}}

The mystery is solved

Simplifying give is:
x = \frac{\sqrt{3}}{2} = 0.8660254037844386
which is where the original 0.866 comes from.

Advertisements