Plane on a coneyor belt
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There isn't any problem with the physics/mathematics behind the problem. The physics say that the plane will take off. The calculations really aren't that difficult. It's the seemingly persistent...and entirely intuitive...notion that the conveyor will have any significant effect on the plane that's the problem.
The friction would be F=uR, where F= friction force, u is friction coefficient, and R= weight. The friction is constant (other than the plane getting lighter as it approaches takeoff speed); therefore the speed of the conveyor is totally irrelevant....nowhere does its velocty (v) appear in the frictional calculation.
This frictional force is not great. There are YouTube videos of people pulling Lear jets with their teeth. They are overcoming not only the friction, but also the inertia.
If the speed of the conveyor is irrelevant...and it is...it doesn't matter whether it's matching the speed of the plane or the rotational speed of the wheels. It's irrelevant in both cases.
As long as the forward thrust of the plane is greater than the friction (not exactly difficult) it will accelerate and take off as normal. It wouldn't surprise me to discover that a fighter jet could take off even with its brakes on. All it has to do is burn rubber for a few seconds. In such a case, of course, the conveyor would be stationary as the wheels wouldn't be rotating at all.
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I was refering to the fact that for the wheels to be moving forward they must spin infinitely fast+the speed of the plane.
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For any matter to reach an infinite speed would require all the energy in the Universe.
Perhaps the second part of the question ought to have asked whether such a model was possible at all rather than asking something more specific like whether or not the plane could take off. If the model isn't valid then whether the plane can take off or not ceases to have any meaning -
But the model does work (to an extent) and the plane cant take off.
Th reasoning: if the plane has any forward velocity this sets the wheels off in to a cycle whereby the wheels keep rotating faster and faster forever, this is impossible so the plane cant have a forward velocity.
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@alan fraser said:
As long as the forward thrust of the plane is greater than the friction (not exactly difficult) it will accelerate and take off as normal.
but the question was what happens, if the conveyor prevents exactly this. and for this question is theoretical we can asume that the conveyor can keep up to the speed (we could create a huge one with incredible speed after all).
is it right, if we say:
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theoretically the conveyor could stop the plane from taking off, because there is no windpressure on it's wings to take off
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practically the speed of the conveyor necessary to stop the plane from moving forward is impossible to reach with todays technology
(I think that is a satisfying solution, because it shows, that we all where right - only approaching the problem from differend ends.
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i like this response, it's clear, concise, correct, and most importantly has a reference to chuck norris.
pav
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Fair point about the conveyor spinning infinitely fast, although the wheels would then be doing the same speed as the conveyor belt, and as theyre going in opposite directions that means no forward velocity.
As for practicalaties, i dont think theres a lot of point discussing them in this situation.
EDIT: to sum up my argument, whatever speed the wheels on the plane are rotating at is exactly matched by the conveyor belt, so the plane cant have a forward velocity as then the the infinite cycle starts. Even if we then follow this through both the speed of the wheel and the speed of the conveyor belt tend to infinity, and so are the same once more, thus no take off.
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yup, Alan is still right on this. Its that simple. Theortically and practically, the plane still takes off.
Remus, ask yourself this: "Where does a car get foraward motion from?" and next "Where does a plane get forward motion from?"
Plot-Paris, the theortical part of you situation is wrong, because a plane doesn't need pressure on its wings to move forward. I thought it did earlier in this thread, but see now that is wrong. It moves forward by sucking air through its propller/turbine. It does not need any lift whatsoever under its wing to move forward. So the engines alone are capable of pushing it forward through the air. Therefore it doesnt matter what the wheels are doing, they just spin. The plane only moves in relation to the air around it, even when its on the ground. That is a very important phrase.
Chris
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@ Remus. No. Acceleration to infinity is impossible. That means that the entire model is invalid, it does not mean that the plane can't take off. You could just as easily argue that the travelator can't accelerate to infinity, therefore the plane CAN take off.
The wheels could only spin infinitely fast if the conveyor was also travelling infinitely fast. In practice the wheels wouldn't spin infinitely fast...they'd go as fast as they could, then they'd skid, adding a slight extra friction, but nothing that amounted to anything serious.@ plot-paris. No. For the umteenth time, the speed of the conveyor is irrelevent. The conveyor can only exert a small amount of reverse force in the form of friction on the plane. This is equivalent to (depending on the size of the plane) between 1 manpower and a couple of horsepower. It is also constant whatever the speed of the conveyor. Admittedly, more work will be done as the conveyor accelerates and the wheels spin faster. However, all that extra energy will be supplied by the conveyor, not deducted from the plane's thrust. It is the conveyor that is making the wheels spin faster than they would ordinarily need to. That work energy is rotational, not directional; it has no effect on the forward motion of the plane.
The plane, on the other hand has its full forward thrust available. It will move forward through the air nearly as normal. Where are you getting this no wind pressure from?
Are you really saying that a man with a rope tied around the tail of a plane can stop it taking off by pulling backwards? Chuck Norris could, of course. -
of course we all know, that chuck norris would never do that, because he enjoys us mortals struggling with this problem
ok. we all agree with the fact, that the plane has to move forward to take off, right (because it needs to move foreward to get air pressure under it's wings). of course we all know as well, that to achieve this foreward movement, the plane doesn't need this upward lift (that is merely a result of the forward movement).
so our whole difference lies in the question, if the conveyor, moving backwards, can transfer any energy in form of motion through the wheels to the plane.
you say that the wheels do not have the option to transfer this energy to the plane whatsoever.
and I completely disagree with that. let me tell you again, why:if you place pav's skate board on the treadmill again and switch it on, it will be transported backwards and fall off.
to prevent this, you hold your finger against the board - it stays stationary
if you remove your finger, the board will move backwards and fall off!that demonstrates, that energy was necessary to keep the board in place.
the kinetic energy was transferred from the treadmill, through the wheels, to the board due to the frictional resistance - thereby was transformed to heat-energy in the wheels.
if you speed up the treadmill, the tyres will heat up quicker, than with low speed. at the same time you need slightly more energy to keep the board in place on the treadmill.
and that demonstrates, that increasing the speed of the conveyor increases the kinetic energy transferred to the plane - therefore the theoretical possiblility to prevent the forward-movement of the plane is proven!
ps: nice argumentation with infinity, remus. would never have thought to take it thus far
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Ok, i think ive come up with a neat mathematical way of explaing this.
First, the question states that the conveyor belt will always match the speed of the wheel, so if we call wheel speed 'x' and conveyor speed 'y':
x=y
For the plane to take off it must have forward velocity of say 100 mph, this can be expressed as:
y+100
if we take to be any speed the conveyor had before the forward velocity.
so at the moment of take off:
x=y+100
Looking back at our first equation
x=y
and
x=y+100
cannot both be true at the same time.
EDIT: i realised a flaw in the original argument, and changed it.
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I am afraid, this is not correct, for you assume that the entire speed of the conveyor (Y) will be transferred to the plane, which is not correct.
we had to introduce the variable of the frictional resistance (f). so the objective speed of the plane (OS) would be (the speed the plane normally would travel is hence called SuS (for Subjective Speed)):
OS = SuS - Y*f
[Edit]
and to calculate Y:
Y = SuS*f
[Edit2]
Sorry, wrong again:
OS = SuS - Y/f
Y = SuS*f
therefore: OS = Sus - Sus*f/f
therefore: OS = Sus - Sus
therefore: OS = 0
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The basic premise behind the argument stays the same, though.
EDIT: (i think
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pav, we two have to agree with one of the answers for this question and then produce a SCF-TV episode.
because we all know: what you see on television news is bound to be true
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exactly!
TV never lies, it's the people in front of the cameras that do.
he he
pav
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There is some really bad maths and physics going on here. Mix that with architecture and it gets a little scary. Make sure you have good engineers guys.
Sorry, but I have now lost the will to live.
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Alan, if you can explain to me in a sensible way, why the skateboard should NOT fall off the treadmill (my example from 8 posts above), when no finger is holding it on it's position, you will have convinced mee...
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yes it is indeed. only that it is work (and my boss) calling at the moment
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If nothing else thats a pretty funny cartoon.
If you ever recover the will to live, i'm interested to know where the flaw is in my argument
You lost me with your thing about friction jakob...
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Remus, The terms aren't defined nearly precisely enough.
I think we can probably agree that conveyor speed would be that speed as observed by somone at a fixed point at the side of the conveyor. I think we can also agree that the plane's speed would be its forward motion through the air (assuming no wind and again relative to a fixed observer).But what the heck is "wheel speed"? Do you mean the speed at the bottom of the wheel...where it makes contact with the conveyor? Is that relative to a fixed observer or relative to the conveyor it is rolling over? Do you mean the speed of the hub (which would be the same as the plane's speed)? Or do you mean the speed of the wheel at the TDC (which would not be simply a negative value of the speed at the bottom, because you would need to add any forward motion of the plane to that).
Unless you are clear about your frames of reference you can make some very basic errors...like the equation. The way it is phrased creates a circular argument that allows the plane no forward motion. All it says is the plane remains static because it's static, therefore it can't take off. No kidding? That's why you end up with two irreconcilable functions...because you artificially injected a plane speed (+100) into an argument that prohibited a plane speed.
Jakob, of course the skateboard will fall off if it has no forward power. I thought we were discussing a plane that was powering forward? The skateboard however will not move backwards at the same speed as the treadmill.
If you don't believe me, I repeat, place a sheet of A4 on the table. Place a roll of tape on top of it and pull the sheet back at something like treadmill speed.The treadmill will transport the skateboard backwards...but only by as much as friction will allow, it's not like transporting it tied down on the back of a truck.
As I said before, this friction will remain constant regardless of the speed of the treadmill. Pulling that paper faster is not going to make the roll of tape go backwards faster...in fact it might even stand still or move forwards.All the business regarding the speed of the wheels is irrelevant...it has no direction, therefore can't counter any movement. All that is happening is that the wheels are storing potential energy so that they would rocket the skateboard forwards if the treadmill were to suddenly stop. There is no equation you can quote that will demonstrate that you need more force to keep the skateboard static as the treadmill increases its speed. The only point of contact between the two is the bottom of the wheels and the only force at play is friction. Friction is a function of the coefficient of friction and weight. Speed has nothing to do with it. Yes the wheels will get hotter with increased speed. That's simply because the amount of kinetic energy being transferred is outstripping the ability of the atmosphere to dispel the resulting heat. It has no more effect on the motion of the skateboard than if you were to siphon off that energy some other way, pass it through a dynamo and point an electric fire at the skateboard.
Increasing the speed of the treadmill has the same effect as increasing the speed of the paper under the tape...none at all.You made an analogy earlier to someone on a bike hanging off the back of a moving vehicle and stated that the force on their arm would increase the faster that vehicle went. No it wouldn't. It would only increase by two means a) if they were subjected to increased air resistance, or b) while the vehicle was actually accelerating. Once it reached a higher speed the pull on the cyclist's arm would be exactly the same as before.
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