Again, not wrong, but depends on the assumptions. You’ve neglected to state (as did the “answer” from the original, not-on-this-board thread) what you assumed about the mass of the tires. If you’re assuming massless wheels, then everything you’ve said is true. However, if the wheels have mass, then the jet and belt are coupled through the spinning tires, and the energy from the jet engines goes into accelerating the wheels.
Ding ding ding! We have a winner!
By the way, I’m surprised nobody has mentioned the air flow created by the treadmill. Even a flat surface will “pull” air. A sufficiently large and fast treadmill may create enough turbulence to make the airplane uncontrollable. Or it may create enough headwind to make it even easier to take off - depends on how large the treadmill is, I think.
Oh good grief. Now you’ve introduced increased airflow to the underside of the wing, resulting in negative lift (yeah, I just made up the term for any physics nazis that are still with us and haven’t strangled thmselves with their own colons). So not only will the thing not take off, it’ll need to go even faster. Or something.
Way to lock this one in to another 6 pages, ace!
Let me repost this with some additions. There are plenty of answers to this question, because the key to the question is what you assume from the beginning. The assumptions are the key. Let’s start off at the top:
A. Suppose we actually built a treadmill like that described in the OP and put a 747 on it. Would the 747 take off? If “exactly matching the speed of the wheels” means that the treadmill matches the hub speed of the wheels (the speed of the wheel center), then yes. The treadmill simply accelerates in the opposite direction that the plane does. The wheels wind up rotating twice as fast as they normally would, but the plane will take off, leaving a treadmill behind that’s rotating in the opposite direction.
B. But, as Manduck says, that problem is trivial. Let’s assume that “exactly matching the speed of the wheels” means “matching the outer diameter surface velocity”–the velocity with respect to the hub. Would the 747 take off? Almost certainly it would, but only because we can’t build a treadmill capable of keeping up with the thrust transmitted to the plane by the engines–in other words, we violate the spirit of the question, because the treadmill isn’t matching the wheel velocity.
C. OK, that’s stupid. It’s a thought experiment. Posit a magic treadmill that can accelerate as fast as desired. And it doesn’t break. I imagine the wheels will skid on the treadmill, because the friction won’t be able to transmit the necessary force. In that case, we again violate the spirit of the question, and–
D. It’s a thought experiment, smart guy. Assume there’s enough friction to rotate the tires. All right. When the engine lights off, the treadmill will accelerate until the force transmitted through the wheel hub to the plane exactly balances the thrust. The plane would stay stationary as the thrust power was dissipated in the wheel bearings (as friction), tires (hysteresis), and in accelerating the wheel to ever-increasing speeds (All of these effects wind up transmitting force to the plane). Since all the power is dissipated in the wheels, eventually either the bearings would overheat, the tires would blow, or the wheel would rip itself apart due to inertial forces. After that, the plane crashes and burns. Then you’ve destroyed a rather expensive magic treadmill.
E. Thought experiment, I said! Let’s posit ultra-strong and heat resistant tires. All right. It turns out the real world is rather complicated. If the treadmill is a long, runway-sized treadmill, it will eventually, running thousands of miles an hour, pull in air at high enough velocity that the plane will lift off at zero ground speed (but substantial air speed). However, now you’re running into trans-sonic compressibility effects…
F. No speed of sound effects! And assume magic air that doesn’t become entrained with the treadmill motion. And don’t throw in any other crazy stuff, either. In that case, the treadmill speeds up (still balancing the plane’s thrust force) and the plane stays in place until the engines run out of fuel. I imagine the treadmill goes pretty fast at that point. The plane stays put until the fuel’s gone, at which point the magic treadmill whips it backwards.
G. Backwards, shmackwards. Now we’re getting somewhere. What if we had infinite fuel? Then the wheels keep going until they’re running near light speed, and relativistic effects take over. The wheels get smaller, I suppose…
H. None of that! No relativity-- Hey, wait a minute. Back up. Suppose we have zero friction bearings and tires. That doesn’t seem so unreasonable for a thought experiment. Well, zero friction tires would mean they just skid on the runway, since nothing turns them. So the plane will take off, tires motionless, and the treadmill won’t move.
I. Hey! Quit it! I already said the tires don’t skid! Sorry. Just friction on the tire/treadmill interface, then, but none in the bearing or sidewall. With zero friction in the bearing, you lose the friction coupling between the treadmill and the jet. But, as treis points out, you still have inertial coupling. The jet power goes into accelerating the wheels, and you have the same case as you do with friction. The jet stays stationary as the wheel accelerates; the wheel just accelerates faster.
J. Well, how about the other way around? Massless wheels, but you still have friction? Here it starts to get complex. As you accelerate the wheels, the bearings will change shape and heat up and so forth, so it’s reasonable to guess that the “friction coefficient” goes up with increasing speed. If that’s the case, then when the engines start, the treadmill accelerates up to whatever speed will give enough friction to balance the thrust. The plane stays stationary, wheels rotating at some reasonably constant (but large) velocity, dissapating the engine power through friction.
K. But I want massless wheels and a constant coefficient of friction. Indestructable wheels, remember? None of this hand-waving “it’s gonna get bigger” crap. OK. It is a thought experiment. With a limited “friction coefficient,” only a limited amount of energy can be absorbed by the friction. When the engine lights off, the treadmill instantly accelerates to infinite speed. It’s never able to counteract the thrust force, and thus plane takes off, leaving the infinite-speed treadmill behind.
L. Ah. OK, one last step. What if we had no bearing friction and massless tires? What happens then? Pretty much the same thing. There’s now no energy losses in the wheels and tires, no coupling between the treadmill and the plane–no bearing friction, no inertial effects, no air resistance, and no way for the treadmill to affect the plane’s motion. The same thing would happen as above, with the plane taking off, leaving the infinite-speed treadmill behind. However, there’s one added interesting thing: This is now an unstable runaway system. There’s no resistance to treadmill motion, and a positive feedback circuit. Imagine the poor mechanic who bumps a wheel, setting it in motion. A very slight roll by the tire is sensed, and the treadmill luches forward. The tire goes faster, the treadmill goes faster, the tire goes faster… Since we’ve posited an instantly-accelerating treadmill and no relativity and no air resistance and no wheel inertia, the treadmill goes from zero to infinity in no time flat. Try to keep your balance on that.
Pick your scenario–they’re all correct.
This is a different scenario than that posited by the OP. If the belt moves forward when the wheels are not spinning, then you have violated one of the conditions of the problem. The correct answer to the original question is the plane does not move and does not take off. Of course anyone can always get to any desired answer by changing the question.
From the OP:
(bolding mine)
Strictly, speaking this means that the plane cannot move because any forward movement of the wheel on a non-moving conveyer belt is not “exactly match[ing] the speed of the wheels at any given time”.
Again, the question sets an impossible scenario. I’m thinking the person who thought this up was thinking that the movement of the jet (and in turn the wheels) would cause the belt to start as if it were a car, for example.
I’ve given up on trying to read all the posts in this thread, as it seems to be an example of the “Zeno’s Paradox” problem - a logic problem so defined (usually by excluding a key element) that the only internally consistent answer cannot match the real world. My WAG take on this is that, if you redefine the test environment by eliminating the effects of friction, inertia, etc. so that the treadmill can actually instantaneously react to any motion of the wheels, then the plane would also instantaneously reach flying speed and lift off, thus removing itself from any influence of the treadmill.
Once the plane is airborne, who gives a flying fuck about the wheels?
This is the most absurd thread I have ever had the misfortune of reading on the Dope. Simply a travesty. You people should be ashamed of yourselves!
zut, good recap.
No, the energy to accelerate the wheels has to come from the belt.
Enlighten us o wise one.
Which part do you need help with?
I don’t need help with any of it but perhaps which ever portion you feel we should be ashamed of ourselves for.
Crap. The jet body is connected to the wheels. As the thrust pushes the body forward the wheels must move. The conveyor rotates the wheels in the opposite direction of the thrust, but the wheels still move in relation to the surrounding land (and air, which eventually will give you your lift). The plane travels further along as the conveyor speeds up and eventually gains enough speed to lift off the damn conveyor and go airborne.
(If the conveyor were able to accellerate to infinity in no time flat) As soon as the thrust of the plane moved it an inch (not even) the conveyor would reach light speed and vaporise.
For the sake of our sanity and the most realistic approach to the problem;
In reality, we can only imagine what would happen if the conveyor matched the speed of the forward thrust velocity of the plane but in reverse. In this case the plane moves forward (in relation to the surrounding land/air) while the conveyor continues to accellerate in the equal and opposite direction. The only resistance to the planes’ forward thrust is the resistance in the wheel bearings (once the thrust causes the plane to move forward in relation to the surrounding land), so the plane continues along untill it has sufficient air speed to lift, meanwhile the wheels are traveling twice the speed otherwise needed and the conveyor is traveling in the exact opposite direction/speed of the plane at takeoff.
That it?
Earlier you said:
So how about providing an answer that we shouldn’t be ashamed of?
This is wrong, the friction between the wheel and the conveyor belt provide a force opposite to the thrust.
You know, those morons in The Navy really ought to think about installingconveyor belts on the flight decks of aircraft carriers. Or would the rolling motion of the sea mess up the aparatus.
Ok,
I really shouldn’t chime-in on this cuz plenty here are more qualified than myself and have already answered this Q. But it seems to me that Aircraft carriers have been implementing similar tactics for quite some time to reduce the overall length it requires for take-off.
I don’t think anyone here said that the plane wouldn’t travel off this treadmill contraption for some distance for the airflow over the wings to be acceptable.
It just seems to me to be (hypothetically) a clever way to reduce runway distance. Just set the treadmill to go only so fast and at this point the plane’s engines would push it off this thing onto the runway where it would travel at a quicker pace since it’s come up to speed at which time the pilot can ensure that there is enough lift to take-off at.
Like one post stated though,* landing* would be the hard part, seems to me you’d have to specifically make landing strips vs. take-off strips. This could in turn complicate the standard layout of an airport, but then again it’s just my opinion on the matter.
There isn’t one, it is a thought experiment… Inherently unsolvable because you can’t keep some duffus from saying ‘what if’ because we voted that we can’t shoot folks just cause they need it so the whole thing collapses because you did not vote to make it mandatory that everyone become a Republican who hates Bush and blames him for everything and throwing Einstein through the window willy nilly so you can say once again, “What if?”
It is a post count thread – nothing more… Not one thing worth while was learned except for those poor folks that thought aircraft wheels were driven by the engines like a car. Some of them may have learned something if they read this whole thing. ( doubtful )
Just think, if you were a Catholic or an atheist and were reading this… you head would explode… The Baptist don’t care…
I think we should blame it all on Tomatoes …
And for the record… the answer is “42”.
*This song’s about the tomato vendetta
And the tale of a man who let a
Hate for tomatoes cause him strife
He lost his job, wife, home, car, kids and life.
He didn’t know why he hated tomatoes
They were just as ugly as far as ugly goes
There in the store, when no one was watchin’
He’d set mellons on them and laugh at their squashin’*