A Plane Problem
Some of you have probably seen this before.
"A plane equipped with fixed horizontal engines and wheel landing gear is placed on a huge treadmill runway. The treadmill has a clever design and always matches the speed of the plane, but runs in the opposite direction. Will the plane take off and fly or not?" A crappy diagram It's confused me for a while but I think I have my answer... |
Re: A Plane Problem
first response: it won't take off, because he has no speed, there is no lift
which is wrong, it will take off, which is explained later. |
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Of course it will. It'd take off in exactly the same way, assuming that the wheels are ideal (ie have zero friction).
The only reason it wouldn't take off (as theamion claims) was if the impetus to get moving was provided by the wheels, which is not the case. |
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There's no forces from the air running against the plane. The plane needs the oppostie resistance from the air to be able to take off.
My understanding of the technical words to describe how it works isn't that great, but I know roughly how planes take off. There's no air circulation over the wings (as it's exactly stationary, due to the treadmill moving at EXACTLY the same speed as the wheels), so it would just keep rolling on and on and on and on. |
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I was hoping Jakiri would take longer to post :(
I agree with him btw... |
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(Or just don't trust that we're correct) |
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I get the feeling it would take off but not very safely as there will be force against the plane to generate lift.
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The engines don't power the wheels and the treadmill doesn't affect the engines. |
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The wheels have next to no power in them. They exist purely so the plane doesn't skid along the ground. The plane's thrust is primarily (if not entirely) from its engines. This "magical" treadmill (which couldn't exist in reality) is the key of the question. It prevents the plane from moving as whatever speed the plane moves at due to the thrust of its engines, it will be counteracted. It'll never accelerate or move off the treadmill as the force pushing AGAINST the plane is exacty equal to the force pushing forward. The engines don't make it take off. The engines give it enough thrust to push through the air, then direct the air underneath the wings. Without movement from the plane through the air, no air will be directed underneath the wings. |
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tomkat is right.
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Strictly speaking, the problem is meaningless if you don't assume it can take off - because if you believe it makes it so the plane can't take off because the air is stationary around it, the speed of the plane is zero and thus the ground isn't moving. Of course, instantaneously, the plane has a speed relative to the air again, and so you get an infinitely fast progression.
The only sensible way of looking at it, if you want to consider what people think it means, is that if the ground has the same speed as the speed of the outer surface of the wheels. Assuming that... Quote:
I was going to do an analysis of different frictional environments, but then I realised that it was trivially obvious that the amount of friction would not be sufficient because airplanes need brakes and whatnot to stop them when landing on a runway. The necessity of the use of these things to stop a plane when the engine isn't running shows that the force is significantly less than half that of the maximum output of the engines when going at a speed sufficient to take off, and thus the friction between the wheels and the track alone in the above example would be not be able to stop the plane moving. |
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Someone explain this please. I can't see how tomkat could possibly be wrong, but i have a sneaky feeling he is. Im confused.
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Oh its ok I get you both now.
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But in answer to the question "would it take off" (ignoring friction issues and speed issues), the answer is simply "no", because there is no air flow over the wings. PS I deleted that post you quoted because it kind of ruins the thread |
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The wheels are simply moving twice as fast.
The plane itself isn't affected if you assume the friction between the wheels and the plane is zero. The plane will move forward and will take off when it reaches the required air speed. |
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I don't think you've understood the problem :( |
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There would be no air speed. The treadmill stops the plane from moving forward. You have to read the initial question again. This isn't a regular treadmill, that the plane can eventually accelerate against and take off. It is being held stationary by the magical super forces of exact speed. |
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It doesn't matter how fast the conveyor is going - it doesn't affect the plane. Just the wheels. If the plane is moving forward there is air moving over the wings. Hence it is possible to fly. |
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How are you modelling friction? Quote:
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Let's assume that (as in idi's example) the plane is being held up by an invisible force, but stationary (on bricks is a fine example). The engines can thrust and thrust and thrust and thrust, but they aren't pushing against anything. They're just blowing hot air out. To generate enough air underneath the wings, the plane needs to be moving through the air at a high speed. We know it isn't moving, as the treadmill is stationary (I assume) and the plane isn't moving off the treadmill. So no air flow exists (and any that does would just be generate by the engines and is insignificant). |
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THE PLANE IS NOT STATIONARY. For the plane to be stationary, the amount of friction from the ground would have to be enormous, which is not the case. |
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Try my toy plane experiment. |
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If you want to assume there's no friction then you should assume there's no airflow either, so I'm still right :( In reality the treadmill couldn't exist. I have said this a few times already :rolleyes: In relation to the AIR, the plane is stationary. How is it supposed to move forward? |
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Lets go over this in detail.
Let us first assume that there is a first order relationship between friction and velocity (that's y=mx + c, where y is friction, x is velocity and m and c are the coefficients of kinetic and static friction respectively, for all you GCSE maths types out there).. This is a pretty safe assumption, because it's true. Next: if you have a static force on an object, then it will take the same amount of time and distance to accelerate to a velocity as the object would to come to a stop from that velocity when the force opposes the movement. For the next bit, I'll explain it as if the forces were constant (which is fine, the net impulses aren't changed by assuming this, and we're only effectively dealing with the plane at two points, when it's stopped and when it's taking off, so we're not losing any particularly useful information). Lets call the force from the engines E. Let us call the (constant in this example) friction force F. The net force when taking off is E - F. The net force when landing, from the wheels alone (no breaks, or flaps or whatnot) is F. Now, let us assume that the plane doesn't need breaks or flaps when landing. It just stops of its own accord. In this case E - F = F E = 2F. Would it take off on a rolling runway made of the same materials? Lets investigate! The friction between the wheels and the track works as if the velocity was twice what it usually is, so y(2) = 2mx + c, not just mx + c. However, if c is greater than 0, which it is, y(2) is less than y, because 2mx + c is less than 2mx + c + c, obviously. So, the friction in that case would be less than 2F, so there would still be some resultant force on the plane, so it would still accelerate. It would still take off, albeit over a very long time if c is small. Remember that this is assuming that a plane will just roll to a halt once you've got it down on the ground. Do I really have to go into the real world, where you have to use breaks and whatnot, in which case E - F = F + K, where K is the retardation force provided by the breaks (etc), and is in all likelihood larger than F? Please say I don't. |
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I just thought of an example that might help people see how I'm thinking (I admit I might be wrong, but then the entire thing is hypothetical anyway).
If you're running on a road, then you feel the air blow against your face. Because you're pushing through it. If you're running on a treadmill, you feel no air blow against your face. Because you aren't pushing against it (ignoring the air being moved round by your arms and legs moving). With no significant air movement, the plane wouldn't be able to take off. I dunno if that helps you see how I'm seeing the problem, but hopefully it might. |
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No. How easy was that! |
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As the treadmill is moving at the exact speed, then it is exerting exactly the same force as the engines are thrusting, so surely E = F. |
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must admit i agree with tomkat here. the treadmill is controlling the speed of the ground the plane traveling on is moving at, it is not controlling the air around it.
as tomkat says the airplane takes off by manipulating the air around it to create lift. the air around the plane whilst on the treadmill is essentially stationary...right? |
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im in agreement with tomkat, there would be no forward velocity relative to the air around the plane, and thus no lift.
Think of it this way. If you are running on a treadmill which matches your speed, do you still feel the same air resistance against your face as you would if you were running on ground? |
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I give up. I prove it with actual proper maths and you still don't believe me.
Did you know that 0.9 recurring is equivilent to 1? |
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I think the main problem we have here is that Mark is thinking about this in a realistic sense, in which case the treadmill could never possibly exert the exact same force back. In that case, yeah, the plane would probably eventually build up enough speed and power to pull off the treadmill, and have accumulated enough to be able to take off.
But in this hypothetical situation, the treadmill can move at an infinite speed, thus preventing the plane from moving anywhere, no matter how much force it exerts from its engines. |
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Where T is the force exerted by the treadmill on top of friction (F) to counteract whatever forces the engines can exert (E). |
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Actually, my analogy has given me food for thought.
In the running on treadmill example, the force is applied against the ground ( running ) , with a plane that force is applied to the air. Im back to the "hmm, not sure" stage. It would need to be shown through experiment for me perhaps the root of the confusion is in the wheels, so lets remove those from the equation. Assume the plane's wheels and the threadmill surface are replaced with magnets of equal charge ( therefore the plane hovers ). Would it take off now? I suspect it would |
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I can't really emphasise my argument without repeating myself again and again so :shrug: .
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