= Minimum-power speed
{title2=$V_{\rm MP}$}
The \b[minimum-power speed] $V_{\rm MP}$ minimizes the mechanical propulsive <power> required to sustain the specified flight state. For steady level flight with <drag> $D(V)$, the required power is $P(V)=D(V)V$, using speed through the air. Minimizing this product is a different problem from finding the <minimum-drag speed>.
With the fixed-coefficient <parabolic drag polar>, write
$$
D(V)=aV^2+\frac b{V^2},\qquad
P(V)=aV^3+\frac bV,\qquad a,b>0.
$$
Differentiation gives
$$
P'(V)=3aV^2-\frac b{V^2},\qquad
P''(V)=6aV+\frac{2b}{V^3}>0.
$$
Thus the stationary point is the unique global minimum for positive speed:
$$
\boxed{V_{\rm MP}=\left(\frac{b}{3a}\right)^{1/4}
=3^{-1/4}V_{\rm MD}\simeq0.7598V_{\rm MD}.}
$$
At this point, $b/V^2=3aV^2$: the decreasing drag contribution is three times the increasing contribution, rather than equal to it. In lift-coefficient form, $C_L^{\rm MP}=\sqrt{3C_{D0}/k}$.
For an original numerical model with $a=0.20$ and $b=512{,}000$ in consistent metre-second-newton units, $V_{\rm MD}=40$ m/s and $V_{\rm MP}\simeq30.393$ m/s. The power minimum is about 22.461 kW, with drag about 739.01 N. At the drag minimum, drag is only 640 N but power is 25.600 kW. The lower power comes from the combination of force and speed, despite the greater force.
In shallow steady unpowered descent, the gravitational energy-loss rate approximately balances $DV$, so minimizing required power relates to <minimum sink speed> when the same assumptions hold. This is consistent with the FAA's distinct minimum-sink and best-glide points.<Glider Flying Handbook by Federal Aviation Administration>[Chapter 5, Glider Polars, Figure 5-11]{ref} A changing <propulsive efficiency> changes the input-power objective: if $P_{\rm input}=P/\eta_p(V)$, a stationary point satisfies $P'/P=\eta_p'/\eta_p$, not generally $P'=0$.
Attainability must also be checked. The model's optimal lift coefficient can exceed the allowed range before <aerodynamic stall>. The formula concerns the stated required mechanical power, not a universal minimum of fuel consumption or an operational speed recommendation.
\Image[/minimum-drag-and-power-speeds.png]
{title=Different minimum-drag and minimum-power speeds in a constructed parabolic-polar model}
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