In respect to post #17, what is minimized is the energy required to travel a given distance, not power (which is a rate of energy production). The article does find the speed that minimizes the sum of both the energy required to keep the drone aloft while covering this distance and the energy required to move the drone horizontally. The distance chosen is 1 metre but the same result is true for any distance. The result looks plausible to me.
Energy per unit distance and power per unit speed are the same quantity. Looking again the author did attempt to minimize that quantity, but with the function (45/v + 0.0235v
²). That's a simplification that calculates the total power as the sum of the powers of two independent horizontal and vertical motor thrusts. That's not correct because the power dissipated by the motors goes is proportional to total motor thrust raised to the power of 3/2, not to the linear sum of the vertical and horizontal components of thrust raised to that power.
Using my nomenclature, as previously, the correct form for total power, combining the force required to keep the aircraft up (its weight) and the force required to overcome drag as a result of horizontal velocity (equation [4] derived in post #6) is:
P = (M²g² + k₁²u⁴)^(3/4)/√k₂
JSR's form for power, expressed in similar terms but with lift and drag as independent, is:
P = (Mg)^(3/2)/√k₂ + k₁u²
Using the values from JSR's article for consistency,
k₁ = 0.0235 and
k₂ = 0.878. Plotting those functions against velocity:
Those result the same power at hover - as expected - but a quite different relationship between power and velocity as the aircraft moves. The much more rapid increase in power predicted by the oversimplification of the motor thrust into two separate forces and then adding the powers is the reason for JSR's optimal velocity for distance being much lower. Plotting that explicitly in terms of velocity per unit power (distance per unit energy) shows the two different maxima of around 10 m/s with JSR's method and nearer 20 m/s with my combined derivation.
Conclusion - JSR's method seems incorrect - you cannot separate forces and then independently calculate powers and add them. That is also consistent with the observation that the larger value for optimal velocity is consistent with observed results. Please critique if you spot any errors.