Plane on a conveyor belt

hywelg

Any interpretation that considers the wheel speed relative to the conveyor or the speed of the periphery of the tyre to be the controlling variable renders the scenario void. 

Say the plane moves at 1m/s. So the conveyor moves in the opposite direction at 1m/s. So the wheels relative to the conveyor move at 2m/s. So the conveyor speeds up to 2m/s. Now the wheels are moving at 3m/s relative to the conveyor. So the conveyor speeds up to 3m/s etc....And all this in instantaneous whilst the aircraft moves at 1m/s. If the speed of the aircraft then increases then the rate of acceleration of the conveyor is increased again. So as you can see the conveyor speed very rapidly tends to infinite velocity. And as we all know conveyors are incapable of much over 50mph. 

The only logical interpretation is that speeds are relative to the ground. The plane takes off, completely unaffected by the conveyor.

CabbageCat

This thread is really cool. Just when it's settled some new joker comes along and makes the same mistakes.

There are some painful bits though.

Sporky

So why does it specify the wheels, rather than the plane?

I agree with your evaluation of what happens if it is the wheels' tangential speed, by the way, but I do prefer the conveyor to reach relativistic speeds and turn into plasma.

viz

It's a paper aeroplane and thus remains stationery. 

HAL9000

Sporky said:

So why does it specify the wheels, rather than the plane?

To deliberately obscure things. If the question explicitly stated that the aircraft was held in place then the answer would have been obvious 14 pages ago. The 'impossible' conveyor belt gets in the way of joined-up thinking (and creates the paradox).

CabbageCat

hywelg said:

The only logical interpretation is that speeds are relative to the ground. The plane takes off, completely unaffected by the conveyor.

It's not the only logical interpretation. It's certainly an interpretation that very few people here have made since it rather spoils the question.

I took the "speed of the wheels" to mean the speed of the outside of the wheel relative to the speed at the middle. The "turning speed" if you will. I can't speak for whether this was the intention but I suspect that it was.

CabbageCat

HAL9000 said:

 If the question explicitly stated that the aircraft was held in place then the answer would have been obvious 14 pages ago.

Likewise if it had explicitly stated that the aircraft was moving. But it didn't. You had to work that bit out.

HAL9000

CabbageCat said:

HAL9000 said:

 If the question explicitly stated that the aircraft was held in place then the answer would have been obvious 14 pages ago.

Likewise if it had explicitly stated that the aircraft was moving. But it didn't. You had to work that bit out.

IMHO if the question had said that the aircraft could move then the paradox wouldn't exist and it would have been able to take off. In the original question the conveyor is a quite simply a mechanism to hold the aircraft in place without saying so. Also, because the aircraft's wheels aren't driven, there's no force to prevent the aircraft taking off - only the constraint that the belt exactly opposes the rotation of the wheels.

CabbageCat

HAL9000 said:

CabbageCat said:

HAL9000 said:

 If the question explicitly stated that the aircraft was held in place then the answer would have been obvious 14 pages ago.

Likewise if it had explicitly stated that the aircraft was moving. But it didn't. You had to work that bit out.

IMHO if the question had said that the aircraft could move then the paradox wouldn't exist and it would have been able to take off.

Exactly. However there would be the paradox of how the plane could move if its wheels were not going faster than the conveyor but I think we would probably just write it off as an unclear question and assume that the wheel speed was just another way of saying the plane speed.

Maynehead

hywelg said:

Any interpretation that considers the wheel speed relative to the conveyor or the speed of the periphery of the tyre to be the controlling variable renders the scenario void. 

Say the plane moves at 1m/s. So the conveyor moves in the opposite direction at 1m/s. So the wheels relative to the conveyor move at 2m/s. So the conveyor speeds up to 2m/s. Now the wheels are moving at 3m/s relative to the conveyor. So the conveyor speeds up to 3m/s etc....And all this in instantaneous whilst the aircraft moves at 1m/s. If the speed of the aircraft then increases then the rate of acceleration of the conveyor is increased again. So as you can see the conveyor speed very rapidly tends to infinite velocity. And as we all know conveyors are incapable of much over 50mph. 

The only logical interpretation is that speeds are relative to the ground. The plane takes off, completely unaffected by the conveyor.

You are absolutely correct that if the plane is "moving" (I take that to mean moving relative to an observer in the same reference frame as the body of the conveyor system) then it does render the scenario void because it breaks the premise of the question.

That is precisely why the plane cannot be moving, as it is the only possible situation that does not render the scenario void, and therefore leads us to the correct solution of the question.

I.e. the plane does not move, the conveyor moves in the opposite direction at x m/s, so the wheels relative to the conveyor move at x m/s also. As soon as the plane tries to move, which would also lead to an increase in the rotational speed of its wheels, the conveyor would increase its own speed to exactly match the new wheel speed. If the plane continues to try to move forwards, the belt will continue to increase its speed to match.

It is correct to assume that in the real world there may come a time when the conveyor is no longer able to increase its speed to match the wheels, and the wheel speed will eventually increase to over and beyond the conveyor speed, at which point the wheels will be rotating faster than the conveyor and the plane will start to move. However, if you are using that as the basis to answer the original question then you are answering the wrong question, because the question specifically asks "can the plane take off IF (and only if) the belt matches the speed of the wheels EXACTLY".

4114Effects

DiscoStu

Emp_Fab said:

Well, I'm happy with my understanding of it.  It all works with Newton's laws, it's absolutely not a paradox.  Thinking it is is just a get-out from working out what's actually going on.  Nothing runs at the speed of infinity and it can be implemented in reality without the universe collapsing to a singularity.  Right, what's next ?  How much wood could a woodchuck chuck if a woodchuck could chuck wood?

And would anyone hear it fall? (Other than the space rabbit)

CabbageCat

Emp_Fab said:

Well, I'm happy with my understanding of it.  It all works with Newton's laws, it's absolutely not a paradox.  Thinking it is is just a get-out from working out what's actually going on.  Nothing runs at the speed of infinity and it can be implemented in reality without the universe collapsing to a singularity.

"I don't like that question so I'll answer one that I do like."

We have to either break the laws of physics, the plane, the conveyor or the question. Of those four I think that the question is the most inviolable since that's the framework of the entire discussion. The laws of physics can be assumed to be accurate so we'll keep them. The conveyor is imaginary and impossible so might as well be indestructible as well. The plane is real and bendy and melty. I vote that the plane gets it.

HAL9000

OK then... Hal9000's definitive final last answer but one....

1. The aircraft cannot move due to the constraint that the conveyor exactly counteracts the rotation of the wheels.

2. Newton's third law states that the action of  aircraft's engines must produce a reaction.

3. Since the aircraft cannot move (relative to the conveyor) the reaction force must go elsewhere to keep Newton happy. So my latest conclusion is that this force will very very slightly change the Earth's rotation. Doesn't break the rules of the conveyor, and doesn't break Newton's laws. QED or something.

Maynehead

HAL9000 said:

Sporky said:

So why does it specify the wheels, rather than the plane?

To deliberately obscure things. If the question explicitly stated that the aircraft was held in place then the answer would have been obvious 14 pages ago. The 'impossible' conveyor belt gets in the way of joined-up thinking (and creates the paradox).

I don't understand why people think that such a conveyor is absolutely impossible in the real world and somehow defies the laws of motion or even physics.

I have stated numerous times that the maximum speed the belt is required to reach is completely dependent on the maximum thrust generated by the engines and the maximum frictional force generated at the wheels.

If the maximum thrust from the engines is not enough to overcome the maximum frictional force at the wheels then then according to Newtons first law the plane will remain stationary.

If a plane used to perform this experiment has a relatively small maximum thrust and a wheel bearing friction that increased linearly with wheel speed, then it is perfectly feasible that even at maximum thrust, the plane does not move on the conveyor, because the conveyor is moving fast enough for the friction to counteract the thrust, resulting in the plane not being able to take off.

However, should a plane with a maximum thrust greater than the maximum resistance at the wheels be used (such as a Boeing 747), then as soon as enough thrust is applied to break the rolling resistance, the belt would continue to speed up until such a point that a) the plane begins to move forwards or b) mechanical failure of the belt or the wheels occurs.

Should either of the above conditions occur, then the experiment terminates immediately, as the behaviour of the system is no longer within the scope of the conditions set out. However, it may be of some significance that at the time of termination, the plane has not yet taken off, so one could argue that the plane cannot take off within the confines of the experiment's parameters.

Drew_TNBD

I've got an even better paradox...

Which brand of mermaid would you bone?


https://dl.dropboxusercontent.com/u/630473/mermaids.png

CabbageCat

Maynehead said:

HAL9000 said:

Sporky said:

So why does it specify the wheels, rather than the plane?

To deliberately obscure things. If the question explicitly stated that the aircraft was held in place then the answer would have been obvious 14 pages ago. The 'impossible' conveyor belt gets in the way of joined-up thinking (and creates the paradox).

I don't understand why people think that such a conveyor is absolutely impossible in the real world and somehow defies the laws of motion or even physics.

I have stated numerous times that the maximum speed the belt is required to reach is completely dependent on the maximum thrust generated by the engines and the maximum frictional force generated at the wheels.

If the maximum thrust from the engines is not enough to overcome the maximum frictional force at the wheels then then according to Newtons first law the plane will remain stationary.

If a plane used to perform this experiment has a relatively small maximum thrust and a wheel bearing friction that increased linearly with wheel speed, then it is perfectly feasible that even at maximum thrust, the plane does not move on the conveyor, because the conveyor is moving fast enough for the friction to counteract the thrust, resulting in the plane not being able to take off.

However, should a plane with a maximum thrust greater than the maximum resistance at the wheels be used (such as a Boeing 747), then as soon as enough thrust is applied to break the rolling resistance, the belt would continue to speed up until such a point that a) the plane begins to move forwards or b) mechanical failure of the belt or the wheels occurs.

Should either of the above conditions occur, then the experiment terminates immediately, as the behaviour of the system is no longer within the scope of the conditions set out. However, it may be of some significance that at the time of termination, the plane has not yet taken off, so one could argue that the plane cannot take off within the confines of the experiment's parameters.

I don't think it's realistically possible to have a conveyor that can exactly matches a plane's wheel speed without the plane moving. For one thing, there wouldn't be any wheel speed since the wheels would not turn without movement and the instant there is movement the experiment is void.

Maynehead

CabbageCat said:

I don't think it's realistically possible to have a conveyor that can exactly matches a plane's wheel speed without the plane moving. For one thing, there wouldn't be any wheel speed since the wheels would not turn without movement and the instant there is movement the experiment is void.

If you collect enough data, to map the engine's thrust against throttle position, and rolling resistance against wheel speed, it is perfectly feasible to design an automated system that increased the throttle and belt speed in unison such that the resistance constantly matched the thrust.

It does not have to be a reactive system if you have prior knowledge of its behaviour.

viz

Back to the trolley example (assuming frictionless bearings).

As soon as you start to push the trolley, according to the question, the conveyor will start moving, attempt to match the trolley wheels, and the trolley would stay in position. The harder you pushed, the faster the conveyor's wheels would spin.

What force would you feel on your hands if regardless of how hard you push, the trolley won't budge? The answer: equal and opposite. It would be like pushing a brick wall.

But would does this opposing force come from? Nowhere. It doesn't. There isn't any. Even if the wheels are spinning at infinite speed, there is no source of opposing force. Even if you are pushing the trolley with a feather, infinite wheel speed will not provide sufficient opposing force. 

However the question implies that the trolley isn't moving. 

Paradox. The problem is in the question. The conveyor wheels cannot keep up with the trolley wheels as soon as the forward thrust is non-zero. They will speed up to infinity instantaneously, but the trolley wheels will be spinning at infinity+1. 


If you bring into the equation the friction in the wheels' bearings, then you do have a resistive force that can present opposition to you pushing, and in theory the conveyor could speed up to match this and you would in effect feel a brick wall-like opposition to your pushing. 

CabbageCat

Maynehead said:

CabbageCat said:

I don't think it's realistically possible to have a conveyor that can exactly matches a plane's wheel speed without the plane moving. For one thing, there wouldn't be any wheel speed since the wheels would not turn without movement and the instant there is movement the experiment is void.

If you collect enough data, to map the engine's thrust against throttle position, and rolling resistance against wheel speed, it is perfectly feasible to design an automated system that increased the throttle and belt speed in unison such that the resistance constantly matched the thrust.

It does not have to be a reactive system if you have prior knowledge of its behaviour.

I guess. Or at least close enough. But without any force working against the plane (air resistance, friction, inertia {OK, I don't really understand inertia but I assume its absence is relevant}) even a tiny bit of thrust will require a massive backward speed as soon as the static friction on the bearings is overcome. It would have to be a crazy-assed conveyor if not an impossible one.