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Airplane on a Treadmill...

So, I've seen this pop up in several places recently, and it's been bugging me a bit.

“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?"

My initial reaction was to think that it could, but the more I think about it (and draw some good ol' force diagrams) the more I'm inclined to think that it doesn't.

However, as some people in various comments have pointed out the problem is a bit vague so I figure I should explain some of my assumptions (some other comments I read also talked about wheels bursting into flames etc... and IMO those suggestions ignore the question of interest):
  • The airplane's engines apply force purely in the horizontal direction
  • The plane can only take off when the lift generated by the wings is greater than the downward force (i.e. its weight) that it's exerting on the treadmill (L>W).
  • Wings can only generate lift as a function of moving in the horizontal plane (L=f(v))
  • All the airplane's engines are being applied equally (no rotation from either side pushing more or less)
  • The wheels, treadmill etc... are all indestructible and all bearings etc... are perfectly frictionless
  • The only friction is that between the wheels and the treadmill (so wheels can't magically slide off)
  • This is a closed system with no external forces (no wind etc...). If it's not the jet or the treadmill it doesn't count (no nudging the plane)
Now assume you fire up the engines. What happens? Well the engines push against the plane in the horizontal and the plane pushes against the wheels and as they start moving the treadmill kicks on such that it eats up the horizontal force from the plane. As you fire up the engines more and more they keep pushing the plane against the wheels and it still goes nowhere because the treadmill provides an equal and opposite push against it.

Even though it's true that the engine is pushing against the air it can't make the plane start to lift off until the the velocity of the plane is great enough that the wings produce lift. If the plane had strong enough engines that could lift its entire weight and those engines were mounted such that they could move or apply force away from the horizontal then it could take off (like a Harrier), but so long as the engines can only provide thrust in the horizontal then all the force that the engines generate goes into the plane (i.e. they're pushing/pulling the plane) and into the landing gear and into whatever the airplane is sitting on. Until a plane reaches take off speed it is still basically an engine (or engines) pushing on parts of a vehicle in order to make the wheels turn upon a surface. If that surface is matching the wheels' speeds in the opposite direction then the plane can't move forward, therefore it can't get up enough speed to lift off (this is why a car can sit on a dynamo with its drive wheels spinning at top speed and it won't go anywhere).

Arguing that it doesn't matter what the wheels are doing in this problem would imply that a plane could take off if the wheels (and the treadmill) were standing still (and that doesn't seem to make sense). Until it takes off a plane's velocity is a function of its wheel's speeds (vw) minus the speed of the surface (vt) it's sitting on (i.e. v=vw-vt=0). Until that speed is greater or equal to the take-off speed (vto) the plane won't take off. You could re-conceptualize the problem as putting a plane on an infinitely long treadmill that's going at half of its take off speed in the opposite direction (0.5vto). If the plane can only move forward it would have to move at 1.5x its take off speed (vw=1.5vto) in order to overcome the speed of the treadmill (v=vw-vt=1.5vto-0.5vto=vto)and if its wings could produce life in either direction the plane would only need to accelerate under its own power to 1/2 of its take off speed in to lift off (v=vw+vt=0.5vto+0.5vt=vto) in reverse (momentarily ignoring the obvious aerodynamic issues)

A real world example of this problem would be seaplanes. Seaplanes can't take off if they're trying to take off against sufficiently fast moving water, however if they are turned around and they move with the water then they can take off much more easily (theoretically with no power if the water they were sitting in was moving fast enough).

So, any thoughts? Does that make sense or did I make an embarrassing 2+2=5 kind of error?

Comments

Maybe I'm too simplistic, but if the plane is not moving with respect to the earth, then air is not flowing over the wings. If air is not flowing over the wings, then the plane cannot take off because it has no lift.

I mean, I assumed it was written to be a simplistic trick question, so that's how my mind solved it.

I assumed the treadmill/wheels do not cause air flow, which I guess means frictionless system and all that. And of course that there is actually a treadmill that could go that fast.
It is a trick question and you fell for it. The "trick" is believing that the plane wouldn't be moving forward. It's a trick because it's easy to fall into the trap of believing that a plane on a conveyor belt would behave the same as a car or a person walking.

If you put a car on the conveyor belt you'd be right -- the conveyor belt would be able to thwart the forward motion of the car by perfectly counteracting its motion.

But planes don't move the way cars do. Planes have engines which apply force directly to the air. When the thrust from the engines starts that plane is going to move forward no matter what the conveyor belt does. And since the plane does move forward it will achieve sufficient airspeed for takeoff.
Me thinks you are correct sir. And I think this guy needs a basic rundown of how planes and, more importantly, their wings work to product lift.

If the plane is stationary there is no air flowing over the wings, so lift would be next to impossible (though I'm sure you could figure out a way with a hammer, bungi cords and a big enough rocket). The wheels are only on a plane to provide movement on the ground. whether it be on land, water or ice, the key is not the wheels, but the wings and the air which needs to flow over/under them in order to produce lift.

It'd be like getting on a treadmill, hitting max speed and trying to pole vault. Without a stationary platform from which to push off of, you wouldn't go very far up (if at all) ...though that'd be something funny to see

I guess it'd be possible though if the plane could be on the treadmill and flap its wings REALLY hard haha
"If the plane is stationary" is a huge, impossible if. There's simply no way for the conveyor belt to impede the forward motion of the plane. The wheels act as bearings which decouple the ground from the plane. As long as the wheels work and the brakes aren't engaged the plane is going to move forward.

You're correct that if somehow a conveyor belt acting on the wheels could prevent forward motion of the plane that it wouldn't be able to take off. But that's impossible. Wheels and axles are very efficient at eliminating friction and kinetic friction is almost immeasurably small compared to engine thrust on a plane.
You're making it way too hard.

1) Your version of the question is worded wrong. The question is supposed to say that the treadmill matches the speed of the plane, not the wheels. Really.

2) Even in your version, the plane can take off, if you assume all components are indestructible.

The engines of the plane put force on the air. This force on the air causes the plane to move forward. This causes the wheels to spin, which triggers the runway to start moving. However the runway does not have any ability to absorb the force provided by the engine. None. That's because the force of the engines is on the air not the tires. So long as the friction from the tires isn't enough to keep the plane from reaching takeoff speed (which, in the "speed of plane" version, it won't be, and it won't be in the "speed of wheels" if you assume the wheels are indestructible), the plane can takeoff.

If you have a Cessna 172, takeoff speed 60mph. If you have this Cessna turned into a 60mph headwind, it can take off with no groundspeed. If you have it in a 60mph tailwind, its groundspeed would have to be 120mph to take off.

The ground has NOTHING TO DO WITH A PLANE TAKING OFF. It's only airspeed that matters. The wheels (and thus the treadmill) are only relevant if they create so much friction that the friction has more force than the engines can give. As long as the plane has wheels, this cannot happen, because wheels just don't have that much friction.

Get a motorcycle with a sidecar. Put the sidecar on the same treadmill the plane is on (only running one at a time ;) ) and drive the motorcycle. The sidecar will keep up with the motorcycle, assuming its wheels aren't destroyed causing more friction than what should be there.
I think the important thing to consider is that the wheels have to move in order for the plane to accelerate. nugget made the claim that the rotation of the wheels was irrelevant, but that implies that the plane could take off if the wheels were not moving which doesn't seem to make sense.

If the wheels have to move in order for the plane to take off then it must use some of it's force to get the wheels moving. Once the wheels start moving the treadmill starts moving in the opposite direction so it takes more thrust to make the wheels moveas much as they were for the first split second. The treadmill compensates and you find youself applying more and more thrust to try and overcome the treadmill which is perfectly matching the speed of the wheels.

The motorcycle example doesn't work because the motorcycle is applying a force to the ground next to the treadmill. I had thought about using the example of how you push a shopping cart and you can feel that it's pushing forward and down (just try being run over by one ;), but I decided not to be because you would either have to be walking along outside of the frame of reference (on the ground, not the treadmill) or you'd be walking on the treadmill which everyone seems to agree would match your speed if you were walking on it.

This is why I chose the seaplane example. If you get on a boat in a fast moving river the river will carry the boat along at the river speed (vr) minus the boat speed (vb). It's possible that you could "stand still" in a fast moving river by applying enough force to make your boat match the speed of the river, but a seaplane couldn't take off because lift is a function of airspeed over the wings, not a function of how fast the plane is trying to move. A seaplane could only take off against the current if it could generate a speed equal to it's normal take off speed PLUS the river's speed.

Basically a plane doesn't act like a plane until it gets enough lift from its wings.
I hate to break it to you but yeah, you've made an embarassing 2+2=5 kind of error.

Here's the key, I think. Nobody is making the argument that a plane can take off without moving forward. It's absolutely correct that the plane needs airspeed in order to generate lift and consequently take off.

Here's the rub, though: It's completely, totally, entirely impossible for the wheels on a plane to prevent it from moving forward. The wheels on a plane serve only one purpose -- they exist to reduce the friction between the ground and the plane. They are not like the wheels on a car which are integral to forward motion.

Imagine that the conveyor belt is made of ice and instead of wheels the plane is sitting on ice skates.

If you want a more physics-derived answer, plot your force diagrams and attempt to explain *how* the wheels would possibly halt or prevent the forward motion of the plane. You have the thrust from the engine on the plane which causes the plane to move forward. In order for movement to be halted the ground would have to be able to apply an equal force on the plane in the opposite direction. How is this possible when the plane and the ground are almost completely decoupled by the wheels? The wheels are like ball bearings. The static friction (and then subsequent kinetic friction) of the wheels is orders of magnitude smaller than the thrust from the plane's engine.

The plane's force is applied to the air, the plane will move forward and take off. The runway could be moving foward, backward, or standing still and it wouldn't have any measurable effect on the takeoff.

Your statement "Until it takes off a plane's velocity is a function of its wheel's speeds" is incorrect. A plane can take off when its airspeed is sufficient. What the wheels are doing is irrelevant. Seaplanes have a little more difficulty taking off in the scenario you describe but this is because pontoons on a seaplane aren't nearly as frictionless as an axle.

The comparisons to cars and dynos is completely off the mark, though. Cars move by applying force to their wheels which is then transferred to the ground. THAT is why a car can sit still on a dyno. On an airplane the wheels are just along for the ride. The airplane doesn't impart any force to the wheels at all.
What the wheels are doing is irrelevant.

So, without wind, using only the engines, an airplane can take off if the wheels are not moving and they're not slipping on whatever surface they're sitting on?

The critical issue isn't that the bearings do or do not have friction, for these purposes that's irrelevant so they can be modeled as perfectly frictionless bearings, but if the wheels can't slip on the treadmill, then the plane has to exert some force on them.

If you do a force diagram of what's going on with the wheels you'd see that when you combine the force from the weight (that's applied downward) and the force from the engines (that's applied forward) the net result would be a force vector that's pointing forward and down (so basically pushing the plane into the ground. Once the wings generate a lifting force greater than the weight, the combined horizontal and vertical force vector becomes forward and up and the plane can take off.

This is why airplanes generally can't take off from a stationary position.
I think the only thing missing from that analysis is boundary layers - i.e. at some point that treadmill is going to be going fast enough to start pulling air. A lot of air. If the engines are powerful enough eventually they could make that air go fast enough to reach takeoff speed. Of course real life engines probably aren't that powerful. But then neither are real life treadmills and if we're imagining frictionless treadmills that can go orders of magnitude times as fast as a 747 (cause in order to counter the force of those engines ONLY with the friction from the wheels it's gotta go damned fuckin' fast) we might as well imagine the 747 has infinitely powerful engines, no? Ok, maybe it's a Cesna and not a 747 but that's still pretty fast.

Spherical Cows

Heh you were one of the people I was deginitely hoping would chime in.

I considered boundary layers, but it seemed like they would create more confusion which seemed to be the primary problem facing most attempts at understanding the problem (e.g. the wheels spin so fast they burst into flames and the plane can't take off). So I played with the problem trying to make assumptions that allowed for an infinite progression.

So ultimately, I'm assuming the treadmill will only generate a force on the wheels (no secondary wind) and that the engines can power up to infinity without running out of fuel etc... (I suspect that the fuel draw on jet engines powering up to infinity would be substantial ;).

Although technically for practical purposes you could also play with this assuming that the engines can produce enough thrust to equal 2 or 3 times the normal thrust needed to accelerate a plane into take-off and you'd probably have the same general problem.
This boils down to a simple question: does the air around an object on a moving treadmill accelerate relative to air around an object on a stationary treadmill, assuming the object does not change position relative to the treadmill in either case? I actually don't know the answer (my google-fu is weak), but if it does, then the airplane takes off, otherwise it doesn't. So there ;P.
Heh, simple question my butt :-p

I like your explanation although I'm still trying to really wrap my head around how it would work exactly :)
I think if we tried this...in a hurricane...it may just take off :-)

100

I couldn't resist