= Parabolic drag polar
{title2=$C_D=C_{D0}+kC_L^2$}
A \b[parabolic drag polar] models <drag coefficient> as
$$
C_D=C_{D0}+kC_L^2,\qquad C_{D0}>0,\quad k>0.
$$
The <zero-lift drag coefficient> supplies the intercept. In a simple finite-wing approximation, $k$ can represent <lift-induced drag>; fitted whole-aircraft values can include additional lift-dependent losses. The model is conditioned on a configuration and operating regime, rather than universally valid through <aerodynamic stall>.
Its dimensional form explains a useful speed tradeoff. In steady level flight let <lift> equal <gravitational weight> $W$, with fixed <mass density> $\rho$ and <wing area> $S$. Substitution of $C_L=W/(qS)$ and $q=\rho V^2/2$ yields
$$
D(V)=aV^2+\frac{b}{V^2},\qquad a=\frac12\rho S C_{D0},\quad b=\frac{2kW^2}{\rho S}.
$$
The first contribution grows with <speed>; the second decreases as the required <lift coefficient> falls. <Differentiation> gives the <minimum-drag speed> $V_{\rm MD}=(b/a)^{1/4}$.
Minimizing required propulsive <power> instead gives
$$
P(V)=D(V)V=aV^3+\frac bV,\qquad
\boxed{V_{\rm MP}=3^{-1/4}V_{\rm MD}\simeq0.7598V_{\rm MD}.}
$$
The <minimum-power speed> is therefore different from the <minimum-drag speed> in this model. For stipulated $\rho=1.2\,\mathrm{kg/m^3}$, $S=20\,\mathrm{m^2}$, $W=10000\,\mathrm N$, $C_{D0}=0.020$ and $k=0.050$, the speeds are about $36.30$ and $27.58\,\mathrm{m/s}$, respectively.
These are idealized level-flight comparisons of required propulsive power. Engine efficiency, allowable <lift coefficients> and a <speed>-dependent polar can move an operational optimum. Relating the result to <minimum sink speed> in unpowered flight requires the additional shallow-glide approximation; it does not identify minimum fuel flow for an arbitrary engine.
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