Airplane-Treadmill problem

(Update: Cecil Adams of The Straight Dope provides a clear explanation of why the plane will indeed take off.)

David Pogue at the NYT has presented this classic airplane on a giant treadmill problem, and people are arguing about whether or not the plane would take off or not. Here's the problem:

"Imagine a plane is sitting on a massive conveyor belt, as wide and as long as a runway. The conveyer belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off?

"I say no, because the plane will not move relative the the ground and air, and thus, very little air will flow over the wings. However, other people are convinced that since the wheels of a plane are free spinning, and not powered by the engines, and the engines provide thrust against the air, that somehow that makes a difference and air will flow over the wing."

I say yes. Let's assume the friction in the wheel bearings is negligible. Putting a plane on a treadmill is like putting it on an icy lake. When you fire up the jets, the plane is going to shoot down the lake and take off just like it would on a runway.

Link

I've added the comments to this link in the extended entry.

Reader comment:

Mark H says:

I believe you are incorrect. The treadmill and icy lake are not equivalent because on the treadmill, there IS friction between the tires and the treadmill. The movement of the treadmill will have the effect of the plane standing still. No motion relative to the air and no lift. Therefore, the plane will just sit there.

Another way to think of it is for every incremental distance, x, the plane moves forward due to the thrust of the engines, the motion of the treadmill will move the plane backward and equal amount.

My reply: If the brakes were locked, then sure, the friction between the tires and the treadmill would be a factor to consider. However, we only need to concern ourselves with the friction in the wheel bearings, which is negligible.

Ian Varley says:

Mark: you are right. A plane doesn't increase velocity by pushing off
the groud; it does so by pushing off the air. The ground is just there
to keep the plane from falling into the center of the earth. (Think of
seaplanes … they can still take off despite lack of significant
friction with the ground). Since the air in this example is no different
from a usual takeoff, the plane would push off it and move forward as
usual. The difference, however, is that as the plane started to move,
the wheels of the plane would turn, and the fictional treadmill would
increase in speed to match … which would cause the wheels to turn
faster, thus causing the treadmill to move faster, etc … a mutually
reinforcing system, until the wheels and the treadmill both turned to
molten lava (and how fast that happens depends on how closely the
treadmill could match wheel speed). Meanwhile, the body of the plane
would be busy taking off as usual, unaware of the drama happening below
(except, perhaps, for the smell of melting rubber).

I think the consequences to the
treadmill (infinite acceleration) are what get people forget about and
get tripped up by in this example.

RH says:

You're right, the airplane will move forward. Unlike a wheel-driven
automobile, aircraft propulsion relies on Newton's Third Law. Thus, when
the airplane starts pumping air backward, if the wheels have any
reasonable freedom of rotation the plane _will_ move forward regardless
of what the damn conveyor is doing.

Given this, the really interesting part of this problem lies in how the
conveyor belt's control is programmed. "The conveyer belt is designed
to exactly match the speed of the wheels, moving in the opposite
direction." If this is taken literally one could argue that, since the
wheels start at zero RPM and are not being forced to turn, the conveyor
should hold the wheels at zero regardless of the plane's motion (-0 =
0). Thus, the conveyor would follow the plane's motion, and the plane
would take off with the wheels at rest.

We could also interpret this as having the conveyor oppose the wheel
rotation caused by the airplane's motion as if it were on a runway, in
which case the plane would lift off with the wheels spinning at -2X
their normal rate. Blown tires and an aborted takeoff are likely.

Or, we could interpret it as the author would mislead us into thinking,
that the conveyor should move opposite the plane's attempted motion at a
rate sufficient to keep it in place. Since the conveyor cannot prevent
that motion, in theory its backward speed will quickly go to infinity.
At which point the plane may fly through a wormhole to its destination,
saving much time and fuel. Or a black hole will form that ingests
plane, conveyor, and eventually Earth. I look forward to the
experiment. I think.

Johndan says:

1. If we're assuming there's no friction caused by the wheels on the treadmill, then Newton's Third Law (for every action, there's an equal but opposite reaction) says the plane should move forward in response to the thrust of the jets.

2. Planes generate lift by movement of air over the wings; since the jets are providing forward movement, the plane should eventually reach lift-off speed.

Peter says:

This thing has been around for years and always gets people quite worked up.
The problem is that already the question is flawed. It says "the conveyor
belt is designed to exactly match the speed of the wheels, moving in the
opposite direction". This can only be true if the plane is standing still
with regards to the ground (or else the speed of the wheels would be greater
than that of the belt). If the plane is not moving, there's no airflow over
the wings and the plane can't take of. So far so good. BUT (and a big one at
that…), there just is no way to design a conveyor belt "to exactly match
the speed of the wheels, moving in the opposite direction". As a conveyor
belt can't transmit any longitudinal forces through the wheels to the plane
(ignoring friction of course, but if you show up with a perfect threadmill
I'll raise you a couple of perfect wheel bearings), there's simply nothing
it can do to stop the plane from moving. So the plane will pick up speed
regardless of what the conveyor-belt is doing, violate the condition of the
question in the process and take off anyway…

Joseph says:

Mark, You are very much right. The trick to solving these problems is take them in steps. Step number one is turning on the belt with the airplane on it. There may be a bit of initial slippage due do the wheel's spinning before they begin to grab fully (relative to the friction between belt and wheel) and the plane moves backwards. Step two is to bring the plane to a speed consistent with that of the belt. That is to say; make it stop. How much force is require to do this? Very little, only enough to overcome the friction between belt and wheel. Picture it this way; You are on a treadmill wearing roller skates. To stop you from moving backwards all I have to do is put an tiny amount of pressure on your back (a pinky finger's worth would suffice), I don't need to apply 150 founds of pressure because you weigh 150lbs. I need only defeat friction. At this point any additional energy applied would propel the plane forward, and the plane's engine would have plenty of power to spare. It would need a longer runway, but it would not be an exceptional amount.

Tyler says:

It's really amazing how many very intelligent people think about this question in the wrong manner. It's not a trick question, just a tricky question. The comment about friction of the ball bearings in the wheels is really negligible to understanding why the plane WILL take off. The key is this: Airplane wheels do not supply any forward motion for the airplane; they are the equivalent of putting roller skates on your feet. When touching the ground, the wheels spin at the exact speed of the airplane, but that is because the engines are pushing the airplane forward, not because the wheels are moving.

All of the forward motion of the plane comes from the planes engines, which are in no way affected by the treadmill. The engines are grabbing air that is standing still in front of them, and pushing it away from the plane, the wheels are only along for the ride. Imagine if you had roller skates on your feet and were standing on a treadmill holding a stationary rope that outstretches in front of you. As the treadmill tries to push you backwards, your grasp on the rope keeps you stationary and your wheels spin. Now, someone begins to reel that rope in. The treadmill can try and compensate, but it only means your wheels spin faster, you will move forward in relative space. The same is true for the plane, only instead of grabbing a rope, the plane grabs the still air in front of it. If you were a passenger on such plane, you wouldn't even notice anything different was happening.

Michael says:

At first, I also thought the plane wouldn't take off, but reading some
of the comments, I understand why the plane WOULD take off now.

Disregard the conveyor for a minute and think of a normal take off
routine. As the jet engines go off, the plane moves forward, right?
Was the jet's forward motion propelled by the wheels in any way?
Nope. The wheels are, as you say, frictionless (or friction is
negligible). They serve only to keep the plane off the ground.

Now, think of the plane in mid-flight with its landing gears down. Is
it possible that as the plane is flying, the wheels could spin? Yes;
due to air resistance on the tires, etc, etc. I'm sure you can
imagine a dangling landing gear with the wheel spinning. Now, imagine
you were able to put your hands to the tire and spin the tire in
mid-flight of the plane. Does that spinning action of the tire affect
the flight of the plane? No. The forward thrust of the plane is
still coming from the engines and not the wheels. No matter how you
spin the wheels, it doesn't affect the forward thrust.

Now, imagine if, instead of your hands spinning the tires in
mid-flight, it was a conveyor belt. Would it affect the plane's
forward thrust? No. The same principle applies here as before. Even
at a full stop (instead of in mid-flight), the plane is "floating" on
the ground because the wheels are frictionless.

Think of an air-hockey table. The puck floats above the surface. Any
forward push sends it going forward. If you move the table underneath
the puck, does it affect the puck's movement? No. The table and the
puck are frictionless to each other, and so do not affect the puck's
movement.

One last image to help you visualize the idea of it. Think of a plane
sitting on the ground. The plane is physically in contact with the
ground only through the wheels. But because the wheels are
frictionless, you can think of the plane as "floating" above the
ground. Thrust from the engines pushes on the plane itself, and not
on the wheels or anything touching the ground. So, no matter how the
wheels are affected, the plane still moves forward when forward thrust
is applied and so it will take off.

Nex says:

You are totally correct, and Pogue is not just wrong, but so majorly confused that he disqualified himself from having an opinion about this subject. If you want to know why, but are in a hurry, skip ahead to the last paragraph of this e-mail.

In case you didn't know, Kottke had posted an acceptable version of "the classic Airplane-Treadmill problem" earlier this year, here.

It spawned a huge debate, which had to be closed down in less than one day. Much to my surprise, it had taken Kottke himself quite a while to figure out the correct solution. I still think he was completely wrong when he claimed that it is "obvious" that the plane would remain stationary. I have no idea how this proposition could appear intuitively right at first sight. In Pogue's article, it is a complete non-sequitur; he just says, "[…] the plane will not move relative the the ground […]", out of the blue, without any explanation, evidence, anything. But keeping the Kottke incident in mind, I'd say, while I can't relate to other people's difficulties with understanding this question, apparently you don't have to be an utter idiot to have such dificulties. So it seems that this problem is worth talking about, after all.

However, Pogue's question, as quoted by you, is NOT even worth debating! It is NOT "the classic problem", but mindless drivel. Behold this excerpt: "The conveyer belt is designed to exactly match the speed of the wheels, moving in the opposite direction." This is different from Kottke's wording. Clearly, Pogue either doesn't even know what exact problem he would like to solve, or fails to put it into words that make sense. What is the "speed of the wheels"; the speed a speedometer connected to the wheels would display, or the speed of the wheels' axles relative to the conveyer belt, or the speed of the wheels' axles relative to the ground on which the belt is resting? Which direction is the "opposite" one; is it the forward direction, thus causing the wheels to turn in the opposite direction (compared to the direction in which they usually turn when the plane goes forward on the ground); or is it the backward direction, opposite to the direction in which the wheels' axles move?

Pogue's question isn't very clear, as it can be interpreted in various ways. (For example, mentioning the wheels and their "speed" [whichever of the at least 3 possible wheel speeds that's supposed to be] is superfluous.) But if we rule out interpretations that don't make any sense to begin with, there are only two scenarios: The speed of the conveyor belt's upper surface, with respect to the ground the belt is sitting on, will match the speed of the plane, with respect to the same ground, in one direction or the other, so the wheels will either turn twice as fast or not at all. In either case, the conveyer belt will have only a minimal impact on the movement of the air above it. No matter in which direction it moves, the plane's airspeed will hardly change, thus it will take off at pretty much the same speed as it always does, on conventional runways. The speed of the plane with respect to the conveyor belt simply doesn't matter.

Jonah asked me to explain why I thought the plane would take off. Here's what I said:

Think of it this way. Say it's a prop-plane. Does
the propeller give
a damn how fast its wheels are spinning? No. The
propellor just pulls
that plane forward until there's enough lift under
the wings.

Jonah replied:

Ha! I get it…

it WILL move foward because the engine is creating
thrust, and the freespinning wheels wont stop it from
moving forward, so you do have wind over/under the
wings.

and a plane on the ground with no wheels, just sitting
there, with its engines burning away is not going to
fly up into the air.

Thats the thing I wasnt thinking about.. which was
that nothing was really stopping the airplane from
moving foward. It won't be stationary. Now I can
sleep.

I have been having an interesting email back and forth with Patrick Smith of Salon's Ask the Pilot.

Me: You know, the problem is that the question is poorly constructed. No matter how fast the treadmill goes, the thrust of the plane will cause it to move forward. If the treadmill was going 1000 miles an hour, the plane's wheels would be going 1000 mph + whatever speed the plane was moving forward. So then some kind of sensor in the treadmill would increase the speed of the treadmill to match the higher speed of the plane's wheels. Of course, when the treadmill's speed increases, the plane's wheels speed up to match the new speed of the treadmill, plus the speed caused by engine thrust. The speed of the treadmill and the wheels both approach infinity, but the wheels are always going to be moving faster. The best the treadmill can do is lag behind.

Patrick: I thought the version of the question that Pogue presented specifically stated that the treadmill would always increase its speed to match that of the plane's tires. How *possible* that might be is one thing, but I believe the point was to emphasize that the plane would not, could not, move forward. I assumed that was part of the scenario. Otherwise, why even bother with the treadmill in the first place? It's not a brain teaser about wheel friction, it's about relative motion, and understanding what gets a plane off the ground.

Me: If it's just about a plane rolling on a treadmill with no air running past it, then it seems like a dull question. Most people assume that if a treadmill were rolling backwards at the normal speed for a jet takeoff, the thrust of the jet would not be enough to counteract the backwards movement of the treadmill. But of course, in the real world, it would. To me, this is more interesting. But I agree with you — if the rule is that the plane is not allowed to move forward because of some kind of magical treadmill that can move as fast as the wheels of the plane (even though they have to move faster than the treadmill because of the speed increase they get from the thrust), then the plane will not fly.

Matt says:

I hate to disagree with the masses here, but it is very likely that the airplane would not take off. Ian Varley basically pointed out why this is so. If the treadmill really can match the speed of the wheels (either instantly or with some relatively minor delay), it is likely that the wheels would very quickly be turning at speeds well beyond what they can handle. When the wheels are rolling against the pavement (or in this case, the treadmill), on a molecular level, the tires are basically grabbing a hold of the surface and then as the wheels turn, the molecules are ripped away from the surface. As speeds increase, this creates tremendous friction. Aircraft tires are some of the most robust tires of all but are still only rated for speeds just above what the aircraft needs to take off (say, 160mph).

The speed of the wheels would increase very rapidly, probably well beyond the limit of the tires (to say nothing of infinity, which would be the speed if the treadmill could instantly match the wheel's speed). The tires would blow out relatively early in the take-off. It is then a contest between the "minimal" friction in the wheel bearings and the friction from the contact between the wheels and the ground to see which part fails next. If the wheel melts (which it surely would, depending, again, on how quickly the treadmill matches the speed) before the aircraft reaches take-off speed, the aircraft would be sliding along on the struts and I don't think there's anyone who will say that the struts will slide along the treadmill with minimal friction.

What happens next? Technically, I guess the treadmill would instantly stop because the wheels are no longer turning and the aircraft struts would fail, landing the aircraft on its belly to slide along the now-stopped treadmill at whatever speed it was travelling before the wheels failed. More excitingly, if the treadmill didn't stop (or took some amount of time to stop), the aircraft would probably be launched off the back of the treadmill at incredible speeds.

In the end, I guess it's slightly up in the air whether the aircraft could ever get.. up in the air.. but chances are it wouldn't, given the scenario.

Quitter says:

I've figured out the misunderstanding.

It took Rob Knop's post to explain where the lift off vs hold still people were diverging.

The hold still people are allowing the flaw of the thought experiment (the bad idea of trying to use a treadmill to hold a plane still to persist) to go unnoticed and are accepting as an assumption you could hold a plane still while it's engines are on full. A better way to do this would be to just chain the damn airplane to the ground.

The take off people are obsessing on the flaw of the experiment, and the necessity of higher rolling friction for a treadmill to have an effect on the airplane. A pilot could actually do this by applying break pressure to the wheels to increase the friction effects.

Depending on which assumptions you accept or reject, a physical impossibility occurs. If you accept you could generate enough backward momentum to hold the plane still relative to the ground's reference frame (which I think was a precondition of the problem as I saw it stated) then the hold-still people are right. If you reject the treadmill as a mechanism of holding a plane still, then the take off people are right.

I think it's funny that you're an engineer and I'm a physicist. This reminds me a lot of most of my arguments with engineers, with the physicist being sloppy setting up the preconditions, and the engineers obsessing over the details. The physicist doesn't care about the details that lead to the end result – the plane being held still in the ground's reference frame by whatever means necessary to demonstrate lift's importance for flight. The engineer attacks the set up of the whole problem and won't let you hold the damn plane still to make the point.

It's really kind of hysterical.

Dr. Paul J. Camp, Spelman College, Department of Physics says:

Whatever physicist gave Cecil his explanation is essentially correct. What bothers me is how many seriously misconceived ideas came up along the way. The ones I particularly noted before I got bored reading the back and forth were:

1. "Planes work by pushing against the air." OK, prop planes work that way. Jet planes do not. Jets operate exclusively on Newton's Third Law. If you provided them with a ready source of oxygen, they would work in space. They push against their own exhaust, which contains some of the air that entered the front of the engine, but you can't think of a jet engine as simply a conveyor belt that pulls air in the front and shoves it out the back. The purpose of the incoming air is to provide oxygen for the combustion of jet fuel, and in fact the air is deliberately slowed down a lot by a series of compressors for that purpose. Combustion produces a large quantity of very hot gas which, being hot, moves out the back at high velocity, but, due to the magic of chemistry, it is by no means the same gas that came in through the front.

2. "We need only concern ourselves with friction in the wheel bearings, which is negligible." Not true either. There is also rolling friction. Rubber is designed to grip the road surface. At a molecular level, the rubber is weakly bonded to the road surface. Rolling the tire involves peeling the rubber off the road, that requires the consumption of kinetic energy, and so there is friction between the tire and the road even when it is not sliding or tending to slide.

I solved this problem in literally less than five seconds by drawing a free body diagram. Anybody who didn't do that as a first step shouldn't be playing the physics game. There is an upward normal force from the road, a downward gravitational force and (eventually) an upward lift force. As long as the plane remains on the ground, these balance because the vertical component of acceleration is zero so the vertical component of the net force is zero. There is a thrust force from the exhaust pushing the engines (yes, I said that right — Newton's third law) that points horizontally forward, and there is friction backward. What form of friction? The wheel is not sliding or tending to slide on the conveyor belt because the belt moves along with it so there is only rolling friction and it is quite small, certainly smaller than the thrust or the plane wouldn't get off the ground even without a conveyor belt. Add the vectors. The net force is forward, so the acceleration is forward (Newton's second law). All else is math.

Mike says:

I just read the post on Boingboing about the Airplane-Treadmill
problem. As much as this topic has been beaten to death, I have yet to
read an explanation (including Cecil's) where people come to the
correct conclusion based on the assumptions that they have made. As
someone who has a background in airplanes and treadmills (don't ask),
here is my explanation as briefly as I can make it that somewhat
skirts the picking of any particular assumptions.

The "treadmill matching the speed of the wheels" terminology is a
classic example of poor specification definition and many lawsuits
have been fought over such wordings being placed into contracts. So
let's look at the end result: does the plane move or not?

The assumptions chosen to make the plane not move (sufficient friction
between the wheels and the treadmill and some rotational inertia of
the wheels combined with a steady acceleration of the treadmill *or*
sufficient friction between the wheels and the treadmill and wheel
bearing rotational friction combined with a really fast moving
treadmill *or some combination thereof) will also have the consequence
of making the treadmill move very fast eventually. That fast moving
treadmill will drag the boundary layer of air just above it (since we
can't "realistically" be choosy about which friction we want to
neglect) causing the air above the treadmill to start moving very fast
eventually. So even if the plane has no "ground speed", it will have
sufficient airspeed to take off eventually. This is assuming that
there are materials that exist which will not tear themselves a part
long before this point. If the treadmill does explode, the fall back
is that the plane will start to move when this happens.

The assumptions chosen to let the plane move will … well … let the
plane move and therefore gain enough airspeed to take off.

So, no matter which set of assumptions you make to have the plane
stand still or move, the plane takes off.

One can always find extreme assumptions that can be made that would
cause us to come to a different conclusions:
1. if we assume that this is all taking place on the Moon where there
is no air and the plane is not rocket propelled
2. if we assume that the wheel rotational friction and friction
between the wheels and the treadmill is so high that the treadmill can
achieve enough drag on the plane to counteract the plane's thrust at
some constant treadmill speed that doesn't cause the air above to go
fast enough to reach the plane's takeoff airspeed
… but finding those friction limits would need more specification of
the engine thrust than the initial problem provides ;]

Elizabeth Greer says:

My father, George Springer, is a Stanford professor and past chair of the Aero/Astro (Aeronautical and Astronautical Engineering) Department, so I thought I'd send the puzzle on to the experts. My father has 50+ years of experience in structures, aeronautics and fluid dynamics, I figured it would be obvious to him. Turns out it is a poorly worded and more subtle problem than it seems at first reading (as many have commented). After a couple of days, and discussion about it with another member of the department who is THE guy – the world's foremost expert – on this kind of thing, the bottom line is that the plane will move, even if the treadmill goes backwards, and will therefore take off. Here is the explanation (with demonstration) that he sent to me:

The problem here, of course, is that the poster (and Neal) cannot disengage themselves from seeing the airplane as a car. The difference between a car and a grounded airplane is that a car uses its wheels to propel itself forward, and an airplane moves itself forward by moving air. They assume that the runway moving backwards would move the plane backwards. This is what would happen with a car (that is in gear), so why not for an airplane? Well, because an airplane’s wheels are free rolling. There is obviously some friction, so there would be some small backwards force, but it would be infinitely small as compared to the forward thrust of the airplane.

You can test this with a piece of paper and a matchbox car (which has free rolling wheels like an airplane… or like a car in neutral.) Place the paper on a table, and place the matchbox car on the paper. Take your hand, and hold the car still with a lightly placed finger on top of the car. At this point you are providing no forward thrust, and the “conveyor belt” is not moving. The car remains stationary. Now, continuing to hold the airplane with a lightly placed finger, and start to pull the paper out from under the car, in the backwards direction. According to Neal’s logic, the car should push back on your finger with the same force that you are exerting on the paper… but this is not what will happen. You will find that your lightly placed finger is not stressed to any noticeable extent. The paper will slide out, and the wheels will spin, but the car will not be propelled backwards. The reason for this is is that the rotation of the wheels is not related to the movement of the matchbox car except by the very small friction component of the axle, which your lightly placed finger can easily control.

So now we have established that movement of the surface beneath a free wheeling object does not exert a noticeable force on the object. Next, we’ll see what happens when the object is trying to move forward. Attach a string to the matchbox car. Place the car at one end of the paper, and use the string to start pulling the car forward with a steady force. As the car moves forward, start pulling the paper out from under the car, backwards. Do you feel increased resistance as you pull the string? Of course not. The wheels are free rolling! Spinning the wheels does not make the object move!

When an airplane takes off, there is one major forward force… the forward thrust. The main rearward force is air resistance. The turning of the wheels provides a small frictional force, but because the wheels are free-rolling, this friction is very small. Unless the wheels are locked, the friction is always going to be less than the thrust, which means that the overall force is still forward, and the plane will still move.

Thanks for all of your work. I always enjoy learning new things on your site.

Dr. Paul J. Camp, Physics Department, Spelman College says:

I thought about this a little more over dinner last night and I think this is probably the canonical direction to take:

There is a thrust force from the action of the engines that is driving the plane forward. If it is not balanced by another force pointing to the rear, then the plane accelerates forward and it doesn't matter what the wheels are doing. That only affects how big the acceleration will be, not the fact that it exists.

From the interactions that the plane is experiencing, there is only one candidate — friction. And this is in fact the force that is providing a torque on the wheels to produce an angular acceleration. If we imagine increasing the thrust, then the friction will increase accordingly, producing a larger torque on the wheels to make them spin at the appropriate speed. So far, so good. Plane goes nowhere.

But the implicit assumption here is that static friction can always supply a sufficiently large force to produce a sufficiently large torque to maintain this static situation no matter what the thrust of the engines is, and that is just not true. Friction is peculiar in that it has an upper bound determined by the normal force and the coefficient of friction. That depends on the material and characteristics of both surfaces (here, rubber and concrete), so the upper bound varies depending on the situation but it does exist.

If the thrust of the engines is sufficiently large, then the torque required to spin the wheels at the appropriate angular acceleration will in turn require a friction force that is larger than the upper bound on friction. At that point, the wheels begin to slide instead of spin. The "rolling without slipping" condition is violated and all bets are off. At this point, the net force is indeed forward and the plane accelerates accordingly. I'm betting that this occurs at a pretty low thrust but I don't have the time to look up data on a real plane and calculate it right now. I have 100 final exams to grade and an equal number of projects, due by Monday. Some time next week, I might take a look at it and get back to you. I think this is going to happen long before any boundary layer effects become involved.

As a side note, any explanations that depend on driving the conveyor belt in some way are pretty irrelevant. While friction with the belt is pretty irrelevant, all you really need to assume in a model of this situation is a belt of negligible mass that is able to spin freely, without friction. The wheels will then drive it at the appropriate speed. It wouldn't be particularly hard to design a practical device that exactly mimics this ideal situation.

Now I've really got to do some actual work.

Coal says:

Sure you're getting loads of these, but it's fun to see how even the scientists don't get the puzzle. Here's my attempt at ending the constant round and round in circles nature of the discussion.The problem lies in that the question can be interpreted in (I count) 4 different ways. All the time that people refuse to agree on an interpretation, their answers will never agree. It's a classic case of a car painted green on one side and blue on the other getting into a crash, and the witnesses all fail to agree on what colour it is so the case gets dismissed. The 4 interpretations I can see are as follows:1) Could a plane take off without moving relative to the ground?Simple answer: yes, providing there's sufficient headwind or it's a Harrier (a Harrier IS a kind of plane, so it's not outside the parameters set by the question). Wouldn't necessarily be a smooth takeoff, or recommended at all, but under such circumstances, the treadmill is a red herring. The plane just needs
to be locked to the spot.2) In real life, would the thrust created by a plane's engine overcome the counter forces of a treadmill on free spinning wheels (enough that the plane could take off)?Simple answer: yes, in principle. However fast the treadmill turns, the wheels will just turn faster until there's sufficient speed to take off because thrust is produced by something other than the wheels. The problem is that this ignores the only condition the question sets. For the aircraft to proceed forwards, the wheels of the aircraft would have to spin faster than the treadmill, something that the question specifically states cannot happen. You're answering a similar, but ultimately different question. (Some argue that the question has been "poorly" worded. I think on the contrary it's been intentionally worded this way).3) Could the thrust of the engine overcome the friction of the tires?Simple answer: depends. *If the treadmill really is as the question states*, then no
amount of thrust will cause either the wheels to turn or the treadmill to move. Some think that they will both start spinning uncontrollably, but this will not happen. They will not move. So again, the treadmill is a red herring that would create the exact same effect as locking the wheels so they cannot turn. Whether or not the force of the engines could push the plane down the runway against the friction of the non-turning tires and gather sufficient lift to take off without the tires bursting, the undercarriage breaking off, and the plane exploding as it slides down the runway on its belly, will depend on a large number of unspecified factors. Scientists would need to know these things to determine the answer.4) Is such a treadmill possible?Simple answer: no. The question doesn't specify that the wheels of the plane have been modified in any way, or that the wheels are directly connected to the driveshaft of the treadmill, so the only way such a treadmill could be buil
t is if it senses the movement of the wheels and then attempts to mimic them. There would be a delay, and in trying to catch up, it would cause the wheels to spin ever faster. There'd always be a difference in the speed of the wheels and the speed of the treadmill, which would exactly coincide with the speed of the plane moving down the runway. Again, we're talking about different circumstances to the ones outlined.As whatever way you look at it, it's a trick question, the only way to win is not to play! Doh… too late!