akclimber wrote:[...
I am trying to apply different equations to come up with a proof of why that is and it's still not penciling out.
I do have my own theory, but can't prove it mathematically (hence my question here), and it's too cold outside to prove it experimentally.
Well, it may help to consider what other documentation might exist out there for similar comparisons.
For example, Part 25 airplanes have much better performance documentation, and ...as this thread is supposed to be about clearing an obstacle.... not particularly about the drag of rolling wheels in muddy turf of undefined depth or flap-induced drag during the takeoff-run....you might consider what AFM recommendations are made with those airplanes. The same principles apply. Their take-off/climb performance is divided into segments.
Back to definitions: the takeoff run is from brake release to 35' in those types...and is called the 1st segment, and the climb to the "close-in" types obstacles they usually deal with (such as tall bldgs, hills, radio towers, etc..) are usually larger than 35'....and farther than the airport boundary...and are addressed in second-segment data.
In high density alitude conditions, the data shows that aircraft can clear those obstacles better if their flap settings are reduced from normal...sometimes to zero. (At the beginning of the takeoff roll, of course.) This is because their second segment (which is the climb to 1500' AGL) considers the loss of an engine at/near rotation, and the fact that deployed flaps hurt climb performance. The more flaps deployed...the more drag suffered....the less climb-angle experienced. I.E., the aircraft is not accelerated to a higher speed and "zoomed" to clear the obstacle because the distance used to "zoom" brings you closer to the obstacle. The obstacle (a real obstacle is not one with room-to-zoom)....is always presumed to be close enough to require maximum effort until cleared. This means one cannot waste distance to accelerate/zoom. Nor can one accept the loss of climb performance experienced due to flap-retraction. (Regardless of what some light plane pilots subjectively believe, retracting flaps damages climb performance to the point of the retraction-completion....as the airplane must be accelerated to a speed above the lesser-flap-stall-margins and one appropriate to continue the climb...which is energy being used to accelerate...instead of climb.)
The penalty of that Pt 25 operation is increased takeoff run. (Lesser flaps means higher stall speeds, etc., and therefore the airplane must be accelerated to a higher speed before flight is attempted at Vr (rotation). Flaps are never retracted until obstacles in the second segment are cleared, to avoid those performance losses.
So, the problem of "take off to clear obstacle" becomes one that involves a comparision between distance available for TakeOff, and what flap setting is necessary to reduce takeoff run to within the runway available....and can we clear that obstacle with the drag of that flap setting? The same principles apply to our airplanes because the same dynamics apply (we just don't have the data available to calculate climb performance after the loss of thrust at/near rotation).
The pilots of Pt 25 airplanes do the same thing light plane pilots do to improve performance to get off runways and clear obstacles. They reduce takeoff weight. The advantage they have over light plane operators is they have the documented data to easily calculate that weight, that distance, that flap setting. LIght plane pilots are limited to the simplified (cheaper) data in their AFMs which (usually) only give one weight: Gross weight.
Flaps are not retracted until obstacles are cleared in either type because flap retraction hurts climb-to-obstacle in all types...Pt 25, Pt 23, CAR 4, CAR 3. So does zoom-to-climb, etc etc. Some things are the same for all types.
As for the takeoff run in light planes: "Popping" flaps avoids the induced drag of the flaps during acceleration. But once deployed, the same principles apply....retracting them again hurts the climb. Either you were too late deploying them (as they developed lift during acceleration they'd have gradually/constantly lifted the airplane out of the mud sooner and thereby continuously reduced drag during the takeoff run and therefore reduced that run-distance...also meaning you'd have started the obstacle-climb segment sooner) or you were too early retracting them prior to the obstacle and therefore cleared the obstacle with too much speed. (I.E. you could have carried more weight...you left the wife and kids ....or, more importantly,...the ice and beer...behind unnecessarily.)
