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The difference between mine and your approach is that I calculate the changed induced drag due to weight change (which means that Cl/AoA change is accounted) while the formula you use, assumes constant Cl/AoA which is a wrong approach for this particular case.
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Of course we are holding CL constant. We are demonstrating the affect of a weight change. If the aircraft does not have that quantum increase in power available, then it must reduce performance to the point Pr = Pa.
A 50% increase in weight represents an 83% increase in power required at the same CL. That is far from insignificant.
It is not a very large change in velocity but it represents a very significant reduction in the aircrafts envelope. For Gods sake, fly a plane with and without passengers to look at your performance or one with 1/2 fuel as opposed to full tanks.
All the best,
Crumpp