ELI5, If a large passenger jet was trying to take off on a free spinning treadmill, would it take off?
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Like most such “brain teaser” questions the issue actually hinges on the specific phrasing of the question and not the physics. The question as it is typically posed is:
> Imagine a 747 is sitting on a conveyor belt, as wide and long as a runway. The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off?
Let us ignore all the practical problems. The treadmill exists, the airplane isn’t going to break, etc. Also let us assume there is no wind (as a sufficient headwind could allow the airplane to lift off without the wheels even spinning) and that the movement of the treadmill causes no wind. **Most importantly let us assume the situation as stated in the question is accurate**; nothing pisses me off like someone saying “Oh, but constructing a treadmill like that would be impossible or impractical,”. The question is the question regardless of practicality.
All this leads to a problem in the interpretation of the situation. If the airplane’s engines push against the air to make the aircraft roll forward at say 10 mph, then the treadmill will spin backwards at 10 mph as well. No problem, the drag of the wheels spinning at twice takeoff speed wouldn’t be enough to stop the aircraft lifting off since it pushes against the air, not the treadmill/ground. But notice the problem there? Once the aircraft is moving 10 mph down the runway the treadmill spins backwards at 10 mph causing the wheels to spin 20 mph in total, but the treadmill *matches the speed of the wheels!* The treadmill would need to spin at 20 mph, which means the wheels are actually turning at 30 mph so the treadmill need to move at 30 mph, etc.
This seems to form an infinitely increasing loop as long as the aircraft is moving down the runway. If the aircraft is stationary with respect to the ground then the wheels could be going any speed matched by the treadmill, but any movement across the ground implies a conceptually infinite speed of the treadmill and wheels.
Now many infuriating people will look at this apparent physical impossibility and decide to change the question itself. (The Mythbusters did this because they needed a physical test.) As above this isn’t in the spirit of the question and any answer produced isn’t really for the question as it is posed. It also isn’t really a physical impossibility; the treadmill won’t need to move at infinite speed in order to prevent the airplane from moving with respect to the ground. It would of course require an absurd amount of speed to produce the drag required to keep the airplane from moving forward, and realistically the engines of the 747 would be able to melt and obliterate the wheel bearings. But saying “the wheel bearings would break” seems like as much a cop out answer as saying “the FAA would never allow such a situation”. Yes, the landing gear assembly would be incandescent at that point but if we assume nothing breaks it should be possible.
At this point the answer is obvious, **the airplane cannot take off**. In order to fly the aircraft needs air moving across its wings, and without wind that means the aircraft needs to be moving with respect to the ground in order to take off. The engines of the airplane push against the air but they still need to overcome the drag of the wheels over the ground; that is why there are wheels on the end and they roll on a smooth runway, to reduce their drag. If the runway was covered in 10 feet of thick mud then the airplane couldn’t take off either. Increasing the speed of the treadmill will at some point produce enough drag to equal the thrust from the engines and the airplane will not move with respect to the ground and thus not with respect to the air, meaning no lift and no takeoff occurs.
It is also fairly common for some first year engineering student to chime in with nonsense like “rolling resistance is the same regardless of speed” but of course it is obvious that a bearing moving at 1 rpm doesn’t have the same amount of drag as a bearing moving at 200,000 rpm. The latter bearing will presumably be getting very hot and that energy has to come from somewhere. Surprise, it is friction! But we don’t even need that proof because even if we assume the bearings are truly frictionless the force to counter the engine’s thrust can come from inertia. The wheels are going to resist being spun up to a conceptually infinite speed, so the acceleration of the engines can be countered by a constant acceleration of the wheels. Think about how an angle grinder jerks in your hand as it starts up, that jerk is the grinding wheel’s inertia resisting it being accelerated to operating speed. Unbreakable, frictionless wheels would still have inertia and could provide that countering force indefinitely as they are accelerated towards infinity to stop the aircraft rolling forward.
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