User:Roeles

Something for the speed-to-fly page later on.

Known variables: $$a,b,c,V_{thermal},S_glide$$ Unknown variable: $$V_{glide}$$

$$P(v) = av^2 + bv + c$$

$$t_{total} = t_{glide} + t_{thermal} $$

$$t_{glide} = \frac{S_{glide}}{v}$$

$$t_{thermal} = \frac{S_{thermal}}{V_{thermal}}$$

Vanaf hier moet ik de -P consequent doorvoeren. Je gaat omhoog in de thermiek, dus minus de polaire klim je. $$S_{thermal} = P(v) * t_{glide} = -P(v) * \frac{S_{glide}}{v}$$

$$t_{thermal} = \frac{-P(v) * \frac{S_{glide}}{v}}{V_{thermal}}$$

$$t_{total} = \frac{S_{glide}}{v} + \frac{-P(v) * \frac{S_{glide}}{v}}{V_{thermal}}$$

$$t_{total} = S_{glide} * (\frac{1}{v} + \frac{-P(v) * \frac{1}{v}}{V_{thermal}})$$

We want to find the minimum of this formula. Since $$S_{glide}$$ only acts as a scaling factor, we can eliminate it:

$$t_{total} = \frac{1}{v} + \frac{-P(v) * \frac{1}{v}}{V_{thermal}}$$

$$t_{total} = \frac{1}{v} + \frac{-P(v)}{v*V_{thermal}}$$

$$t_{total} = \frac{1}{v} + \frac{-(av^2 + bv + c)}{v*V_{thermal}}$$

$$t_{total} = \frac{V_{thermal}}{v*V_{thermal}} + \frac{-(av^2 + bv + c)}{v*V_{thermal}}$$

$$t_{total} = \frac{-(av^2 + bv + c) + V_{thermal}}{v*V_{thermal}}$$

$$t_{total} = \frac{-a}{V_{thermal}}*v + \frac{-b}{V_{thermal}} + \frac{-c+V_{thermal}}{v*V_{thermal}}$$

$$t_{total} = \frac{-a}{V_{thermal}}*v + \frac{-b}{V_{thermal}} + \frac{-c+V_{thermal}}{V_{thermal}} * \frac{1}{v}$$

$$t_{total}' = \frac{-a}{V_{thermal}} + \frac{-c+V_{thermal}}{V_{thermal}} * \frac{-1}{v^2}$$

$$\frac{-a}{V_{thermal}} + \frac{-c+V_{thermal}}{V_{thermal}} * \frac{-1}{v^2} = 0$$

$$\frac{-c+V_{thermal}}{V_{thermal}} * \frac{-1}{v^2} = \frac{a}{V_{thermal}}$$

$$\frac{-c+V_{thermal}}{V_{thermal}} * \frac{-1}{v^2} = \frac{a}{V_{thermal}}$$

$$\frac{-1}{v^2} = \frac{a}{V_{thermal}} * \frac{V_{thermal}}{-c+V_{thermal}}$$

$$\frac{1}{v^2} = \frac{a}{c-V_{thermal}}$$

$$v^2 = \frac{c-V_{thermal}}{a}$$

$$v = \sqrt{\frac{c-V_{thermal}}{a}}$$