User:Roeles

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

Known variables: $a,b,c,V_{thermal},S_{g}lide$ 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:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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