= Minimum sink speed
The \b[minimum sink speed] is the <airspeed> at which a <glider (aircraft)> has its smallest steady downward speed relative to the air, for a specified configuration, loading and flight condition. It maximizes time aloft from a given height in still air. It does not generally maximize horizontal range, which defines <best glide speed>.<Glider Flying Handbook by Federal Aviation Administration>[Chapter 5, printed p. 5-10, Glider Polars and Figure 5-11]{ref}
For a differentiable <glider sink polar> $w(V)>0$, an interior minimum satisfies $w'(V)=0$, with a positive second <derivative> sufficient for a strict local minimum. For the original shallow-glide model developed in <best glide speed>,
$$
w(V)=\frac{AV^3+B/V}{W},\qquad
w'(V)=\frac{3AV^2-B/V^2}{W}.
$$
With $A,B>0$, this gives
$$
\boxed{V_{\rm MS}=\left(\frac{B}{3A}\right)^{1/4}=3^{-1/4}V_{\rm BG}.}
$$
Indeed $w''=(6AV+2B/V^3)/W>0$ for $V>0$. The speed ratio is about $0.7598$, within this stipulated <parabolic drag polar> approximation. A real polar need not follow that numerical ratio, especially near <aerodynamic stall>.
Using the same original $W=3000\,\mathrm N$, $A=0.12$ and $B=67500$ gives \b[$V_{\rm MS}\simeq20.81\,\mathrm{m/s}$ and $w_{\min}\simeq1.442\,\mathrm{m/s}$]. From a stipulated $1000\,\mathrm m$ height, the model's times are about $693.6\,\mathrm s$ at minimum sink and $608.6\,\mathrm s$ at best glide. Yet the approximate range ratios are $14.43$ and $16.67$, respectively. The slower optimum buys time rather than maximum distance.
A horizontal wind changes range without changing air-relative sink at a fixed condition. Vertical air motion changes ground-relative height loss as well: descending at $w$ through air rising at $u_z$ gives ground-relative descent $w-u_z$. Observing constant height therefore does not establish zero air-relative sink. A <bank angle> also changes the required <lift> and hence the appropriate polar; the straight-glide optimum cannot simply be reused for a tight turn.
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