Airplane on Conveyor Belt – Deconstructed
February 5, 2006 11:02pm by swood |
Riddles |
I have been doing a lot of thinking and research into the "Airplane on a Conveyor Belt" conundrum. For some reason, I find this thought experiment to be extremely compelling. You can tell a lot about a person’s thought process (and background) by the way they defend their position. It really is amazing.
As I’ve said previously, this problem tends to divide people radically. There are some people (very few) who just "get it" right from the start. Then there are those who believe that their initial, intuitive answer must be the right one… there’s just no other way. I must admit that I was in the camp that immediately said "no way, that thing can’t possibly fly… it’s obvious!" Eventually, after giving the problem at lot of though, I finally understood what was going on.
The Problem:
An aircraft is standing on a runway that can move (a conveyor belt). The aircraft moves in one direction, while the conveyor moves in the opposite direction. This conveyor has a control system that tracks the aircraft’s speed and tunes the speed of the conveyor to be exactly the same, but in the opposite direction. There is no wind. The pilot begins to add thrust to the engines…
The question is:
Will the plane take off or not?
The Result:
Yes, the aircraft will proceed down the conveyor-belt runway in exactly the same way it would down an asphalt runway. It will take off as normal. If you were watching from the sidelines, the take-off roll would look identical to any other that you may have seen. The only difference being the rate at which the wheels are spinning. They would be spinning approximately twice as fast.
The aircraft does not suddenly lift vertically (like an elevator) as some proponents of the "No, it won’t fly" camp seem to think we are proposing. That would be physically impossible, assuming no wind.
The Assumptions:
In order to avoid any messiness, let’s make one basic assumption.
- Let’s assume no friction at the wheel hubs. ie… the aircraft wheels can spin as fast as they want.
This isn’t a deal breaker or a shortcut, in fact, it’s not really even necessary. It is merely a way to dismiss the completely irrelevant argument that "there’s no way an aircraft’s wheels could spin 500mph without burning up." Yeah, that’s fine… but it has no relevance in our thought experiment. Nobody is going to build a giant runway conveyor belt either.
The Solution:
Here are the keys to understanding this problem:
- The wheels of an aircraft are "free-wheeling". They do not provide propulsion, and therefore do not "push against" the action of the conveyor belt.
- The thrust for aircraft movement comes from jet engines or propellors… not the wheels. Therefore, the thrust being applied to the aircraft body is completely decoupled from how fast the wheels happen to be spinning.
- Thrust acts according to Newtons Third Law of Motion – every action has an equal and opposite reaction. The thrust of the engines is acting against the air.
Because the wheels are free-wheeling and we have assumed zero friction at the hub, it follows that the conveyor belt, no matter how fast it is moving, CANNOT EXERT ANY FORCE on the aircraft with respect to forward motion! There is no force in our experiment that can oppose the thrust vector of the aircraft.
If the conveyor belt cannot exert any relevant force on the aircraft, you can completely ignore it. Ergo, the aircraft takes off as if nothing unusual is happening.
Addressing Some Common Arguments:
1) "The conveyor belt will cancel out any forward motion of the aircraft. The plane will not move at all."
Short Answer: The belt has no way to exert force with respect to the forward motion of the aircraft. All it can do is make the wheels spin faster or slower.
Long Answer: Your conditions are illogical.
- You claim that the aircraft will not move
- If the plane doesn’t move, then the conveyor belt doesn’t move either
- If the belt is not moving, then how is it cancelling any forward motion of the aircraft?
2) "The plane will remain stationary, but will lift into the air… thus taking off."
Answer: This is an aerodynamic impossibility (assuming no wind), which should be obvious.
3) "But if you said the the conveyor belt matches the speed of the WHEELS, it wouldn’t be able to take off."
Short Answer: See Argument #1.
Long Answer: Once again, your conditions are illogical. The conveyor belt can never "match" the speed of the wheels unless the aircraft does not move. With a tremendous thrust vector behind it with no opposing force, the aircraft will move. Once the aircraft begins to move, we enter into a paradoxical situation.
- X = Wheel Rotational Speed
- X = Conveyor Belt Speed, as per your conditions
- Z = Speed of Aircraft = Some non-zero positive number
The equation is: X = X + Z, which is illogical.
Example. The aircraft is moving 10mph (X = X + 10) with the wheels rotating at 10mph. Therefore, the belt must react and accelerate to 10mph. But now the wheels are rotating at 20mph… and so on to infinity.
4) "You can’t just ignore the conveyor belt as you claim. Take this situation for example…"
A guy is standing on the conveyor. He sees a plane moving forwards away from him at 10mph on the conveyor. He also knows that the conveyor itself is also moving at 10mph in a forwards direction. The total velocity of the plane in relation to the ground must be 20mph.
If the conveyor can be ignored then why is the plane’s total velocity twice what it would normally be if it was moving along on tarmac.
The fact that you cannot explain this indicates that you’ve either ignored or overlooked some of the forces at play between the conveyor and the plane.
I don’t see any violations.
You state that the aircraft is moving 10mph relative to the conveyor (perhaps as measured by a speedometer on the wheels). The conveyor itself is moving 10mph relative to the ground in the same direction. The total speed of the aircraft relative to the ground (tarmac) is 20mph.
I don’t see any problem with this. Ignore the conveyor by making it pop out of existence and you suddenly have an aircraft traveling down the tarmac at 20mph.
You can change your point of view as much as you like, but you still end up with an aircraft traveling 20mph with respect to the ground.
References and Further Investigation:
The Straight Dope – "An airplane taxies in one direction on a moving conveyor belt going the opposite direction. Can the plane take off?"
AVWeb.com – Pilots Lounge #94 – "Conveyor-Belt Runway"
Airliners.net Tech Ops – "If A Plane Took Off A Conveyor Belt…"
Tempus Fugit Blog – "Airplane on a Conveyor Belt"
PhysOrgForums – 275 pages of discussion!!!
Argument three…
“But you said the conveyor belt matches the speed of the wheels..”
The problem actually states that the conveyor belt matches the speed of the plane. Isn’t that different than the speed of the wheels?
If this was truly the case, why don’t we simplify airports and replace the long, huge, plots of runway land w/ conveyor belts? I bet there is a business model here somewhere, perhaps to the goverment, making aircraft carriers smaller.
The actual argument reads “But IF you said…”
Some people have said that if you worded the problem differently by having the belt match the speed of the wheels rather than the speed of the aircraft… it would make a difference. It doesn’t.
I was just trying to point out that it is, in fact, illogical to ask the belt to match the speed of the wheels. It can’t be done. And even if it could be done it would still remain completely and utterly irrelevant.
Keep in mind that the length of the runway would be unaffected as the plane still travels the same distance (relative to the ground not the belt).
You could even change the direction of the belt (which may help you wrap your head around this or confuse the situation even more). Theoretically, I think the plane could take off with the wheels rolling backwards (and the belt rolling forwards fast enough). That’s not fun to think about, actually.
Yep! That’s right! It doesn’t matter which way the wheels are spinning, or how fast.
Am I the only one who thinks you are spending WAY too much time on this?
David
I spent about an hour trying to explain this to a friend, and she still insists that we could make runways shorter by using conveyor belts as runways. Even after I pulled out the model airplane.
It was funny because she came up with the shorter runway idea while I was trying to exlain to her that the airplane would not stay in one spot, and then a few minutes later, we read Jeffs first comment stating the exact same idea!
I just need a video camera, a model plane, and a treadmill….
it says in there that it will take off, and the wheels will be going twice as fast as the coveyor belt, but thats the point, the conveyor belt is always matching the speed so it wont move!!
Poor Joe. I’m not sure exactly what you are trying to say, but you are wrong. How fast the wheels on the plane are rotating has nothing to do with the forward motion of the plane. They are completely independent of each other.
I still find it hard to believe that the wheels have nothing to do w/ the forward motion of the plane.
Forgetting about the conveyor belt for a moment, what happens if you remove wheels from the plane? Will the plane still move forward and take off? According to your logic, yes!
The only purpose of wheels on an airplane are as friction reducers. They attempt to eliminate any friction between the airplane and the ground. In real life, they cannot eliminate all friction… but they do eliminate nearly all of it. This tiny bit of left-over friction, which exists in the bearings of the wheel hubs is an insignificant force for the engine thrust to overcome.
In our example, to make things simple, we are assuming no friction at the wheel hubs (not that it would make any difference anyway).
In your example, yes… we can certainly remove the wheels as long as we replace them with some other friction-reducing device. How about covering the conveyor belt with ice and putting skates on the plane? That would be roughly equivalent and the plane would still take off.
Or remove the wheels and replace them with frictionless Teflon pads. That would work too.
The problem people have with this scenario, is that they make one wrong assumption.
The assume the plane remains stationary. Then they go into the explanation about how the plane requires forward motion, and go into detail about lift and air over the wings, etc, etc.
But the idea that the plane remains stationary is not given in the problem. The problem does not state that the plane remains stationary…it simply says that the conveyor belt moves at the same speed.
When people read the question, their mind tells them, “okay, so that means the plane remains stationary.”
But that is the heart of the problem. The real question is not wether or not the plane can take off. The real question is, can the plane have any forward motion (which is required before takeoff is allowed)?
The answer is yes.
“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?”
It amazes me how some people apply logic to a hypothetical question. The aircraft never achieves forward momentum and dispite physics the questions says; “tracks the speed of the plane and tues the speed to be exactly the same (but in the opposite direction.) No forward motion, no air over the wing, no flight.
Jet engines and propellers provide forward thrust so the aircraft can reach a speed to allow lift off. The thrust of a jet engine is not sufficent to accomplish flight by it self. Some aircraft such as the Harrier, Osprey, and rotory wing aircraft (helicopters) redirect thrust to allow lift off and stationary flight.
Fixed wing aircraft depend on speed to fly. Fixed wing aircraft have stall speeds, fly too slow and the airfoil fails causing the aircraft to fall from the sky.
This is a hypothetical question with parameters beyond the control of physics. Despite claims the conveyor cannot counter act the the engine’s thrust the question says just the opposite: 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).
The question does not address the speed of the wheels. The question addresses the speed of the aircraft. Zero forward movement, zero lift, no flight.
Have fun folks and no, this is not rocket science.
Thanks for commenting Mike, but you haven’t quite got it.
The speed of the conveyor belt is irrelevant. It could be moving 1000 mph and the plane will still take off normally assuming no friction at the wheel hubs.
In fact, picture that situation in your mind and the truth will become clear.
Mike,
“The aircraft never achieves forward momentum “. I agree with you completely – if the conveyor belt is REALLY tracking the exact speed of the plane, it will have to adjust its speed to that of the wheels on the plane. If that’s not the case, then this is just a trick question, with the conveyor belt put in there to throw you off.
David
I’m not sure how many times it must be said…. but the speed of the conveyor belt is irrelevant!
Pretend you’ve got a brick sitting on a frictionless surface. Now start moving the surface at 1000mph. Does the brick move? No. The velocity of the moving surface causes no effect on the brick at any speed.
Now stick a jet engine on the brick and throttle it up. The brick will begin to move opposite the thrust of the jet engine. It still doesn’t matter how the surface moves.
This example is the exact same situation as the aircraft on the conveyor. The wheels and bearings of the aircraft act as the frictionless interface between the aircraft and the conveyor.
It’s that simple.
Yes but what most people dont understand is that most planes will not be equipped with wheels that can rotate at such high speeds, and therefore will fail before rotation.
You are correct about it taking off if you assume a frictionless wheel bearing however if you then imagine your perfectly frictionless world then the it would be impossible to build the plane in
the first place, therefore if the plane does not exist then the plane cannot take off.
Okay, let’s NOT assume frictionless hubs. The plane will still achieve speed through the air and take off with no problem. The hub assumption was just introduced to quell those who would sidestep the problem at hand by claiming that the hubs would "burn up" or some such.
@bobthebob: You’re right but it’s a stupid thing to say regards this thought experiment, it even mentions this in the original article!
@Alan: You’re right but it’s a stupid thing to say regards this thought experiment. The idea of assuming no friction is to remove the overcomplication of having to explain that the conveyor will have a slight affect on the plane so it will reduce the planes forward speed ever so slightly. Thankfully the amount of friction in a system is constant regardless of speed so even if the conveyor were moving at extremely high speed it would still only have the same small affect on the plane. Unless of course you want to take it further and say that the speed of rotation causes excess heat which causes the bearings to expand and eventually resulting in the wheels failing.
It just doesn’t make sense to go into all that because it’s a thought experiment and it’s not what the question is getting at.
Someone posted this quiz on our website, and i was amazed how many people got the wrong idea.
In my eyes it has a very logic answer, wich is displayed nicely on this page!
A couple of points, though I agree the plane WOULD take off in almost any variation of this and that the speed matching argument is illogical,I think you cant totally discount wheel friction (takeoff roll might be affected a little) while no conveyor belt is likely to be fast ,enough bearings can only take so much speed and you could conceptually at least have a device that prevents an aircraft taking off due to wheel friction but it would be due to bearing failure from heat/vibration not from drag (we are talking about some pretty extreme forces here you would have catastrophic failure long before it would be able to absorb the energy of thrust!) another thing is that there are some aircraft designed for very short takeoff rolls where the propellers pull enough air directly over the wing to achieve takeoff (in real life there is always a takeoff roll but if you were to somehow old the aircraft back until the moment of takeoff… ) lightly loaded (very small fuel quantity/no payload) VSTOL aircraft may still get off the ground, they would instantly start to accelerate forward when released .these are very high power light weight machines with wings designed for extreme lift in low airspeed and some of them cover a very large percentage of the wing with a powerful prop wash! (ah screw it you know lets just say the plane is a harrier) anyways I basically agree, with the exception of catastrophic bearing failure the plane will fly.