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#1




Plane on a Treadmill  SOLUTION
MODERATOR EDIT: Please note that this thread was started in 2006, has been revived several times since. Please be careful about what you're responding to, since it could be way (way!) old.  CKDH
************************************************************** Cecil is wrong. The plane does not take off. This is why: Original Statement: "A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?" Keep in mind that no system is frictionless. We are assuming real systems, and the conveyer as described is well within present capabilities. It can be built. Fact 1: The force generated by an engine is limited. Fact 2: Friction is a force Fact 3: The mass of the airplane is an "inertial force" Fact 3: The frictional force of the wheel assemblies against the ground, whenever referred to below, contains the friction of the tires against the ground, as well as the wheel assembly (bearings, axle, etc). This also includes the downward force of the mass of the airplane upon the entire assembly. This entire frictional force is not inconsequential. Fact 4: The conveyer, as it moves faster, exerts more and more backwards force upon the wheels, and this force is not inconsequential. Cecil dismissed this, but this is considerable and is a significant part of the frictional force of the wheel assemblies. Unfortunately, this is Cecil's big physics flaw. The backwards force of the conveyer is equal to the force generated by the conveyor system. THIS is what everyone has ignored, and this data is required in order for the system to be closed. You cannot ignore the force generated by the conveyor system itself. It is equal to the backwards frictional force against the wheels. You cannot ignore the force of the conveyor system itself. The conveyor is moving, a force is being generated to create that motion, and that force, the force to move the conveyor upon which a big hulking mass of airplane is sitting, is large and considerable. As the below shows, it is the force generated by the conveyor system that prevents the plane from achieving lift. Fact: The plane has no horizontal motion. Item: For the plane to have horizontal motion, the force of the engines must be greater than: [The big hulking inertial mass of the plane] + [frictional force of the wheel assemblies with the ground (including conveyor system)] If the control system tracks the rotation of the plane's wheels exactly, then it follows that the full, force of the engines at that moment have force exactly equal to the frictional force of the wheel assemblies against the ground (bearings, ground frictional force, etc). Therefore, as long as the treadmill tracks as in the original statement, the full force of the engines are exactly equal to the frictional force of the wheel assemblies against the ground. (it may take some time to realize this). As the constant force of the engines are equal to the frictional force of the wheel assembly and the ground friction, no force is available to counteract the inertial force of the mass of the airplane. Fact: In the original statement, the velocity of the plane relative to the conveyer, is not the same as the velocity of the plane relative to the air. Item: No matter how fast the conveyer moves, with or without anything on it, it won't make the air move any faster six feet above it. Therefore, velocity of the air over the wings is independent of the conveyor speed. Fact: A plane takes off not by it's speed relative to the ground, but by the velocity of the air relative to it's wings. Item: We can agree that, in the original statement, a plane without wings will not take off. See the next Fact for a further explanation. FACT: THE AIR FLOW OF THE ENGINES ALONE ARE NOT SUFFICIENT TO PROVIDE LIFT! Proof: An airplane at the end of a runway as it's about to take off, regardless of whether the airplane is propeller driven, or has engines mounted on it's wings, is at full throttle/full thrust. At the beginning of the runway, this full throttle/thrust is not sufficient to pull enough air over the wings for the plane to take off  if it did, planes would take off immediately and would not require a runway. This exercise can be compared to an airplane sitting at the beginning of a runway with it's engines at full throttle. A plane's engines provide forward motion to increase the flow of air relative to the wings. The air flow provided by the engines alone are not sufficient to provide lift. Forward motion is required. If the force of the engines is equal to the frictional force of the wheel assemblies and the conveyer assemblies, then there is no forward motion. Without forward motion, the airflow generated by the engines alone are insufficient to create enough airflow across the wings to provide lift. (If we can talk about frictionless surfaces, infinite thrust engines, etc. we change the experiment.) Last edited by C K Dexter Haven; 01032014 at 08:48 AM. 
#2




Please change:
Proof: An airplane at the end of a runway as it's about to take off, regardless of whether the airplane is propeller driven, or has engines mounted on it's wings, is at full throttle/full thrust. At the beginning of the runway, this full throttle/thrust is not sufficient to pull enough air over the wings for the plane to take off  if it did, planes would take off immediately and would not require a runway. To: Proof: An airplane at the start of a runway as it's about to take off, regardless of whether the airplane is propeller driven, or has engines mounted on it's wings, is at full throttle/full thrust. At the start of the runway, this full throttle/thrust is not sufficient to pull enough air over the wings for the plane to take off  if it did, planes would take off immediately and would not require a runway. 
#3




Welcome to the Straight Dope, Paradoxic. Your conclusion is certainly reasonable, but it depends quite a bit on the assumptions that you make about the situation. In fact, I listed a lot of those possible assumptions in this post relating to Cecil's earlier thread.
In particular, you are correct that, according to the original problem (at least, some versions of the original problem) the plane will be held stationary with respect to the ground, and thus the conveyor belt will apply the necessary force to counteract the thrust. And this force is, by definition, not inconsiderable. The real question that a lot of people asked was if it was even possible for the treadmill to apply such a force. Again, it depends on your assumptions. However, a few things: First, mass is not an "inertial force." It's a component of the inertial force, but the force only occurs when the mass is accelerating (F=ma). Since we're assuming no motion of the plane at all, there's no inertial force associated with the bulk mass of the plane. Second, assuming that the frictional force associated with the wheels will balance the engine thrust is problematical. Clearly, this frictional force is nowhere near the engine thrust when the plane takes off from a concrete runway. It's possible that the frictional force increases with increased velocity, but it would have to increase substantially, and this is by no means assured. A more significant force, I would argue, is the force resulting from the rotational acceleration of the wheels (as Cecil mentions). Third, you are clearly correct that the airflow from the engines won't lift the plane, and that bulk flow over the wings is necessary. However, it is certainly possible that if the conveyor were long enough and ran fast enough for a long enough time, air could become entrained with the moving belt, moving fast enough to lift the plane off the belt. 
#4




The Plane Will Not Take Off
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The force generated by the conveyor system itself is what everyone has ignored, and you are still ignoring it. Clearly, the physics show that the conveyor system itself has to be responsible for a force equal to the thrust of the engines. That's all it takes. If the force of the conveyor system equals the thrust of the engines, the aircraft can go nowhere. Suppose, for example, you created a conveyor system powered by multiple turbine engines similar to the ones used to power the aircraft? And the frictional force HAS to be considerable because the physics dictates that the force generated by this turbinedriven conveyor system can not go anywhere else. You must look at this as a closed system and consider the force of the conveyor. You are making assumptions, and using words like "clearly" and "it's possible", because you are still considering this to be a thought experiment. You have to look at this as a closed system, and consider the force generated by the conveyor system itself. Quote:
You need to stop treating this as a thought experiment and start looking at this as a closedsystem physics equation. You cannot ignore the force generated by the conveyor. The conveyor system must be generating a lot of force to move the belt, and that force must go somewhere, right? Quote:
I know that, as a matter of business, Cecil is supposed to be always right, but in this case, the excuses must stop because the plane cannot take off. You need to look at this as a closed system, and stop making assumptions about friction and airflow. 


#5




Congratulations, you've jumped straight to BR #2: If the conveyor belt exerts backward force via friction et al. on the axles of the jet plane equal to the thrust generated by the jet engines, the plane will not move, and thus not take off.

#6




Friction is Our Friend
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To put it simply: In this closed system, if we've got a turbinedriven conveyor system, such as the turbines used to drive the aircraft, and if this conveyor system uses the same force as the aircraft engines, then this force has to go somewhere. Friction is not as trivial as most people make it out to be. 
#7




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For example, if the frictional drag in the wheel hub is constant with velocity, the treadmill will continuously accelerate the wheels in order to apply the appropriate force. Quote:
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#8




The Plane Does Not Take Off
O.K., then, I guess we agree. Actually, what I am objecting to is your use of the word "clearly". This problem is difficult as a thought problem, and is more obvious when viewed as a closedsystem physics equation.
If the plane does not move horizintally, then according to the rest of my analysis, the plane does not take off because the airflow generated by the engines alone is not sufficient to create enough lift with the wings. Sorry, Cecil. 
#9




Let’s say the airplane needs 100mph ground speed to take off.
Let’s also assume that the landing gear can support the plane at 200 mph (forward speed of the plane, plus the backwards speed of the conveyor). If the engines can overcome the additional forces against the landing gear of 200mph over 100 mph (not a big deal) and the landing gear can withstand twice the limit of ground speed (again, not a big deal) The plane will take off. The question states that the conveyor matches the speed of the plane. The plane moves 100mph forward (ground speed) The conveyor moves at 100mph backwards (ground speed). The wheels turn as if the plane was traveling 200mph. What’s to understand? The plane is moving 100mph down a moving runway moving in the opposite direction. So what if the runway is moving? The plane has 100mph air over the wings. It takes off. To put it another way  What is the frictional force against the airplanes wheel at 100mph? What is it a 200mph? Three times the resistance? Four? Now, we also have to deal with the resistance created by the rotational velocity of the tire spinning up to 200mph, instead of 100mph. Race cars overcome that routinely. Consider the wind resistance as a plane approaches take off speed. It must be 100 times the resistance from the landing gear. The landing gear resistance is nothing. It’s gonna be there, for sure, but hardly something that couldn’t be overcome. How about this? Would it be possible for a plane to take off with a 100mph tail wind? Same thing. Not something anyone would dare to do, but something that would be about as easy as building the hypothetical treadmill. 


#10




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That is the question that Cecil (correctly) answered. 
#11




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Note, Paradoxic, that we're not subject to 1,000 mph winds here on the surface of the earth. This isn't because the atmosphere has rotational inertia; it's because the atmosphere is pulled along with the big spherical "conveyor belt" we call the earth. 
#12




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#13




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This is something like trying to keep a freight train from moving forward by applying a belt sander to one side of the engine. It's theoretically possible, sure, with a magic belt sander and an impervious train, but there's no way you could build such a critter. I think it's much more likely that the question is asking about the plane's speed relative to the ground. It doesn't make sense to me to say the speed of the belt is "exactly the same" as the speed of the plane when the speed of the belt would have to be much much faster than the plane's top speed could ever be (~600 mph for a 747). The belt is no longer matching the plane's speed, in this thought experiment; it's moving fast enough that friction prevents the plane from moving, which is not the same thing. 
#14




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It's like the question, "A car travels 50 miles of a hundred mile trip at 30 mph. How fast does it need to go for the last fifty miles to average 60 mph?" Implicit in the question is an impossibility. I guess a more accurate answer would be 'mu,' but I'm more inclined to say you can't make a treadmill that stops a plane's forward movement, therefore the plane takes off. 


#15




Simple & Elegant Explanation  Plane Takes Off
The treadmill is irrelevant because it doesn't matter whether the plane is on it's wheels, or suspended from above by a set of steel chains hanging from a curtain rod that runs the length of the runway.
The people who refuse to believe the plane takes off are confusing how engines push an aeroplane forward with the idea of how motorized wheels would pull the plane forward. Planes are not pushed to takeoff by motorized wheels. If you imagine a curtain rod running the length of the runway, from which the plane is suspended by a chain (which can move along the length of the rod) you see that the treadmill below is irrelevant to how the engines push the plane forward. Regards, Vic P.S. An ordinary car with wings would *not* take off because the car is driven by wheels, which push against the runway for speed. 
#16




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as for why a plane can or cannot take off on a treadmill... the math is here http://boards.straightdope.com/sdmb/...1&postcount=45 enjoy. the nonmath summary: if you assume the treadmill matches the plane's airspeed then the plane clearly takes off. if you assume the treadmill matches the planes's tire's speed then the plane clearly does not take off. but the reasons why not are usually confused. the treadmill does not apply any force to the plane. it can't. not significantly anyway. the plane is physically able to move. but if it does move then the belt speed is no longer equal to the tire speed. and a premise of the problem is violated. therefore the correct conclusion is that the premise is wrong. ie the belt speed cannot be set to the tire speed when the plane is moving relative to the air. which would be required for takeoff. the incorrect conclusion is that the treadmill holds the plane still. presumably by magic. or equivalently, closed physics models. or two. ;> 
#17




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It’s two closed systems working against each other. With the landing gear as the connection. Sheeessss. Put a Cessna 172 on a runway. What type of force is exerted on it’s wheels and landing gear at a take of speed? Compared to the force that the engine needs to pull the plane forward, and get air over its wings it’s nothing. Nothing at all. Now. The force on our 172 landing gear certainly counts for something. Double the speed on the gear and it’s going to be more. But not much. Does any one believe that ANY airplane could not take off because of the effect that the landing gear is spinning twice as fast as normal? IT WON’T MATTER. It would be a ***** to steer before transition, but the plane would take off. It has ground speed. Air over the wings. That’s all that matters. 
#18




I leave geeking to party now.
Ah, now I see that each of us sees here what we wish to see. I hang my plane from shower curtain rods and others believe wheel drag shall pin the plane down. Some do math and others draw pictures. I think there is much more utility (and truth) in inebriation and celebration. If any of you are in NY, you can join my krewe at the Alt.Oscar Party Sunday in chelsea at http://www.costumeculture.org/
It is very politically incorrect. That is easy for me because I was not intelligently designed. I definitely evolved from a monkey. I'm done geeking out now. Time to go rock star. Regards, Squid Vicious 
#19




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As far as I can tell, there isn't anyone in this thread who disputes the fact that, if you interpret the question as meaning that the plane fuselage moves forward and the wheels spin at double speed, the plane will take off. There's nothing wrong with that interpretation, and nothing wrong with that answer. However, much of the rest of the discussion in this thread centers around the alternate interpretation. There's nothing wrong with this alternate interpretation, either, but it does lead to different conclusions. Quote:
If you assume a treadmill capable of large velocities and accelerations, and assume that it tracks the wheel speed of the plane, the belt will supply a force to the plane to counteract the engine thrust. However, this requires some coupling through the wheelsthe wheel inertia, for example. The scenario you're giving is a bit oversimplified because it skips the step describing the coupling mechanism. 


#20




Jeeze, not again...
The first problem is, as I see it, that the original problem set up impossible circumstances. Setting up a problem with impossibilities is kind of like dividing by zero... it sort of invalidates the whole thing. There is no way for a conveyor belt to match exactly the speed of the rotation of the wheels, plus even if there was, that would set up a feedback loop with both the wheels and the belt going infinite miles per hour. Also, no matter what anyone thinks, the friction from the wheels turning would still be much less than the thrust of the engine... unless they were spinning at an infinite speed, I guess. I just though of something... what if you locked the wheels altogether and poured water on the treadmill... then froze it. Hey! Then the plane could take off! 
#21




[QUOTE=VicSarjoo]The treadmill is irrelevant because it doesn't matter whether the plane is on it's wheels, or suspended from above by a set of steel chains hanging from a curtain rod that runs the length of the runway.
No, this is NOT AT ALL like this sort of situation. To create the situation of your "set of steel chains" the plan has to move through the air, to create the lift. Until the plane reaches a speed relative to the air mass around it (or, equivalently, to a stationary observer also not being moved by the treadmill) the wheels sit on the conveyor belt and tend to get moved backward alolng with the surface of the conveyor belt. This is not a forcevector calculation (except ina VERY abstract way) and it is not a closedvsopen environment. It is a semantic/logic exercise and subject to the same problems of any other logic problem. The key difficulty is sometimes called "assumptions contrary to fact" wherein we have put forth assumptions (that it is possible to build a conveyor belt that can move at a speed opposite to (and synchronize with) a forward motion of a plane) that are contrary to fact. 
#22




OBTW, when the pilot stands on his brakes and revs the engines, I don't THINK he's generating a force equal to the total thrust of the engines, just enough force to stop the wheels from turning. Maybe it has something to do with inertia, et al, but it is beyond my aging Physics education.

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#24




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__________________
Time travels in divers paces with divers persons. As You Like It, III:ii:328 Check out my dice in the Marketplace 


#25




Sorry about being late to this forum. I was going over the archives and found that Cecil's conclusion on the 'plane on a tredmill' question is in error. The plane if it remains stationary to the air surrounding it, will not not fly no matter how fast the prop spins and the wheels turn. Lift is provided by airspeed. The problem described only mentions groundspeed. Without sufficient air flow over the wings there is no lift.
Sorry folks, this turkey won't fly. 
#26




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You don't comprehend that the issue is what happens to the force created by the turning of the propellers or the igniting of the jet fuel. That force acts upon the plane. If you can explain how the plane can stay motionless relative to the air under those circumstances, then we will accept your answer. 
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#28




I hate arguments semantics. Here's what is really going on here. The original question was badly phrased. This created two factions, one that took the question literally as is, and one that instead tried to assess the intention of the original question. Arguments over semantics are stupid, it's a lot of wasted energy to prove nothing except that you interpret something differently. Instead lets focus on what we do agree on.
You need relative motion of a plane to air to create lift. That's it, we're done, lets move on. 
#29




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#30




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As for the version asked on the Internet, I've yet to see it asked in a way that allows the friction component to actually overcome the force of the engines. I suppose it might be buried in one of the many threads on the subject here on the board, but I've not bothered to find it. 
#31




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As you point out, one problem here is the wording of the question. Your version straightforwardly states that the conveyor moves backward at the same rate that the plane moves forward. If the plane's forward speed is 100 miles per hour, the conveyor rolls 100 MPH backward, and the wheels rotate at 200 MPH. Assuming you've got Indycarquality tires and wheel bearings, no problem. However, some versions put matters this way: "The conveyer belt is designed to exactly match the speed of the wheels at any given time, moving in the opposite direction of rotation." This language leads to a paradox...I've truncated it there because the rest of Cecil's argument is, to be charitable, not well thoughtout. Luckily, though, he did a bit better job of it in the second column on this subject. The point here is that you needn't rely on friction to balance thrust from the engine, but rather the inertia of the spinning wheel. 
#32




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Further, and this is VITAL, there simply is no way to reach the type of result in what Cecil calls BR2 without completely changing the wording of the problem. As long as the problem is worded to deal with the velocity of the aircraft, or the wheels, you cannot achieve the solution advocated and discussed in the second column. All of which means: As originally formulated, the problem was intended (CLEARLY) to make people stop and think that airplanes are not cars. Any other approach to the original question is simply an attempt to be a contrarian. 
#33




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__________________
Time travels in divers paces with divers persons. As You Like It, III:ii:328 Check out my dice in the Marketplace 
#34




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If we are to assume that the treadmill can and will try to speed up to keep the plane stationary, friction does allow it to. Quote:



#35




I think I can offer some info for those looking at a realistic approach to this problem. Aircraft have max rated wheel speeds, on the ground obviously, above which the tires and bearings may overheat. So, even if it was at twice normal wheel speed, the tires and bearings will likely fail before the aircraft could reach take off airspeed, assuming a large aircraft. There is a surprising amount of friction on aircraft wheel assemblies even when not loaded. Even on jacks a wheel takes some effort to rotate. Just my 2c

#36




Now that this thread has been resurrected for a while, let me repost something that I've posted a few times before:
There are plenty of answers to this question, because the key to the question is the wording and your interpretation and what you assume from the beginning. And these answers can all be correct, but the assumptions are the key. Let's start off at the top: A. Suppose we actually built a treadmill and put a 747 on it, and had the treadmill match the speed of the plane. Would the 747 take off? If the treadmill matches the plane fuselage speed, 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. Let's reword the question. Suppose we actually built a treadmill and put a 747 on it, and had the treadmill match the speed of the wheels. Would the 747 take off? Depends. 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, which is the same as the fuselage speed), then yes. Just like in the last scenario, the treadmill accelerates in the opposite direction that the plane does, the wheels rotate twice as fast as they normally would, and the plane takes off. C. But 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, or the "speedometer" speed. 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 enginesin other words, we violate the spirit of the question, because the treadmill isn't matching the wheel velocity. D. 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 E. 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 everincreasing speeds. 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. F. Thought experiment, I said! Let's posit ultrastrong and heat resistant tires. All right. It turns out the real world is rather complicated. If the treadmill is a long, runwaysized 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 transsonic compressibility effects... G. 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. H. 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... I. 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. J. 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 you still have inertial coupling. The wheels *accelerate*, and that acceleration takes force. Now you have the same case as you do *with* friction. The jet stays stationary as the wheel accelerates; the wheel just accelerates faster. K. 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. L. But I want massless wheels and a *constant* coefficient of friction. Indestructable wheels, remember? None of this handwaving "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 infinitespeed treadmill behind. M. 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 planeno 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 infinitespeed 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 instantlyaccelerating 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 scenariothey're all correct. 
#37




Excellent. There is one additional scenario. If the magic treadmill has zero friction in the direction opposite the plane's moement then it presumably has zero friction in the smae direction the plane is moving. In that case as the plane moves forward the whelels will not move (they have intertial resisitance) and the treadmill will move with the wheels. Since the wheels are not rotating they have no speed according to your scenario C, and the magic treadmill will not provide any force to resist them. In this case the airplane takes off and the wheels do not rotate.

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#39




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Nonetheless, unless the treadmill is allowed to accelerate without limit, put me in the "plane takes off" camp. 


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I don't understand why the wheels wind up rotating twice as fast as they normally would. Lets say the plane needed air traveling over the wing at 2 miles per hour in order to take off. Normal "real world" scenario. Plane's engine is pushing the plane forward at 1 mile per hour. Plane rolls onto a treadmill, and the treadmill comes up to a speed of 1 mile per hour. Plane stops moving forward. Plane's wheels are moving at the same 1 mile per hour. Plane accelerates to compensate.. such that were it not on a treadmill, it would be traveling at 2 miles per hour. Treadmill increases speed at the same rate to 2 miles per hour to compensate. Plane never moves forward. Plane's wheels now moving at the same 2 miles per hour that the treadmill is moving backward. No air is moving over the wing, plane remains stationary on the ground. I don't see where anything in the treadmill needs to be able to be capable of infinite acceleration (thus taking this out of the real world, into the realm of ideal physics) or the wheels move at twice as fast as they normally would. Can someone please explain to me the part of this mechanism that's like a rope, tied to a wall beyond the treadmill that the plane is pulling on to move forward? I don't think I'm moving from the real world (BR1) to the ideal/theoretical (BR2) unless you account for the treadmill's ability to compensate smoothly and perfectly matching the plane's acceleration . I am seeing where there might be an issue if we were talking about a prop driven plane where the engine is pushing air over the wing surfaces, but i am not even sure that these forces would have enough effect. (Yea, sorry folks, it got linked on BoingBoing to article to an article in the Times, which lead back here.. the discussion is coming back.. and the hamsters are gonna get a work out.) 
#41




What is the sound of one thread awakening?

#42




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So. How can the speed of the treadmill match the speed of the plane if the treadmill is moving and the plane isn't? It depends, of course, on the interpretation. In your scenario, the speed of the plane with respect to the treadmill matches the speed of the treadmill with respect to the ground. Nothing wrong with that. However, other people interpret the question as requiring the speed of the plane with respect to the ground to match the speed of the treadmill with respect to the ground. In that case, the plane's speed with respect to the treadmill is twice its speed with respect to the ground. Since (we assume) the plane needs to hit a certain speed with respect to the ground to take off, the wheels (which are running against the treadmill) must be going doublespeed. Quote:
In this case, ask yourself this: what limits how fast the plane is trying to go? You arbitrarily cut off your example when the treadmill goes two miles an hour. However, unlike a plane on a nonmoving runway, resistance to forward motion doesn't increase with increasing speed. So, with a particular amount of engine thrust, nothing prevents the plane from trying to go faster and faster and faster. Quote:
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#43




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If the plane has sufficient speed relative to the ground (or more accurately the speed of air over the wings), then it has to be able to take off period. If the plane's take off velocity is air moving over the wings at 2 miles per hour, it doesn't matter if the wheels are moving 2 miles an hour or 4 miles an hour, it's going to take off. The only way I can see the "problem" being any sort of a puzzle at all is if the plane is kept stationary relative to the ground by the speed of the treadmill. And I think that's the scenario you were explaining when we got to.. Quote:
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To elaborate, it seems like you're telling me that even without increasing the thrust, the plane will continue to accelerate and the treadmill will have to go faster and faster and that's not making sense to me. OK.. I just reread that last quote. I'm pretty sure you're discussing an element of fluid dynamics that comes into effect if the plane is actually moving relative to the ground, but can't then follow that to a point that supports any supposition we've mentioned. 
#44




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When the plane's engines light off, they produce thrust that pushes the plane forward. In order for the plane to remain stationary, the treadmill must do *something* to counter that thrust force. About the only thing the treadmill *can* do to counteract the thrust force is to accelerate the wheels, thereby transmitting a force. How else would the treadmill hold the plane stationary? Quote:
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(Note that there could be many different answers here, because it's possible friction increases with bearing speed, and tires will ultimately fail, and so forth and so on. But this is a thought experiment, and it's up to you to outline your assumptions.) 


#45




zut is awesome. I hope, at this point, we all agree that when the belt matches the plane's air speed/relative ground speed, the plane takes off. But I think zut has made the "belt matches wheel speed" answer far too complex. And Cecil? Well he's just plain wrong.
Think of a toy airplane with freespinning wheels, just as a real airplane has free spinning wheels (like rollerblades, but unlike a car, a concept we all should agree on at this point). If I were to hold the plane in my hand and set it down on an actual treadmill traveling at 10 mph, the treadmill itself would push the wheels at 10 mph in the opposite direction, with zero thrust applied. And the plane, being pinched steady in my hand, would have an airspeed of 0 mph. Increase the speed of the treadmill to 20 mph, and the plane’s wheels would travel in the opposite direction at 20 mph, and the plane would still be held steady in my hand with an airspeed of 0 mph. Therefore, the paradox does not exist as Cecil suggests. There is no requirement whatsoever that the belt speed would force the wheel speed to double. Now, of course, I could simply move my hand forward independently of the wheels and overcome the speed of the treadmill — the way thrust would work, the way pulling the rope would work if I were wearing rollerblades on a treadmill — and the plane would move forward, attain lift and take off. And in the real world, that’s exactly what would happen, and the plane would fly. Aha? I got myself? No. The instant my thrust overcomes the friction and the speed of the treadmill, the wheels would be moving forward faster than the treadmill is moving backward, thus violating the spirit and rules of the question that the treadmill and wheels are always moving at identical, opposite speeds. In order to adhere to the rules of that question, thrust would never be allowed to overcome the friction, the wheels would never spin faster than the treadmill, and any actual forward motion is disallowed by the rules themselves. Therefore, so long as the treadmill can INSTANTLY adjust for any change in wheel speed and always match that wheel speed, the plane will continue to have an airspeed of 0 mph. 300 mph belt, 300 mph wheels, 0 mph airspeed. No airspeed, no lift, no flight. It may be overly strict, and I agree it’s a poorly worded, unintentional variation of the REAL question where the belt matches the plane’s speed. It's no longer about physics, it's about semantics. The question itself dictates that forward motion — any airspeed or relative ground speed greater than 0 mph — simply cannot occur, because that would require that the wheels move faster than the belt. Forget all the other complexities outside that simple argument. The plane cannot, and will not, take off under these rules. 
#46




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Well done. 
#47




Let me paste in my email to BB.
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If the wheels have friction, then the friction from the belt on the wheels provides backward force to counteract any thrust, and the velocity changes so that these balance out, keeping the plane stationary. In the first case, the universe explodes in a puff of logic. In the second case, the plane cannot move. End of story. 
#48




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__________________
John W. Kennedy "The blind rulers of Logres Nourished the land on a fallacy of rational virtue."  Charles Williams. Taliessin through Logres: Prelude 
#49




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This is one of the ways on interpreting the “the speed of the treadmill matches the speed of the plane”, and it’s perfectly valid. But you then make this assumption: Quote:
Consider a plane going down the runway at 5 mph. Lets say the wheels have to rotate once per second at this speed (I have no idea of the real speed). Now the plane drives up on a treadmill running backward at 5 mph. What happens? Does the plane stop relative to the ground? Almost certainly not. What happens is the plane continues to move forward at about 5 mph relative to the ground, but the wheels are now turning twice as fast (twice per second instead of only once). Now imagine a car going 5 mph drives up on that same treadmill. What happens? Does it continue forward like the plane? No, unlike the plane, it stops moving forward relative to the ground. Why the difference? With the car, the thrust of the engine is transmitted through the tires pushing back on the ground. By counteracting that movement with the treadmill, the car no longer moves forward relative to the ground. But the plane is pushed by the engines directly, not through the wheels. The thrust is not transmitted through the tires. They just freewheel. It’s as if the plane is pushing directly against the air. It’s not, it’s an action/reaction kind of thing, but in this case the effect is the same. So all that happens on the treadmill is that the wheels freewheel at a higher speed to make up for the fact that the treadmill is going backward, while the plane continues moving forward relative to the ground pretty much as if nothing had changed. So it takes off normally. Now if you make one of the other assumptions about “the speed of the treadmill matches the speed of the plane”, things can be different. But in your specific example, I think this is where you go wrong. 


#50




A different Problem with this...
Sorry if this has been explained elsewhere, but I haven't seen this covered...
This is a different look at why this seems to be highly problematic... Okay, a typical commercial jet needs to reach a speed of 150 to 180 miles/hour to lift off from the ground. If the plane achieves this speed on the treadmill (which is standing still relative to the earth), and it achieves takeoff speed, how does the plane suddenly go from 0 mph (again, relative to the earth) to 180 mph... This would be less of a takeoff and more of a launch. And even in scenarios where a rocket launches, the physics are much different. Someone please explain to me how the plane would make this seemingly miraculous jump from 0 to 180mph. cheers! 
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