This is what happens when you consider this as a thought experiment and not a physics equation. What if I’ve got a 100mph tail wind?
No airplane requires ground speed to take off. It requires airflow across it’s wings of sufficient velocity to take off - or, to put it another way, it requires air speed, not ground speed.
The answer is not intuitive, and this fails as a thought problem unless you actually do the math.
[ul]Force generated by the engines directed in one direction.[/ul][ul]Force generated by the treadmill in the opposite direction (how about a treadmill driven by the same engines as the plane uses?)[/ul][ul]The forces meet and cancel out, at the wheel assembly (w/hot tires) and ground friction.[/ul]
No it’s not, really. Changing the treadmill from one that reacts to the plane wheel speed to a simple constant-speed treadmill is changing the meaning of the question.
I already explained in my previous post about the two different interpretations of the question. As did flight. As did timmerov2.
As far as I can tell, there isn’t anyone in this thread who disputes the fact that, if you interpret the question as meaning that the plane fuselage moves forward and the wheels spin at double speed, the plane will take off. There’s nothing wrong with that interpretation, and nothing wrong with that answer.
However, much of the rest of the discussion in this thread centers around the alternate interpretation. There’s nothing wrong with this alternate interpretation, either, but it does lead to different conclusions.
[quote=Paradoxic]
The answer is not intuitive, and this fails as a thought problem unless you actually do the math.
[ul][li]Force generated by the engines directed in one direction.[/li][li]Force generated by the treadmill in the opposite direction (how about a treadmill driven by the same engines as the plane uses?)[/li][li]The forces meet and cancel out, at the wheel assembly (w/hot tires) and ground friction.[/ul][/li][/quote]
I’m still not sure if I agree with you, or you agree with me. Agreed that the plane requires airspeed, and the answer is not intuitive. However, you keep mentioning a treadmill “driven by the same engines as the plane uses,” which makes no sense to me. It makes no difference what the power source driving the treadmill is, as long as it’s large enough to provide the power necessary for accelerating the belt.
If you assume a treadmill capable of large velocities and accelerations, and assume that it tracks the wheel speed of the plane, the belt will supply a force to the plane to counteract the engine thrust. However, this requires some coupling through the wheels–the wheel inertia, for example. The scenario you’re giving is a bit oversimplified because it skips the step describing the coupling mechanism.
The first problem is, as I see it, that the original problem set up impossible circumstances. Setting up a problem with impossibilities is kind of like dividing by zero… it sort of invalidates the whole thing.
There is no way for a conveyor belt to match exactly the speed of the rotation of the wheels, plus even if there was, that would set up a feedback loop with both the wheels and the belt going infinite miles per hour. Also, no matter what anyone thinks, the friction from the wheels turning would still be much less than the thrust of the engine… unless they were spinning at an infinite speed, I guess.
I just though of something… what if you locked the wheels altogether and poured water on the treadmill… then froze it. Hey! Then the plane could take off!
As you can see, I’ve already taken that into account. I’m looking at this as a physics question. The people that don’t believe the plane will take off (BF#2) see this as a thought experiment.
Perfect. See below.
Where does the question state ‘Wheel Speed’?
[sub]bolding mine[/sub]
Sorry, but you’re the one changing the question.
And in my analogy, I assumed that it was understood that as the toy car speed up, so would the treadmill. Still works.
To expand on my example of a toy car pulled down the runway……
Lets say the treadmill has rollers 6” in diameter. Attach a string from the toy car to a pully the exact same diameter, that is fastened to the ground. When we turn the pully, the toy car moves forward.
Now. Connect that pully to the 6” roller of the tread mill. But cross the belt that connects them. As the pully move clockwise, the roller moves counter clockwise.
Now, crank on the pully. It pulls the car forward. It moves the treadmill backward at the same speed the car is moving forward.
I don’t like this analogy as much as the person pulling the string, since this is a closed system. And the real world stuff would be two closed systems.
[quote= “The Question”)This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same.[/quote]
Actually, my example above with the pulley that pulls the toy car, and at the same time drives the conveyor backwards answers the question. And satisfies the constraints of the question as well.
The plane moves. If the plane can move, it can take off.
Hell, tie a cable on the back of the plane to the rear roller of the conveyor (you’ll have to wrap it underneath. See it?) As the plane moves forward, it pulls the conveyor backwards under its wheels. Here is another system that satisfies the constraints of the question and the plane takes off.
You’re assuming one version of the question. Nothing wrong with that. Paradoxic is assuming another. Nothing wrong with that either.
I suppose you could argue that your interpretation is more semantically correct for the given problem statement. However, note that in response to the same problem statement, Cecil himself explicitly states that “if the treadmill continues to accelerate” then “the plane stands still and doesn’t take off.” Admittedly a confusing flip-flop on his part, but it does illustrate the different interpretations of the question.
I went way way back to the original question. Before Cecil addressed it.
I was wrong about the ‘plane speed/wheel speed’ thing. The original question states wheel speed.
I think someone in that thread showed that it is impossible for the treadmill to match the wheel speed unless everything was at a full stop. Meaning not movin/spinning at all. Unless, the conveyor could continue to accelerate and keep the plane from moving. To look at the question that way, the question answers itself. The only way for the planes wheel speed can match the conveyor speed is if the plane is not moving relative to the ground. So obviously, it would not take off.
Not sure what the question was really asking. If it concerned the ability of a plane to take off while not moving but with the engines at full power, the answer is so obvious that I can’t imagine anyone seriously asking it (as it turns out some folks didn’t get that either :stunned: ). Why even include the conveyor part at all?
Anyway, I thought that my example of the plane actually hooked to the conveyor pulling it underneath itself as it went forward at least showed a system where the conveyor matched the speed of the plane. Does this also mean the speed of the wheels?
I’m quite sure the last statement is true. The wheel speed is always the aircraft speed plus the conveyer speed so unless the plane’s speed is zero the conveyor and wheel can’t have the same speed. it appears to be an unstable system. In order for the conveyer belt to move, the plane has to move. Once the plane moves in the slightest then the conveyer rapidly accelerates to its limiting speed, or to destruction whichever occurs first.
For me, personally, I thought the answer to the question where the plane is allowed to move (with wheels spinning) was so obvious that the question wasn’t worth asking–the answer is clearly what you stated, and how can anyone argue with that? In any case, interpretation of the question is key.
Unstable in the sense that the system quickly bumps up against some real physical limitations. Stable, though, in the sense that the conveyor could theoretically hold the plane in position to within an arbitrarily small distance, and small perturbations in thrust only require small changes in belt acceleration to hold it in place. Until the system bumps into the real physical limitations, of course.
You’ll have to go through this process for me step by step. If the wheel speed (in the conveyer matches wheel speed case) is the plane speed plus the conveyor speed, how can the conveyor ever be going fast enough? It seems to me that if the plane moves at all, the converyer will just go to its maximum speed and stay there.
Plane and conveyor speed are relative to the ground. Wheel speed is the peripheral speed of the wheel.
wheel speed = plane speed + conveyor speed
conveyor speed = wheel speed - plane speed
Until the conveyor is going fast enough for the force required to accelerate the wheel equals the engine thrust the plane speed won’t be zero. The conveyor speed then can’t match the wheel speed because it is slower by the plane speed. Based on computations in this post I don’t see how the required wheel acceleration force can be achieved before any reasonable, or even unreasonable, system limits out.
There is the problem of bearing friction. Such friction is usually taken to be a constant times the velocity. This will make the force opposing engine thrust grow faster than just the force required to accelerate the wheel alone. That reduces the required wheel speed for balanced forces. However the forces tending to throws the tire off the rim and the tire speed limit will probably be reached before force balance is reached even with friction helping the retarding force.
But remember that the conveyor responds to engine thrust with a change in acceleration, not just a change in velocity. For example, you calculated a nominal required acceleration in your linked post, based on some mass and thrust assumptions.
If we ignore physical limitations for a second, then what would happen in a perfect system would be that the plane’s engines would light off, the treadmill would respond with the constant acceleration that you calculated, all the forces would balance, and the plane wouldn’t move. Right?
In a less-than-perfect control system, there would be some lag in the response. So the plane’s engines would light off and the plane would surge forward slightly before the treadmill had a chance to respond. The control system would register the increased wheel speed, and respond with a larger treadmill acceleration than the steady-state acceleration you calculated. The increased acceleration would transmit more force to the plane, slowing it down, until plane speed = 0. Then wheel speed = conveyor speed (even though both are still accelerating!) and everyone’s happy.
Can’t argue there. Interpreting the problem this way (with an ever-accelerating belt keeping the plane stationary) is pretty clearly a thought experiment, and not practical at all. Unless…I recall someone in the previous thread suggesting an RC model outfitted with very heavy wheels. It’s possible something like that would be test-able.
I would like to see a computation. I think you are describing a Type 1 velocity system that will track a constant velocity with zero error and I’m not sure that this setup fits that description.
I’m assuming a system where the force from the engine thrust is balanced by the force transmitted through the wheels due to the acceleration of the treadmill (and the inertia of the wheels).
I’m not sure what calculation to do here, because the most applicable calculation is the one that you did yourself in the other current thread. Just take a look at that calculation and ask yourself what would happen if the treadmill accelerated more than the acceleration required to keep the plane stationary.
I think you’re right. No matter the details of the mechanization, an acceleration of 1245 rad/sec[sup]2[/sup] is needed to balance a thrust of 500 lb given the assumed wheel inertia. That means that the wheel and the conveyor will almost instantly, say within 1.5 sec., reach entirely unrealistic speeds. As far as I’m concerned that makes BR#2 a fantasy world scenario.
Well, I prefer “the realm of a thought experiment” to “a fantasy world scenario,” but, yeah, we’re talking about a situation that isn’t really testable on a real aircraft given real-world limitations.
This never ceases to amaze me. I think, in the first article he wrote, Cecil pointed out that this was intended as a thought experiment, not to find out whether or not the plane would take off, but to find out how people look at puzzles and try to solve them.
Everybody’s right.
Everybody’s wrong.
Interpetation #1: Assuming real-world materials and a vaguely-stated supposition that the runway moves backwards as fast as the plane moves forwards, the plane takes off, because making the wheels spin twice as fast doesn’t exert enough force to stop the plane. BR #1.
Interpretation #2: Assuming non-real-world materials and the iron-clad fact that somehow the treadmill exerts enough force to keep the plane from moving, the plane does not take off, because non-moving planes don’t take off without a lot of wind, and that’s not in the puzzle. BR #2.
So, for all of you arguing that real-world materials behave a particular way and even if you melted the tires off the plane, it would still screech down the treadmill and take off, congratulations. You’re right.
And for all of you arguing that the question states that the treadmill keeps the plane from moving, congratulations. You’re right, too.
And for all of you arguing that the treadmill, moving fast enough to satisfy group #2, generates enough wind due to the laws of contagion such that the wind passing over the wings causes the plane to lift off, you’re right, too, but you’re cheating.
I see at as asking a question which contains a paradox, like Cecil’s first column says.
It’s like the question, “A car travels 50 miles of a hundred mile trip at 30 mph. How fast does it need to go for the last fifty miles to average 60 mph?”
Implicit in the question is an impossibility.
I guess a more accurate answer would be ‘mu,’ but I’m more inclined to say you can’t make a treadmill that stops a plane’s forward movement, therefore the plane takes off.
I’ve been thinking over this thought experiment in my spare time. Since I am retired I have a lot of spare time.
I think it’s a lousy thought experiment. Such experiments are those that you can do theoretically but not practically. I don’t think you can even do this one theoretically.
Take the case where the only force opposing the engine thrust is the acceleration of the wheels. In order to stop the plane indefinitely you must constantly accelerate the wheels and that is not possible.
In the case where viscous friction is also included the scenario would be to run the wheel speed up until wheel acceleration force plus the viscous friction force balances the thrust. If the acceleration is then stopped the force immediately becomes merely the friction force and the plane moves forward. If the acceleration is not stopped the plane moves backward but here again you can’t maintain constant acceleration indefinitely.
Why not? The realistic limiting factors that I can think of are: a) material limitations of the tire and bearings and belt and so forth, b) velocity limit of the motor or whatever power source moves the treadmill, c) compressibility effects (sonic shock waves) when the conveyor exceeds the speed of sound, and d) speed of light.
To turn this into a thought experiment, I’d be comfortable ignoring all of these (and Chronos pointed out that the last one–the speed of light–isn’t a problem).
In any case, a thought experiment is useful in exploring the relative magnitudes and effects of various causes. To my mind, this airplane question is similar to “what happens when you approach the speed of light?” When you ask that question, you don’t get hung up on how long it takes, or what kind of vehicle you travel in, or where the fuel comes from, or any of those kind of questions.
[QUOTE=VicSarjoo]
The treadmill is irrelevant because it doesn’t matter whether the plane is on it’s wheels, or suspended from above by a set of steel chains hanging from a curtain rod that runs the length of the runway.
No, this is NOT AT ALL like this sort of situation. To create the situation of your “set of steel chains” the plan has to move through the air, to create the lift. Until the plane reaches a speed relative to the air mass around it (or, equivalently, to a stationary observer also not being moved by the treadmill) the wheels sit on the conveyor belt and tend to get moved backward alolng with the surface of the conveyor belt.
This is not a force-vector calculation (except ina VERY abstract way) and it is not a closed-vs-open environment. It is a semantic/logic exercise and subject to the same problems of any other logic problem. The key difficulty is sometimes called “assumptions contrary to fact” wherein we have put forth assumptions (that it is possible to build a conveyor belt that can move at a speed opposite to (and synchronize with) a forward motion of a plane) that are contrary to fact.
