I think the simple answer to the airplane on a treadmill problem is that once you understand what the problem is REALLY saying, the problem becomes uninteresting and trivial. If the problem wasn't intended to give the impression of a stationary (with relation to the surrounding air) plane taking off, no one would've cared in the first place and there would be no discussion. No one is getting confused by math or aerodynamics or physics or anything. They're getting confused by a problem intended to be confusing.
Physics is not intuitive for most people. The aircraft problem works for the same reason you can yank a table cloth from under a wineglass without knocking it over. But watch someone do that and it will seem like a magic trick.
Edit: Perhaps a better explication might be how a tail wind increases a modern cars top speed at almost a 1 to 1 ratio.
There's an obvious intuition that the author seems to be missing with the 'two aces' paradox.
If you have a hand with two aces, you have two chances of getting the "Ace of Spades." Having a hand with two aces doesn't make it any more likely to have "an ace" than having a hand with one ace does.
You know those big giant things sticking out from the side of the fuselage? They're called wings. Air flowing over the and under the wings creates different pressure; hence, lift. There is a magic speed for each aircraft that over which the lift will be sufficient to move the entire aircraft vertically. For a 747, this is 180 mph/290 km/h at 80,000 lbs. That's pretty fucking fast.
Any movement relative to the ground that is under that speed will not result in flight. This is true even if the aircraft is actually in the air.
If, as the problem states, the treadmill can equal the forward thrust of the engines, no lift will be produced by the wings, ergo, it will stay right where it is.
This is why aircraft have an airspeed and a ground speed. It's also why stalling is a major concern.
EDIT: the fact that finding a treadmill that can act in a manner as suggested by the problem is impossible is another issue completely.
EDIT 2: Think people: They have breaks on aircraft wheels do they not? (They do) You can sit on a runway with the engine revved equal to your break ability and not move. If the engine in the problem can outperform the treadmill, you've solved a different problem.
Airspeed over the airfoil, not groundspeed is the primary factor involved in lift. The logical fallacy in your argument is thinking that the wheels provide the forward momentum, but they don't, it is the propeller that does.
That being the case, the plane will be pulled forward down the runway irrespective of how fast the treadmill under it is moving because the speed that the wheels rotate has nothing to do with how much airspeed is generated, other than the negligible friction that is transmitted to the airframe through the wheel bearings.
Said another way: a car traveling on the treadmill will be slowed becasue its forward momentum is transferred through the wheels. An airplane will not be slowed because the forward momentum is generated by the propeller and has nothing to do with how fast the wheels are spinning.
The only way to achieve airspeed required for flight when you are sitting on the ground is to move the wheels. (ie, roll forward).
Your argument suggests that if the aircraft was attached to the ground with steel beams it would somehow break free into glorious flight. It would not.
The treadmill removes the ability of the aircraft to achieve correct airspeed, regardless of whether it is powered by prop, jet or ion engine.
Any assumption that the aircraft can move forward is predicated on the fact that the treadmill cannot match the engine's output. This is accurate, but like I said, outside of the scope of the problem.
Try this:
1. Aircraft has floats, not wheels.
2. Aircraft's normal take off speed is 100 kts. Let's say the engine can produce a take off speed of 100kts at 2000rpm.
3. Aircraft is on a river moving the opposite direction at 50kts.
Q: Will the engine be able to achieve the required 100kts airspeed at 2000rpm? If not, if I can somehow match the river to the airspeed, is it not sound theory that I can prevent the aircraft from ever reaching that 100kts take off speed?
The treadmill does not remove the ability of the plane to move forward. If it did, you are absolutely correct, the plane could not fly.
The wheels on an aircraft are not attached to any drive mechanism, they are free spinning. In the scenario described they would act as a bearing and mitigate nearly all of force applied by the treadmill. There simply isn't enough friction transmitted to the airframe to counteract the force created by the propeller.
An airplane with no airspeed sitting on a treadmill would not fly, as you say. An airplane generating enough airspeed will. The treadmill does not prevent the airplane from moving forward with sufficient airspeed.
The wheels not being powered is irrelevant. The engine produces forward power. The plane moving forward causes the wheels to move in relation to the forward power of the engine.
By your logic, putting breaks on aircraft wheels would be irrelevant. But guess what, they can and do stop the aircraft from rolling forward.
If you counter that movement - of the wheels - you counter the forward movement of the aircraft, because the movement at this point is against the ground, not the air.
Isn't this the difference between the way an automobile moves and an airplane? The automobile applies force through friction of a rotating tire against the ground, a non-moving surface. The fact that the car moves through air is secondary, hence aerodynamic designs...
The aeroplane applies force through movement of air via propeller or jet/turbine engine. The fact that the aeroplane interacts with the ground is a secondary (hence bearings to reduce friction) while the primary thrust is derived from air movement.
Therefore, wouldn't the brakes remain relevant for providing the friction against another form of energy (inertia) of contact with the ground to help "fight" the plane's movement through air as it lands? Now a plane is harnessing both mediums - Air flaps/reversed engines to help create braking push against air flow, and wheel brakes to push against the ground...
This isn't true becasue moving the wheels in the oposite direction does little to slow the plane becasue the bearings in the wheels don't transfer any appreciable force back to the airframe.
Try it with a hot wheels car on a treadmill sometime - it take very little force to keep the car stationary because the wheels act as bearing and keep the surface of the treadmill from applying much force to the car body.
Edit: or sandwich a hot wheels car between your palms, move the bottom palm backwards to simulate the treadmill effect -you wont feel much force acting on the body of the car that is resting on your other palm, no matter how fast you move your lower palm in the other direction.
Either way, this is a really interesting debate :)
When you take off with a tail wind you have basicly the same setup, the wheels need to roll faster relative to the ground to take off. All that happens is you need a longer runway, it does not take significantly longer because the largest limit on acceleration is drag from the airspeed not wheel drag.
Wheel drag is a function of weight, at 1/2 take off speed you have 1/2 the weight friction so you can get closer to take off speed. (The limit works out so you can still take off.)
The less obvious and more important factor is stall speed of an aircraft = takeoff speed, but because of ground effects you can lift the wheels off the ground below take off speed. You don't do this because it creats the posiblity of bouncing the aircraft as you stall once the height is above the ground effects.
"Any assumption that the aircraft can move forward is predicated on the fact that the treadmill cannot match the engine's output. This is accurate, but like I said, outside of the scope of the problem."
If this is assumed then you are absolutely correct. This, however describes a situation that is different than the reality. If the problem is phrased "Assuming that the treadmill can provide opposite forces sufficient to prevent the airframe from achieving enough airspeed to fly, will the airplane take off?" Then no, the airplane won't take off.
Your example of the floats is a good one. The friction created by the water flowing over the floats would be much greater than that of wheels on a treadmill, so in that case the plane may or may not be able to achieve enough airspeed.
What if a big fan blew air straight into the airplanes wings while it was stationary and those fans were attached to the plane so that they would continue to blow air if the plane gained lift?
If the problem is pursued as described to keep the plane stationary on the treadmill (with the treadmill blazing so fast that the bearings in the plane's wheels eventually melt from friction), the plane will never take off.
However, since a treadmill cannot be constructed as described in the question, testing it is an impossibility. A plane on a normal treadmill will quickly move forward on the thing and take off -- but that's not the question being asked!
I've witnessed this debate a few times now, and every time it is absolutely two groups of mostly-correct people arguing past each other about the definition of the problem. Half of them are assuming the runway is a treadmill going backwards, but otherwise the same, and the other half are assuming that something is physically preventing the plane from going forward (and, presumably, backward) while we run a treadmill beneath the plane and wait for it to take off. A much smaller group of people want the treadmill to run backwards fast enough to counteract the forward force of the engines, but this just involves destroying the plane or the treadmill so I tend to ignore it as an uninteresting problem.
Long, tedious, mostly-correct explanations of why the other side is wrong tend to follow.
Personally, I give the nod to the first group of people since the problem is never specified with anything holding the plane, and consequently that's an added entity not in the specification. However, you are free to frame the problem any way you choose, and it's perfectly valid to say there is such a thing. But you should be aware that you're not arguing the same thing as the other guy.
This one is unique among the "world-killing" problems I know, in that there is generally a right and a wrong side. In this case, both sides are usually fairly right, just talking past each other.
Which, IMHO, makes this a rather intensely boring HN debate, all ye in the other thread.
I think the intent of the problem is that it wants to show that an equal and opposite force is being applied to the engine, therefore no forward movement is applied.
The way it is worded however causes those who a reading word for word to conclude that there is less than equal force being applied (because 1:1 with the wheel motion is that) and therefore the aircraft moves forward.
You should read the links I provided (in the blog post). Your argument requires that the treadmill prevents the plane from moving. It cannot do this unless the plane operator cooperates.
Matching the speed of the wheels at any given time DOES NOT ensure that the acceleration of the plane (relative to the ground) is equal to zero.
the fact that finding a treadmill that can act in a manner as suggested by the problem is impossible is another issue completely.
Fine, but the reader's acceptance or non-acceptance of the existence of such a magical treadmill pretty much exclusively determines how they decide the issue, so you can't claim that it's irrelevant. Nobody that accepts that such a treadmill exists (in the problem) says the plane will take off, and nobody that thinks it doesn't exist says it won't.
Which makes it a brilliant problem from a psychological standpoint - everybody gets to feel like they're smarter than the other side, but really they're all just being asses and arguing about different problems.
Any time you can prove both P and !P in a system, you can prove literally any statement. The airplane on a treadmill problem asks us to assume "normal physics," but then postulates some magical treadmill, vitally important to the "paradox," that violates everything we know about physics. It's no wonder the arguments degenerate into foolishness quite rapidly.
Think people: They have breaks on aircraft wheels do they not? (They do) You can sit on a runway with the engine revved equal to your break ability and not move. If the engine in the problem can outperform the treadmill, you've solved a different problem.
The brakes on the airplane are not supposedly engaged during the treadmill experiment; if the problem specified that there was some specific amount of residual friction from the wheel joint, then we might have a problem to discuss, but it doesn't. So the best we can do is suppose that we're talking about a normal plane, and for a normal plane with normal wheels the treadmill would have to be moving so rapidly as to rip the wheels right off the plane before it could overcome the force of the engine.
At that point, the plane would fall on its belly onto the treadmill, and the backwards force the treadmill could exert on the plane would be dominated by the coefficient of friction between the fuselage and the treadmill. But that becomes far less fun to discuss, because it's now a standard physics problem and we actually have to find the parameters involved.
> If, as the problem states, the treadmill can equal the forward thrust of the engines
If the problem stated that you would be correct. But the problem doesn't state that. The problem states:
> This conveyor has a control system that tracks the plane speed and tunes the speed of the conveyor to be exactly the same (but in the opposite direction).
So you are wrong. What happens is that the plane moves one way, the conveyer moves the other way at the same speed, and the wheels spin twice as fast as they normally would. But the rolling friction of the wheels is negligible, so the plane takes off just fine.
And if you still don't believe me, the Mythbusters actually did the experiment full-scale.
> the treadmill can equal the forward thrust of the engines
Did anyone really phrase the problem so incorrectly? All I find is:
"This conveyor has a control system that tracks the plane speed and tunes the speed of the conveyor to be exactly the same (but in the opposite direction)."
or
"The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction."
Actually, most planes have brakes that are strong enough to keep the plane stationary at maximum engine thrust. If they didn't you couldn't do an engine runup. But no plane is able take off with the brakes set, treadmill or no treadmill.
You are not limited by the breaks only the tires friction which depends on weight. So in many cases, you can start to move if you don't have enough cargo and fuel on board.
PS: You really do need to tie down an F-15 to do a maximum engine thrust test. Edit: As I said "high preformance"
Of course you can, because the break isn't strong enough to counter the engine. The treadmill in question though is designed to keep the aircraft stationary.
The limit is the friction your tires have with the ground. In a car your breaks are actualy to powerful to provide maximum breaking. Once they start to skidd the friction is greatly reduced which is why you have anti lock breaks.
That's the crux of the issue. If you could somehow make a treadmill that keeps the plane stationary then it wont take off. However, any treadmill built in the real world will be unable to keep a normal aircraft stationary.
This is not completely on-topic, but in the nicest way possible, I'd like to bring up the fact that downmodding shouldn't be used to disagree with somebody. (Not happening to me, but I notice that some people are getting downmodded for apparently that reason). This isn't meant to start a debate, just a friendly observation.
My issue with this problem was that it was presented rather poorly to me the first time. Instead of saying that the treadmill matched the negative of the velocity of the plane relative to the ground, the questioner instead stated that the treadmill "matched the speed of the wheel." I took this to mean that the treadmill matched velocity of the wheel as would be given by a conventional speedometer.
Of course, this is prima facie ridiculous. Consider, at any nonzero speed, the speed of both the treadmill and the wheel would quickly approach infinity. Given that the speedometer speed would essentially be the velocity of the wheel + the tangential velocity of the wheel touching the treadmill, the entire thing would just blow up (remember, non-slip == velocity of the wheel == velocity of treadmill). Therefore, I basically answered c) it's a stupid problem and the plane wouldn't move.
Of course, if you define it this way, than it is obvious that the plane would move.
Last year Vaughan Pratt (of automata theory and Knuth-Morris-Pratt algorithm) asked on a hardcore theoretical physics group blog how does a light mill (device allegedly proving corpuscular nature of light, http://en.wikipedia.org/wiki/Light_mill) really work. To my knowledge this question after 150 years still awaits consensus over a satisfactory anwser (while having far too plenty seemingly convincing ones).
The crux of the issue is that no matter how fast the wheels spin in the opposite direction they exert almost no force on the airframe, thus failing to counteract the forward pull of the propeller.
