= Lift-to-drag ratio
{title2=$L/D$}
{wiki}
The \b[lift-to-drag ratio] $L/D$ compares the <lift> supporting an <aircraft> with the <drag> opposing its motion. For $L>0$ and $D>0$, a larger ratio means less <drag> for a given <lift>. If both coefficients use the same <dynamic pressure> $q$ and reference <wing area> $S$, then $L/D=C_L/C_D$. Ratios using mismatched reference areas do not have this interpretation.
A <parabolic drag polar> provides a useful worked model:
$$
C_D=C_{D0}+kC_L^2,\qquad C_{D0}>0,\quad k>0.
$$
Here $C_{D0}$ is the <zero-lift drag coefficient>. In a simple finite-wing model, $k=1/(\pi e\,AR)$ relates <lift-induced drag> to <wing aspect ratio> and a <span efficiency factor>.<Induced Drag Coefficient by NASA Glenn Research Center>[Induced Drag Coefficient and total drag decomposition]{ref} A fitted whole-aircraft $k$ can also incorporate additional effects; it need not be determined by geometry alone.
The <maximum lift-to-drag ratio> follows by <differentiation>:
$$
\frac{d}{dC_L}\frac{C_L}{C_{D0}+kC_L^2}=\frac{C_{D0}-kC_L^2}{(C_{D0}+kC_L^2)^2},\qquad
\boxed{C_L^*=\sqrt{\frac{C_{D0}}{k}},\quad (L/D)_{\max}=\frac1{2\sqrt{C_{D0}k}}.}
$$
At this point the two modeled contributions to <drag coefficient> are equal. With the stipulated $C_{D0}=0.020$, $e=0.8$ and $AR=8$, $k\simeq0.049736$, $C_L^*\simeq0.6341$ and \b[$(L/D)_{\max}\simeq15.85$]. The optimum is usable only if the actual wing can attain that <lift coefficient> in the regime where the model is valid; <aerodynamic stall> can intervene.
For steady unpowered descent in still air, the <force balance> gives $\tan\gamma=D/L$, where $\gamma$ is the <glide angle>. Thus the stipulated optimum corresponds to $\gamma\simeq3.61\,{}^\circ$ and about $15.85$ units of horizontal travel per unit of height lost.<Glide Angle by NASA Glenn Research Center>[Flight Path and Tangent Angle]{ref} Wind and changing flight conditions alter travel over the ground. An <airfoil> section's ratio also should not be substituted for that of an entire <aircraft>, which includes other sources of <drag>.
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