See, it isn't a clear answer at all. That's part of why it's fun. ;^) It's such an absurd theoretical example that it's hard to put all the pieces together.
By the way, I'm cheating by way of things other people already looked up in the discussion, so I claim no credit or victory.
Too much of this boils down to the treadmill's goal. "Matching the speed of the plane" is silly when the plane's force is applied to itself, not the ground. It's not pushing off the air, but it's not pushing off the ground, either. To simplify trying to define things that happen in response to each other simultaneously, I'm operating under the assumption that the treadmill follows a program similar to the following:
10 TV = CURRENT_TREADMILL_VELOCITY()
20 WV = CURRENT_WHEEL_VELOCITY()
30 IF (TV > WV) THEN TV++
40 GOTO 10
Of course, the treadmill makes the measurements and necessary speed adjustments thousands of times per second or more. It's going to need to.
This link helps a bit to visualize the "rollerblade" theory:
http://hyperphysics.phy-astr.gsu.edu/hbase/frict2.htmlLike I said, if the treadmill our rollerblader is on goes very slowly, the wheels won't move at all. But once you hit a certain threshold, you not only slow down, you virtually stop moving.
The plane is generating its own force forward, and the treadmill intends to counter it. The problem is once you've reached the threshold of motion, and the wheels start turning, the amount of force the treadmill can provide backwards remains near-constant,
regardless of how fast it moves. This constant level of frictional force is only true up to a point, but that means that during those critical first few seconds, once the tires start moving, it's all about the thrust of the plane vs. the kinetic friction of the wheels (which the treadmill cannot increase by speeding up, at least not until reaching several orders of magnitude of speed).
Someone posted that a 1937 study of airplane tire friction. The worst result they received was a coefficient of 0.035. Making the assumption that current tires have as much friction or less, that would provide a rolling resistance against an 850,000 pound plane approximately 30,000 pounds of friction. The four engines, on the other hand, each produce about 58,000 pounds of thrust (for a total of 232,000).
So, putting this in snapshots of tiny fractions of a second:
The treadmill does nothing until the plane moves, as it has no wheel velocity to match.
When the wheels start moving and have a velocity, the treadmill can only counteract with kinetic friction, which is a near-constant. As such, no matter what speed it goes (up to a point), it can only generate 30,000 pounds of friction to slow the plane down, which is no match for the plane's 232,000 pounds of thrust.
Thus, the plane continues to accelerate, the wheels gain velocity, the treadmill tries to counteract with additional velocity, but does not generate enough friction to overcome the engine thrust. Effectively, the treadmill goes much faster, and the wheels go much faster, but the wheels are still going slightly faster than the treadmill (because the engine is overcoming the kinetic friction of the wheels).
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Hopefully this does a much better job of putting it all together.
The one theory that is central to how I understand the scenario presented, is that once an object starts moving, kinetic friction is near-constant. That is, you can't generate additional friction by going faster. It's the same reason a box is easier to push while it's moving than if you let it stop, and it just gets easier the harder you push (because your force gets greater and the kinetic friction remains the same).
If kinetic friction is near-constant, the treadmill will fail in its goal of making wheel speed equal treadmill speed, no matter how fast it goes. As I explained, it'll make both wheel and treadmill go really fast, but since virtually no additional friction is being generated (and the engines are still wining by about 232,000 to 30,000), the wheel will continue to travel faster than the treadmill. As such, the plane moves forward, because it's having virtually no more trouble overcoming wheel friction than it does any other day of the week.
At the point where this breaks down, we're certainly no longer talking about standard operating procedure for any of the equipment involved. I'm not going to begin to decide, or even guess, where the wheels fail, or lock up, or simply start slipping off the treadmill... or what the limits of the treadmill's acceleration are. Besides, when we go an order or two of magnitude above the speed a plane normally goes, doesn't the air above the treadmill catch a significant degree of friction itself, and start moving backward itself? In Ludicrous Speed mode, this might be enough to generate lift (though again, we're talking silliness again).
So, my guess remains: Within boundaries of speed, thrust, and acceleration that have any degree of feasibility for the equipment involved, the treadmill cannot overcome thrust (since kinetic friction is near-constant, and the engines overwhelm that friction), and the plane will successfully move forward relative to the air, generating lift and taking off despite the treadmill's best efforts to go really really fast. The first point of failure I expect (after choosing to ignore how a plane got on a giant treadmill and how it can accelerate so much mass to such a velocity in such a short time) will be the physical integrity of the wheels.
But I could still be wrong. That's why I posted.
