Plane on a conveyor belt

Emp_Fab

Sporky said:

Paul_C said:

So . . . if it was a small, four wheeled trolley with a bar sticking out either side and two people standing on the floor started to push it, as the wheels rolled forwards, the conveyor belt would move in the opposite direction, which would turn the wheels but not stop the trolley moving forwards. If the conveyor belt had a fixed length they should be able to push it off the end, but would the rotation of the wheels accelerate or remain constant if the pushing was at a steady pace?

If you have the same constraint that the conveyor belt exactly matches the speed of the wheels then you can't push the trolley - basically you're performing the same function as the engines on the jet.

@Paul_C has it exactly.  @Sporky - I can see where you are going wrong now.  I absolutely, cast iron, guarantee you that I could push that trolley (or me and Paul_C could !) off the end of that conveyor.  I am all ears to hear how you believe that you cannot push the trolley.

mike_l

monquixote said:

Mythbusters actually tried this and showed it could take off.

Mythbusters did it with a microlight which needs very little airflow to get lift. A 747 needs considerably more lift, created by air flow over the wings.

Sporky

Emp_Fab said:

@Sporky  The question, as posed, is not the same wording as used in all of the other instances of the 'Plane on a conveyor belt' conundrum I've seen whilst searching the web.  In all the other examples, it's the plane's speed that is matched by the conveyor, not the plane's wheels.  I'm working on the assumption that we are trying to solve the conundrum as posted elsewhere, and not the misquoted version used here.  However - I can't (yet) see any difference between them.  Could you clarify what the constraint is that you feel challenges the puzzle ?

At first glance I can't see a difference between them either. It is tricky though, because how each person interprets the question affects which answer is obvious.

As far as I can see, that the thing on the conveyor is a plane is made irrelevant. It could be anything on wheels, whether propelled via the wheels or not. Given the statement that the conveyor matches (but opposes) the movement of the wheels, I can't see how the plane can be permitted to move.

However, I also agree that the force the conveyor can exert on the plane via the wheels can't approach the thrust available from the engines. Which means that the whole thing is a paradox. The plane can't make progress because of the conveyor, but the conveyor couldn't possibly stop the plane.

Sporky

Emp_Fab said:

@Paul_C has it exactly.  @Sporky - I can see where you are going wrong now.  I absolutely, cast iron, guarantee you that I could push that trolley (or me and Paul_C could !) off the end of that conveyor.  I am all ears to hear how you believe that you cannot push the trolley.

Sorry, missed this while typing.

The trolley isn't permitted to move (because of the conveyor constraint). At the same time, it obviously can't resist you pushing the trolley. Paradox, see?

If you reword the original slightly then there's a clear answer either way:

1. Can you make a conveyor belt that could stop a plane on it from taking off? No.
2. If you lock a plane's wheels in position, can it take off? No.

As the XKCD thing says, it's a poorly worded question.

Emp_Fab

But - how can it be a paradox ?  If you stick a real plane on a real conveyor, it will take off.  Where's the paradox ?

For anyone else that thinks the plane won't move, I've made a highly technical drawing of @Paul_C ;'s explanation :-)

4114Effects

I've never wanted to buy someone a pint and shake them by the hand as much as I do @Sporky right now ...

Sporky

Emp_Fab said:

But - how can it be a paradox ?  If you stick a real plane on a real conveyor, it will take off.  Where's the paradox ?

I already posted it, over and over.

The conveyor exactly matches the speed of the wheels, but in the opposite direction. That's in the original question.

That means that the position of the wheel cannot change relative to the conveyor.

So you can't move the trolley. If you could, you'd move the wheels relative to the conveyor, and as above, that is precluded by the setup of the scenario.

However, you obviously could push the trolley forwards, and the conveyor couldn't stop you (barring hitting relativistic speeds, turning into plasma and destroying the whole setup).

Thus it is a paradox - the scenario results in the trolley being unable to move without violating the scenario, but any of us could easily push the trolley. The two can't be reconciled.

sinbaadi

Emp_Fab said:

But - how can it be a paradox ?  If you stick a real plane on a real conveyor, it will take off.  Where's the paradox ?

For anyone else that thinks the plane won't move, I've made a highly technical drawing of @Paul_C ;'s explanation :-)

Now, instead of wheels, fix the box to another identical conveyor (inverted).  The paradox is that you have imposed a "rule" which is a 1:1 relationship between the "surface" (the belt) and the "wheel".  (It could just as easily be another belt, or anything, even one hexagon sat on top of another).  

The point is that they must cancel one-another out.  10cm roll of "wheel" equals 10cm roll of "surface".  I.e, the axle of the "wheel" cannot be moved without breaking the rule because if you move the axle along the "runway" there is no way that the relationship between "wheel" and "surface" can be 1:1.

Of course you can move the axle easily along the "runway", but if you break that rule, why impose it?  

monquixote

Sporky said:

Emp_Fab said:

@Sporky  The question, as posed, is not the same wording as used in all of the other instances of the 'Plane on a conveyor belt' conundrum I've seen whilst searching the web.  In all the other examples, it's the plane's speed that is matched by the conveyor, not the plane's wheels.  I'm working on the assumption that we are trying to solve the conundrum as posted elsewhere, and not the misquoted version used here.  However - I can't (yet) see any difference between them.  Could you clarify what the constraint is that you feel challenges the puzzle ?

At first glance I can't see a difference between them either. It is tricky though, because how each person interprets the question affects which answer is obvious.

As far as I can see, that the thing on the conveyor is a plane is made irrelevant. It could be anything on wheels, whether propelled via the wheels or not. Given the statement that the conveyor matches (but opposes) the movement of the wheels, I can't see how the plane can be permitted to move.

However, I also agree that the force the conveyor can exert on the plane via the wheels can't approach the thrust available from the engines. Which means that the whole thing is a paradox. The plane can't make progress because of the conveyor, but the conveyor couldn't possibly stop the plane.

In pretty much every other instance of this question being asked I've ever seen it's been phrased as "the conveyor belt travels at the same speed as the plane" not the same speed as the planes wheels which makes it a very simple question in which the plane takes off with the wheels rotating at double the speed. 

The question is designed to confuse people who don't get that a plane is not a car. 

This thread is the first time I've heard of anyone interpreting that the conveyor travels as the same speed as the wheels, but it explains a lot of the confusion that occurs.

Maynehead

ICBM said:

Sporky said:

As you say, it is a thought experiment, and the experiment - by stating that the conveyor matches the speed of the wheels - locks the plane in position.

No, read the question. It does not say that the plane is locked into position, only that the conveyor matches the speed of the wheels. That is true at *any* speed if the tyres aren't skidding, including an infinite speed.

@ICBM From what you said here I still have doubts as to whether you understand the mechanics of this problem.

As I have said countless times in this thread, for the plane to achieve net forward velocity, the wheels MUST move faster than the belt.

E.g. If the whole plane is moving forwards at 1 m/s, and the belt is moving backwards at n m/s, after 1s the belt would have travelled n metres backwards whilst the wheels would have travelled the same n metres forwards (if we assume no slippage between the wheels and the belt), PLUS the extra 1 metre due to the fact that the position of the plane has shifted forwards by 1m. Therefore during that second the wheels must have been travelling at n+1 m/s, thus breaking the condition that the belt speed always matches the wheel speed.

It is mathematically impossible for the plane's position to shift unless at some point during the experiment, the speed of the wheels differ from the speed of the belt. However, since it is stated that the belt speed always matches the wheel speed EXACTLY, one must accept one of the following: a) no such belt can exist in reality therefore the question is invalid (actually, with the addition of friction, such a belt could exist, at least for a short period of time), or b) the plane cannot move therefore cannot take off.

Sporky

monquixote said:

In pretty much every other instance of this question being asked I've ever seen it's been phrased as "the conveyor belt travels at the same speed as the plane" not the same speed as the planes wheels which makes it a very simple question in which the plane takes off with the wheels rotating at double the speed.

I've only seen it presented twice, and both times with this wording. That's not to say I don't believe you!

It is that relationship between the wheels and the conveyor that breaks it. Almost any other wording and the plane takes off quite happily for all the reasons ICBM and Emp have posted.

PolarityMan

It says the speed of the conveyer matches the speed of the wheels but that doesnt imply that the plane cant move relative to the ground, the conveyer might itself be sitting on a truck which is driving backwards at a speed equal to the speed of the planes wheels.

sinbaadi

"It's conveyors all the way down!"

Sporky

PolarityMan said:the conveyer might itself be sitting on a truck which is driving backwards at a speed equal to the speed of the planes wheels.

Or there might be a space-rabbit with laser eyes that will vapourise the plane if it even twitches.

Maynehead

PolarityMan said:

It says the speed of the conveyer matches the speed of the wheels but that doesnt imply that the plane cant move relative to the ground, the conveyer might itself be sitting on a truck which is driving backwards at a speed equal to the speed of the planes wheels.

Whilst true, it clearly circumvents the intent of the question. In your GCSE maths exam, if you had a question that said: "How far does a car travelling at 60mph in a straight line go in 90 mins?" Try answering with: "Depends if the car is driving on a massive long conveyor belt carried by a massive long truck." and see how far that gets you...

PolarityMan

Sporky said:

PolarityMan said:the conveyer might itself be sitting on a truck which is driving backwards at a speed equal to the speed of the planes wheels.

Or there might be a space-rabbit with laser eyes that will vapourise the plane if it even twitches.

It might be dark.

Sporky

PolarityMan said:

Sporky said:

PolarityMan said:the conveyer might itself be sitting on a truck which is driving backwards at a speed equal to the speed of the planes wheels.

Or there might be a space-rabbit with laser eyes that will vapourise the plane if it even twitches.

It might be dark.

Space rabbit ain't scared of the dark.

Maynehead

monquixote said:

If you add in the assumption that the wheel bearings are capable of generating sufficient friction to dissipate all of the energy of the conveyor and the jet then the sensible conclusion is that the wheels would lose traction with the belt and the plane might potentially skid and be able to take off without invalidating the experiment.  

You'd have to show me calculations based on the coefficient of friction derived from the wheel material and topology, plus the belt material and topology, and the maximum thrust of the engines, and the length of the runway, the gross loaded weight of the aircraft, the minimum takeoff speed, the tolerances of the tyres before failure, the weather conditions at time of takeoff, and a myriad of other determining factors, for me to be convinced that it definitely will take off.

I still don't think it's the intention of the question to force the questionee to consider all of the above factors in order to produce a definitive answer. However, I do admit that the omission of any premises regarding traction of the wheels does open up the possibility of more, albeit a lot less elegant, solutions to the problem.

monquixote

Maynehead said:

monquixote said:

If you add in the assumption that the wheel bearings are capable of generating sufficient friction to dissipate all of the energy of the conveyor and the jet then the sensible conclusion is that the wheels would lose traction with the belt and the plane might potentially skid and be able to take off without invalidating the experiment.  

You'd have to show me calculations based on the coefficient of friction derived from the wheel material and topology, plus the belt material and topology, and the maximum thrust of the engines, and the length of the runway, the gross loaded weight of the aircraft, the minimum takeoff speed, the tolerances of the tyres before failure, the weather conditions at time of takeoff, and a myriad of other determining factors, for me to be convinced that it definitely will take off.

I still don't think it's the intention of the question to force the questionee to consider all of the above factors in order to produce a definitive answer. However, I do admit that the omission of any premises regarding traction of the wheels does open up the possibility of more, albeit a lot less elegant, solutions to the problem.

Actually looked it up as I was interested. 

Word on the street seems to be that a loaded 747 should be able to hold on brakes, but if it's not loaded or it's slippy then it will slide. 

Failure conditions of the tyres don't need to be considered as previously mentioned. 

Maynehead

monquixote said:

Failure conditions of the tyres don't need to be considered as previously mentioned. 

I was hoping you wouldn't notice that, as I'd already written it and couldn't be bothered deleting it