= Best glide speed
{title2=$V_{\rm BG}$}
The \b[best glide speed] is the <airspeed> giving the greatest horizontal distance per unit height lost in a specified steady glide condition. In still air this is the speed for <maximum lift-to-drag ratio>. It differs from the <minimum sink speed>, whose objective is longest time aloft for a given height loss.<Glider Flying Handbook by Federal Aviation Administration>[Chapter 5, printed p. 5-10, Glider Polars and Figure 5-11]{ref}
Write a <glider sink polar> as downward rate $w(V)>0$. The horizontal air-relative speed is $u(V)=\sqrt{V^2-w(V)^2}$, so the exact still-air range per unit height is $u/w$. For a shallow <glide angle>, $u\simeq V$ and maximizing range is approximately minimizing $w/V$. At an interior optimum,
$$
\frac{d}{dV}\frac{w(V)}V=0\quad\Longrightarrow\quad Vw'(V)=w(V).
$$
Geometrically, a line from the origin is tangent to the positive-downward-rate polar. This is not the horizontal tangent defining <minimum sink speed>.
For an original calculation, take a <parabolic drag polar> $C_D=C_{D0}+kC_L^2$ and approximate $L\simeq W$. At fixed air <mass density> $\rho$, <wing area> $S$ and <gravitational weight> $W$,
$$
D(V)=AV^2+\frac B{V^2},\qquad A=\frac12\rho SC_{D0},\quad B=\frac{2kW^2}{\rho S},\qquad w(V)\simeq\frac{D(V)V}{W}.
$$
Minimizing $w/V\simeq D/W$ gives
$$
\boxed{V_{\rm BG}=\left(\frac BA\right)^{1/4}.}
$$
Stipulate $W=3000\,\mathrm N$, $S=10\,\mathrm{m^2}$, $\rho=1.2\,\mathrm{kg/m^3}$, $C_{D0}=0.020$ and $k=0.045$. Then $A=0.12$, $B=67500$ in the corresponding SI units, \b[$V_{\rm BG}\simeq27.39\,\mathrm{m/s}$], and the model gives $w\simeq1.643\,\mathrm{m/s}$. Its shallow-glide range ratio is $V/w\simeq16.67$. These parameters are invented; an attainable <lift coefficient> and a valid <drag polar> are prerequisites, not consequences of the optimization.
\Image[/glider-sink-polar-optima.png]
{title=Minimum sink and best glide are different optima of an original shallow-glide polar. Positive vertical values denote downward speed; the curve is an invented model, not measured aircraft performance.}
Wind changes the ground-range objective to $(u+U_w)/w$ for an along-track wind component $U_w$ and no vertical air motion. Therefore the still-air optimum need not maximize ground distance. The FAA constructs the corresponding shifted tangents on its performance polars.<Glider Flying Handbook by Federal Aviation Administration>[Chapter 5, printed pp. 5-10–5-11, wind and sinking-air polar constructions]{ref} A controlled performance measurement records steady <airspeed> and height loss at several speeds; wind, vertical air motion and changes of loading must be distinguished from the polar itself.
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