I've done some research on the "POAT" question.
First, from various related discussions, a summary of the question and what I think are reasonable assumptions:
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There's no trickery going on, i.e. the pilot is not intentionally holding back to help prevent takeoff. One of the pages I'll be citing wastes a lot of words on this point.
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The brakes aren't being applied.
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The treadmill's reverse speed is based on the forward ground ("solid" ground, not treadmill ground) speed of the aircraft (including its wheels as seen from the wheel's centre), not on "the speed of the wheels" as very ambiguously stated in the question as we saw it on FB.
To understand why this would be a problem, consider that so long as the wheels are not slipping, the velocity of the bottom of each wheel will be exactly equal to the velocity of the treadmill surface, while the velocity of the top of the wheel will be (plane's forward ground speed + treadmill's backward speed), in the "forward" direction. That way lies madness.
The "state of the art" is represented by two publications, at least according to the claims of their respective authors:
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someone named "Chris" has somehow assumed a position of final authority on the question and authored this page: http://www.airplaneonatreadmill.com/ . There is some good discussion there, but it doesn't satisfy my criteria for convincing proof as the author seems to be relying more heavily on forceful assertion than on concrete evidence, including example numeric values and appropriate physical formulae.
Something for which I'm indebted to "Chris" is a blog commenter's obligatory objection that while no treadmill could be powerful enough to stop an airplane from starting, Chuck Norris certainly could.
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the Mythbusters have taken the question on experimentally (as they do) in one of their episodes and this comes down on the side of "the plane takes off." The author of airplaneonatreadmill.com has a broken YouTube link that probably used to point to this episode as well. In discussions including "our" FB post, some people have questioned the validity of the Mythbusters' experiment. I'm mildly concerned about how well matched the speeds were by the driver and pilot both operating by the seat of their pants, but if that's recognized as not terribly important then I'd say the 'busters successfully busted. I enjoyed the irony of their test pilot having been convinced they'd fail. It's actually (perhaps inadvertently) good scientific procedure to set up an experiment under conditions favorable to the less expected outcome.
The discussion on "our" FB post was necessarily among (mostly) laypeople, most of whom have only a smattering of knowledge on the relevant topics, which would include avionics and physics. You and I may thus be among the best qualified taking part in that particular discussion.
Because it's a popular thought experiment in physics, however, the question has been taken up by more qualified people, some of whose contributions are happily available online. Here are two sources which pre-select for at least a minimum amount of expertise:
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Discussion on physics.stackexchange.com
Stack Exchange is a forum for questions and answers on specialized topics; I am one of the more "senior" people in the software corner of this site, and the "physics" department features questions by people of various qualifications and answers from people self-selecting as interested and knowledgeable enough in physics to answer such questions. Many of them helpfully cite relevant degrees. Answers receive upvotes that give a vague indication of how helpful people found a given answer.
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Quora is a broadly general question and answer site. Again, people self-select as experts, again there is voting on answer quality. I used to respond to physics questions on Quora but fielded only the easier question and deferred to the opinions of better qualified experts.
I think I can safely summarize both of these sites as showing a strong concensus for "yes, it will take off." I find it a bit reassuring that at least some of the explanations there go into considerable depth, with examples, numbers and formulae that make the problem and its solution amenable to "real" analysis.
If you're not worried about getting bogged down in even more reading, I invite you to skim those discussions and the explanations given there. While none are as detailed as this here, they may augment the ideas I'm presenting.
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Discussion on XKCD Blog (includes some silly rants regarding interpretations of "wheels")
XKCD, in case you don't know it, is a comic popular among geeks drawn by (I think) a mathematician and touching on many topics of science, computing and mathematics. In my experience, the "XKCD guy" is smart and well educated. He's sick and tired of the discussion of POAT, but his gripe is mostly with people arguing against different possible interpretations of the question. With regard to the question in the form that you and I agreed on, he also supports "yes, it takes off."
The foregoing collection of links presents something of a mix between "argument from authority" and "argument from evidence and work shown in some degree of detail." The Mythbusters experiment is as close as we come to a "real" empirical result, with the caveat that their MO is science and stunt related entertainment rather than strictly scientific.
I'm going to try to pull the above together with an explanation based on my own basic expertise in physics. To compensate for my lack of credentials I'll back myself up with other sources.
My own approach is based on physics, which is the science dealing (in a subdiscipline called kinematics) with why and how bodies move. In a sense, of course, your contributions from avionics on airflow, lift and drag are also based on principles of physics.
We had already agreed that the plane needed to move forward relative to air (and ground, i.e. resting ground, not treadmill ground) in order to lift off. The point of contention was that you insisted that a plane standing on a backward-moving treadmill couldn't be expected to move forward.
We had discussed a couple of analogies to make the question more approachable. You augmented my unpowered hypothetical trundle-cart with a fan - and I agree, intuition doesn't give one a sense that such a contraption would make headway against a backward-moving treadmill. I tried to help that intuition by giving the fan something more solid to push against - yielding either a boat on wheels or a pushcart with a proper boat propeller (screw?) rather than a fan.
The question of where the plane moves is answered authoritatively by Newton's second law: The plane, like any object, will move exactly where the net sum of forces acting on it ends up pointing, accelerating at a rate proportional to the magnitude of that force.
You had agreed that the plane's engines push not on the surface under the wheels (as a car would work) but on either the air behind the plane (certainly this will be the case for a propeller craft) or on a bunch of hot gases used as reaction mass in a jet or rocket. We know this force is substantial: in the "normal" case, there's enough force to accelerate the plane from a standstill to something like 60 mph within the length of the runway while overcoming both the plane's air resistance (a combination of unavoidable structural drag and the drag intentionally created by the wings to provide lift) and the rolling resistance of the plane's wheels. Just for the sake of having numbers to throw around, a 747-100's engines during takeoff provide a force of 4*208 kN = 832 kN = 187,000 lbf for you Imperial types (from www.flugzeuginfo.net; Wikipedia had a lot of specs scattered around in its text but not a compact summary for one particular type).
In order for the plane not to move forward, something would have to be pulling the plane backward with the same force. So says Newton! I asked you what would be doing the back-pulling. You answered: Air resistance and wheel resistance. I suggested discounting air resistance but I take that back: Air resistance is actually an important part of what's going on.
We both forgot a very important factor: inertia. A major part of the engine's work is that of accelerating the impressive mass of the plane. 747-100: 710,000 lbs MTOW. Pencil check: 187,000 lbf applied to 710,000 lbs gives a thrust-to-fuel ratio of 0.26, the same ballpark as the Airbus 380. Absent any of the other forces, this would accelerate the plane at 0.26 g. A noticeable but not uncomfortable push-into-the-seats for the passengers. But then, a 747 is one of the more ponderous of aircraft. The much smaller Cessna Citation CJ4 has a ratio of 0.426 . The F-16, Harrier and Typhoon boast ratios around 1.1, and that's without the use of afterburners.
We can ballpark how much of the available power is going into acceleration: Apparently the take-off speed for a 747 (probably a different model, I'm having trouble finding comparable specs for a single model) is 180 mph = 290 km/s = 80.6 m/s. Given an acceleration of 0.26 * 9.81 = 2.55 m/s^2 and plugging speed and acceleration into the formula d = v^2 / 2a, we should reach takeoff speed after accelerating for 1273 meters. Meanwhile, Narita Airport's Aircraft Profile for the 747-100 says that the required Take Off Distance is 3050 m. We know this includes various safety margins and obviously not all of the engine power translates to acceleration - we have non-negligible friction, of course. I just wanted to be sure to be looking at plausible numbers! Yesterday I thought that my quoted figure of 190 kN was the total thrust for the plane, when it's actually the figure for just 1 of 4 engines.
So anyway, to keep the plane from pushing itself forward you need to pull it back with a force matching the engines' thrust. If you apply the brakes then you've (for practical purposes) attached the plane (via its grippy tires) to the tarmac, the treadmill or whatever, and the craft will be forced to match the speed of said tarmac or treadmill. On the other hand, if you release the brakes then the wheels will do what wheels were made for: provide solid support in the vertical direction while providing negligible force in the horizontal roll direction.
The author of the planeonatreadmill blog gives an example I wish I had thunk up: A skateboard. I tried skateboarding at the uni and experienced firsthand the effect that a skateboard is extremely "slippery" in the direction of roll. It was probably a good thing that my skateboard ended up being rolled over by a truck - while I wasn't on it! Wheels do an excellent job of disconnecting a skateboard from the ground as far as horizontal roll direction (in either direction) is concerned. Good rubber wheels are as good as or better than locked brakes when it comes to "sideways" movement. But in the roll direction? All you have are the tiny amounts of friction the engineers couldn't design out of the wheels and bearings.
The blogger's useful analogy substitutes a powerful leaf blower for your fan. It's not hard to imagine standing on a skateboard with a leaf blower, switching it on, and seeing oneself starting to roll.
Here's the important bit to take note of: Wind resistance and the force against acceleration are the same on our hypothetical treadmill as for a conventional takeoff. The only means the treadmill has of exerting a backward force on the plane is via the wheels, and if that's in the roll direction, only the wheel friction is a candidate for transmitting that force.
How much force is that? I'll provide some substantial real-world numbers in a moment, but before I do I'd like to point you at the Guiness Book of World Records, in which one burly Canadian serviceman pulls a 188 ton CC-177 Globemaster cargo craft over a distance of 28 feet. In other news, a professional strongman named Shaw has dragged a C-130 for hundreds of yards. How much force does that require? I'm not sure, but the record for bench pressing, in which a person applies maximum force in a less awkward direction, is a bit above 327 kg, or about 3200 Newtons. Given a clean smooth runway, it seems that much force is sufficient to overcome airplane tire and axle friction (albeit for a plane of about 1/4 the mass of a 747, so let's say 13,000N for the big one). How does this compare with the 800,000+ N the engines are able to put out? Simple - at 1.6%, it's negligible.
Now this human-pulling is taking place at rest and very low speeds. We know that wind resistance increases sharply with speed. How about rolling resistance? planeonatreadmill claims that rolling friction doesn't increase with speed. I was a bit skeptical of that (unsubstantiated) claim but I was fortunate to find a site that discusses friction in great detail: http://www.tribology-abc.com/abc/cof.htm . There's a section on rolling resistance and they even obligingly provide a little calculator applet where you can plug numbers into their formulae. Their sample inputs are for a bicycle, but I set the weight of the bike to 400,000 kg and gave it the best automobile tires (Cr = 0.01). For air resistance I looked up the 747's drag coefficient (0.3) and wild-guessed its frontal surface area at 100 square meters. I ended up with a plausible rolling resistance, at 1 km/h, of 39244 N - just 3 times my estimate for the human puller data. Presumably, airplane tires have better friction coefficients than car tires. At this low speed, air resistance is a paltry 1.5 N. When I go to 10 km/h, air resistance increases sharply to 149 N; at takeoff speed, air resistance is 125,000 N, a significant and plausible fraction of the available force of 800,000 N. But significantly, rolling resistance through all speeds stayed at 39 kN!
What this means: No matter how quickly the plane is moving forward / the treadmill is moving backward, the backward force from rolling resistance remains roughly between 1.6-5% of the forward force and negligible.
I'm going to give a nod to other effects that would come into play at significantly higher accelerations and speeds (and outside the parameters of the original question), but those are completely unrealistic. Realistically, there's no way a surface moving in either roll direction under the starting plane can have a significant effect on its takeoff, much less prevent said takeoff.