The case of the plane on a conveyor belt has been extensively discussed in the online world, and still doesn't have a definite conclusion. Or rather, there are two opposing sides which believe wholeheartedly in their explanation, and these sides will *never* agree. The question is this: 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? I've read a thousand people's opinions about this and heard every comparison ranging from a skateboard on a treadmill to a weightless car on a sheet of paper. Taking into account some major assumptions (a plane on a conveyor belt is actually plausible; frictionless wheels, bearings, conveyor belt; no wind; ideal/instantaneous control system), my take on it is this: The plane won't take off [**Edit:** I changed my mind]. Here's my reasoning:
1. In order for a plane to take off, it needs to have air passing over its wings at a certain speed. This air can come from the plane moving down the runway, or it can come from wind. Theoretically, a plane can take off while sitting completely still, as long as there is a *significant* amount of headwind. However, since there's no wind in this example, the plane *must* be moving forward at a considerable velocity.
2. A plane's wheels are "dumb". In other words, they're only there to reduce friction. A plane could just as easily have no wheels and just rest on its belly on the runway. It could still take off because its motion is produced by thrust from its engines or the movement of air from its propellers. The wheels will spin when the plane is in motion and in contact with a surface.
3. As many people have pointed out, the question's wording can be confusing. a. If "plane speed" means "angular velocity of the wheels with respect to a stationary/ground observer", the conveyor belt would spin at an infinitely increasing rate, which is logically impossible. For example, if the wheels started to spin at 100 rpm, the conveyor would ramp up and spin at 100 rpm. But this would cause the wheels to actually be spinning at 200 rpm from the point of view of a stationary observer because the ground is no longer stationary but is moving at 100 rpm in the opposite direction. This would force the conveyor belt to spin at 200 rpm, 400 rpm, 800 rpm, etc., ad infinitum.
b. If "plane speed" means "angular velocity of the wheels with respect to the conveyor belt", the velocity of the conveyor belt would always equal the velocity of the wheels, no matter what. This means that the plane wouldn't move, no matter what. If the wheels started to spin at 100 rpm, the conveyor would ramp up and spin at 100 rpm. From the point of view of the conveyor belt, the wheels would still be spinning at 100 rpm even though they're actually spinning at 200 rpm from the point of view of a stationary observer. No matter what speed the wheels spun, the conveyor would *always* be spinning at the same speed as the wheels. This would prevent any forward motion of the plane.
c. If "plane speed" means "linear/horizontal velocity of the plane with respect to a stationary/ground observer", the plane speed would *always* be zero because the conveyor belt would *always* cancel out any forward motion of the plane. For example, if the plane started moving at 100 mph to the right (with motion derived from thrust), the conveyor belt would immediately begin moving at 100 mph to the left. Although the wheels would be spinning at an incredibly high rate (the wheel diameter doesn't equal the conveyor belt diameter, so the conveyor belt speed of 100 mph would translate to a wheel speed of something like 100,000 rpm [total guess, but the concept is there]), the plane would not change position from the point of view of a stationary observer. If it started at point A, it would stay at point A. This is the same result as part b.
d. If "plane speed" means "linear/horizontal velocity of the plane with respect to the conveyor belt", it's the same as part a. The conveyor belt would spin at an infinitely increasing rate. So in conclusion, the plane wouldn't take off because it wouldn't move from its original location.
Part of the reason this whole thing gets me so riled up is the attitude of the people who think they're right. Cecil Adams said, "Everything clear now? Maybe not. But believe this: The plane takes off." Thanks for your mediocre and confusing explanation, followed by an unqualified, unproven conclusion. Michael Buffington said, "Jason's Case of the Plane and Conveyor Belt riddle is confusing very smart people, so I thought I might explain it." Thanks, Michael. Obviously you know everything and everybody else knows nothing. Without you, we'd be nowhere.
This is my explanation. I put a lot of thought into it, and I even lost some sleep over it last night. I sort of think I'm right, but I'd be willing to be proven wrong if somebody has a good explanation. I'd also love to see this on MythBusters. #science |

Comments:Dave BrownDaveDave BrownDaveMr. Dave Brown said, "If the wheels are perfectly frictionless, the conveyor belt can be moving at any speed forward or backward and the plane will not move." I agree with this point. If the plane's engine is off and it's just sitting on the conveyor belt, if the belt were to move to the right or to the left, it wouldn't cause the plane to move. This is because the conveyor belt doesn't have the ability to impart velocity to the plane because the points of contact between the belt and the plane are frictionless. If it can't impart velocity to the plane, it also can't take it away. So if the plane starts to move at 100 mph by pushing on the air, and the conveyor belt moves in the opposite direction at 100 mph, this will have no effect on the speed of the plane. The plane will continue to travel at 100 mph because the conveyor belt can't possibly have any effect on the speed of the plane.

So part 3b and 3c from my explanation weren't completely true: The motion of the conveyor belt won't affect the motion of the plane. Therefore, the plane will move, it will gain enough velocity to become airborne, and it will fly.

I totally changed my mind.

WendyRichI could be really annoying and give the quick useless answer, or I could be long winded and give a more complete answer. I think I'll be even more long winded and give both!

1. Useless answer: You did not define what type of plane this is. I therefore define the plane to be a harrier. It can take off because a harrier can take off vertically. Nyea

2. And here we go...

The original prompt did not specify frictionless wheels, and this matters a great deal, so I will address both the case of frictionless wheels and wheels with friction. I will assume that the plane's engine can exhaust air at an arbitrary speed and ignore transonic effects if such speeds are necessary. I will also assume that the conveyor belt can move at an arbitrary speed, is of infinite length, and that its control system can adjust its speed instantly. (And if you want the quick answer, yes, the plane takes off.)

2a: With frictionless wheels:

This premise is almost untenable. Such a control system could not exist as to keep the belt moving at the same speed as the wheels. All of the linear momentum that the belt transfers to the wheel is used up by the wheel as angular momentum. Nothing is transferred to the plane. In this case, the only force on the plane in the x direction is forward, and is from the exhaust air pushing on the back of the fan blades (I'm assuming a standard turbofan, but this works for whatever). Thus, the plane will begin to move forward (relative to a stationary observer not on the belt), and smoothly accelerates to a takeoff. Before takeoff, though, interesting things are happening to the wheels and belt: This process spins the wheels faster, forcing the belt to move faster (though at any point in time it simply CANNOT actually move fast enough to match the speed of the wheels rotation). The belt will then continually accelerate trying to compensate for the fact that the wheels will always be spinning a little faster due to the plane's forward motion. However, the acceleration is not without an upper bound. What will happen before too long is that the coefficient of static (if you think this should be kinetic, you should read up on how a wheel actually works) friction will be overcome, and the wheels will skid. I originally thought that this would slow the plane down, but, again, the frictionless nature of the wheels means that the dragging from the belt only serves to slow the rotation of the wheels. What happens next depends on how this perfect control system works. If it is measuring the rotational velocity of the wheels, then it will slow down since the wheels slowed during the skid. As the belt and wheels both slow, they will reestablish static friction, and the wheels will roll again, and the belt will speed up again. This will continue to modulate, not affecting the plane's forward motion, but putting a ceiling on the speed of the belt. Then, the plane takes off, and the belt and wheels can finally slow down to rest. (Yes, I know I overanalyzed what was happening to the wheels and belt, but I thought it mattered for a while, and it comes into play below.)

2b: The wheels have some friction:

The plane takes off, but not for the reasons you most likely think. The forward force from the exhaust air causes the plane to start to roll forward, which causes the belt to speed up, spinning the wheels faster. Since there is friction in the wheels, this

doescreate a backward force on the plane. Since the belt is controlled to match the speed of the wheels, then this backward force will always be equal and opposite to the force applied by the air. By definition, the controller will not allow the wheels to outpace the belt, so they cannot roll forward. The plane will stay stationary with respect to an observer not on the belt. If the observer is anywhere nearby, they are likely to be deafened, as the belt's motor and plane's engines scream as they continually put out more power seemingly without bound. However, this actually does not go on forever. At some point, the wheels will skid for much the same reason as in 2a. The coefficient of static friction will be overcome by the force applied by the accelerating belt. Since the coefficient of kinetic friction will be lower, the plane begins to move forward, dragging its wheels the whole way. (I'm assuming the tires won't be worn out, and that the coefficients of static and kinetic friction remain constant despite the drastic heating that is likely occurring at this point.) At this point, the rotational speed of the wheels determines what happens. The forward motion of the plane, and the rearward motion of the belt are both contributing to the rotation of the wheels. The efficiency with which their linear momenta are translated into rotational momentum is dictated by the coefficient of kinetic friction. This spawns 3 cases: 2b1, 2b2, and 2b3. (In each case, the plane takes off, but I'm trying to be thorough)2b1: The coefficient of kinetic friction is so low that the wheels slow down. This causes the belt to slow, until the point that static friction is reestablished, and the wheels roll again. However, the belt and wheels are going far too slow to balance the thrust that the plane is putting out at this point, so the plane continues forward, causing the wheels to roll faster, which once again overcomes static friction, and we are sliding again. All this serves to do is possibly momentarily hinder the plane's acceleration, so it still takes off.

2b2: The coefficient of kinetic friction is at exactly the right value that the wheels spin with constant speed, so the belt moves with constant speed as well, maintaining the regime where the wheels are skidding. The plane continues to accelerate smoothly and takes off.

2b3: The coefficient of kinetic friction is high enough that the combined effect of the plane's forward motion and the belt's rearward motion cause the wheels and belt to accelerate without bound until the plane takes off. During this time, we are still skidding, so the plane will accelerate smoothly and take off (putting the wheels and belt out of their misery).

And with that, I put you out of your misery as well. Sorry it was so long, but I think I've covered all the main sticking points.

DaveRich, thanks for being long-winded. That's what I was hoping for. I think your most important point was, "All of the linear momentum that the belt transfers to the wheel is used up by the wheel as angular momentum. Nothing is transferred to the plane." That's basically the conclusion I came to before, but without the fancy explanation.

RichDaveWendyAt Last, Correct Ray-Traced Depth of Field in LW9! - Page 7 - NewTek DiscussionsDavePeterWhether an airplane can fly or not is all about airspeed, that is, the speed of the plane relative to the air. If a plane is sitting still on the runway, but is facing into an 80 MPH wind, it could take off without moving relative to the ground because its airspeed is sufficient for takeoff. If the wind is going the opposite direction, then the plane has an airspeed of -80 MPH. To take off, it would have to reach a groundspeed of 160 MPH.

When you think about the models designers use to test the aerodynamics, you think of them as being stationary in a wind tunnel. So assuming these tests aren't useless, it shows that the plane doesn't have to move relative to the ground; it just has to have the appropriate airspeed.

The conveyor belt is, as someone else said, a red herring. Groundspeed, whether real or apparent, is always irrelevant when it comes to figuring out whether a plane can take off. It is relevant when the plane practically wants to take off or land, because you have to start from (or finish at) 0 groundspeed, not airspeed. That's why planes take off and land into the wind.

DaveDavid PontzerSharaSharaDaveBut David Pontzer, I'm gonna have to disagree with what I think you're saying. I agree that if the plane doesn't move relative to the air (or the air relative to the plane), no flying happens. But I think the plane does move relative to the air because the conveyor belt doesn't have the ability to add to or take away from the plane's motion. Sorry to keep restating my point, but it's my main point and I think it's right.

And yes, I've gotten comments from Dave Brown, David Pontzer, and me. I'm waiting for Dave Chappelle or Dave Coulier to comment. They probably read my site.

RichI think I was pretty clear, if somewhat lengthy, in my explanation of exactly what physical processes come into play, and why the plane takes off. I also pointed out the different conflicting assumptions that are being made, and how those could affect the outcome. If we were debating this, someone might have actually cited a point someone else (namely me) made and refuted it. This has not happened. So, I will restate the quick answer from before: the plane takes off. If you don't agree, find a flaw in my logic above, and point it out. Dave, I think our opponents are just intimidated by our obvious nerd prowess. You shouldn't have to restate your point. The person who disagrees with you should, and then explain why they think you are wrong (which you aren't, since in my humble opinion we are right ;-p). Then a debate really could ensue.

DaveAnd for the ignorant yet true statement of the day: Just because you prove something with science doesn't mean it's true. Some people need to see physical proof. I'm still holding out for MythBusters.

RichI conditionally agree with your ignorant yet true statement of the day. It all depends on the definitions of "prove" and "true" (and unfortunately, in the wrong hands, "science"). To my mind if you "prove" something, that means it's true becasue that's my definition of prove (and dictionary.com's #1 definition as well: "...to establish the truth or genuineness of, as by evidence or argument: to prove one's claim.") "True" gets real interesting when we get into the area of opinions, which is interesting, since truth only deals with facts. Facts can be true or false; opinions are not nearly so constrained.

The actual logical problem with the statement is that science does not, has never, and never claimed to "prove" anything. (Mathematics on the other hand does so all the time.) In science, we have hypotheses, theories, and even laws. On the surface, one would figure laws are proven, but the best definition of a law in science that I've ever heard is simply a theory that is supported by all known experimental data. If you ever hear a scientist saying "...and that proves..." he's either embellishing to make a point, speaking informally (and will admit if questioned that there can be no

proof), or, if he really means it, is full of hot air.SharaThe_dudeThat said I have heard people try to argue that the questions was worded improperly and that it is sometimes stated that the conveyor belt matches the speed of the plane. Then the only required understanding is that once the planes thrust exceeds the friction coefficient then plan will move forward. You and your buds can prove this with a pair of roller blades, a treadmill, a water-ski rope, a fishing scale, and a 4 pack of really good Russian imperial stout. Attach the scale to the wall, the ski rope to the other end of the scale. Have your friend don the roller blades, climb on the treadmill, and power up the treadmill. The beer should be obvious. Note that only a small amount of pressure will be exerted on the fish weight scale. That will be true no matter what the setting is on the treadmill or even if he pulls himself forward.

Plane move through the air in relationship to the ground.

MikeI think people are reading too much into the test/myth. The point of the question revolves around the fact that if the airplane was to have no relative ground movement would it still take off at full power? The friction this, friction that, and whether or not it will move relatively was too much insight from the original idea.

The fact (as it has been mentioned before) is that if there is no relative air movement...then there is no lift generated by the wings...thus no flight...unless you have a rocket with enough thrust to weight ratio to become airborne. You mean to tell me that if we were to suspend an aircraft in the air to hold it in place with full power it would still fly? I don't think so. The air movement produced from the propeller (or jet engine) is not signifcant enough to get the plane of the ground...thus you must have a proper "airspeed" not ground speed to fly. We all know how an aircraft flies right? so this should be a moot point thus pointless to philosophize about.

If it somehow got airmovement over the wings an airspeed relative to the takoff performance of the specific aircraft it would fly. But if the conveyor belt kept it from achieving that airspeed...which in a no wind situation would be somewhat true...then no flight.

DaveDaveDave McHowever, as Rich pointed out, that coffecient of friction is not possible,nor is it claimed in the scenario. So at some point the propulsion force against the airframe through the air overcomes the force from friction against the wheels and the plane moves in relation to the static air, and you get wind speed over the wings and lift = flight. If you look at the question in the form of wheel rpm, at some point the wheels will hit the point where they spin no faster, but the plane keeps moving forward with respect to the static air because of the propulsion force thru the air. The wheels rpm is not tied directly to the speed of the plane thru the air, but to their relative speed against the conveyor belt, via friction of the tires against the belt. The force against the plane is related to the friction of the tires wheel bearings against the airframe. Given wheels on airplanes won't generate that much force against the plane, the treadmill hits a speed where the wheels are no longer gaing rpm as the airplane moves forward thru the air because the wheel bearings are not imparting enough frictional force against the wheels to make them gain rpm. The bearings are just skidding around the hub, which allows the airframe to move forward through the air, because the wheels themselves gain no rpm due to the airframes increasing forward motion. But that's just my opinion given my understanding of physics and wheels.

RichExactly!

Here's a prediction: Mythbusters will say the plane does not take off, because they will overlook the fact that the friction between the conveyor belt and the wheels must inevitably be overcome if the thrust the plane produces increases without bound.

JeffbvbellomoEither way, it is obvious (as mythbuster's proved) that a plane can take off from a conveyor belt if the conveyor belt moves very slowly.

The way I interpreted the question, is if the conveyor is capable of putting enough force and exactly the right amount of force on the plane to keep it completely stationary relative to the air, can it take off? The obvious answer is no. But doing the experiment is nearly impossible (a ridiculous amount of force from the conveyor to keep the plane still is needed).

Another interpretation is if the plane is moving forward relative to the air at the same speed the conveyor is running. Obviously, the plane still takes off.

I am confused so many people are interested, but really felt all the mythbuster's proved is they didn't think about the question.

JorgeSo, I would say the answer is:

- The plane takes off if and only if you accelerate the plane's engine, otherwise, it will not take off, the velocity of the wheels has absolutely nothing to do with about the plane taking off or not, they are just "friction reducers".

Jorge

DaveRichSammyIn reality, their premise would not be significantly different from that of a plane revving its engine & propellor to full flight speed with its brakes fully locked. If the thrust of the propellor is all that's needed for flight, we'd have no need for a runway - planes could sail right off into the sky from a stationary point. But they don't - you've got to have forward movement, with air moving over the airfoil of the wings, generating lift. Bernoulli's Principle and all that, you know...

JasonKevinI think the problem is that the other posters haven't come to the realization that the conditions of the myth itself are impossible. If the INTENTION of the myth was to ask if a plane could take off without moving relative to the ground (and without the assistance of wind), then putting a plane on a conveyor belt does not prove anything. A better test would be what another poster mentioned: tie the plane to a wall so that it CANNOT move forward, and see if it lifts off of the ground. However, if the intention of the myth was to test whether enough friction can be exerted on the wheels of a plane to keep it from taking off, then it has been busted by the Mythbusters.

JasonMLLet's take this even more theoretical. What if the airplane was bound to the treadmill. So, the airplane was not moving with respect to the treadmill surface. What is your prediction then?

This removes (or perhaps furthers the idea) of the friction being infinite and allows one to remove the worry about skidding reducing the friction at some point

TriffidThe analogies of attaching a rope to the back of the aircraft, or holding a toy car stationary with your hand do not apply. THE PLANE WILL MOVE FORWARD AND TAKE OFF! Luckily, I realized my mistake before I posted any letters to the contrary.

I cannot call it a trick question, like the rooster laying an egg on top of a barn, but at first glance it can mislead.

Kind of reminds me of this one...If a space ship traveling at the speed of light turns on its' headlamps, how fast does the light from the headlamps travel!

Well, maybe not.

Anyway...Naysayers, for a short time I was one of you. YOU ARE INCORRECT !

RoySo I thought it wouldn't take off, but I didn't think about the fact that the treadmill basically can't keep the plane stationary because the plane doesn't depend on its wheels. The Mythbusters subtly changed the conditions to the treadmill going backwards at the same speed as the plane was going forwards..of course the plane just accelerated to twice the speed relative to the treadmill and took right off.

If you designed a plane which took off by using a car engine driving the wheels to get up the necessary speed, then the treadmill could prevent it from taking off.

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