# So, you have a plane on a conveyor belt...

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Thank you @Snark for the wonderful explanation, as always.

As I understand it, the treadmill in the traditional problem is not magical, rather it is set to whatever acceleration the plane achieves, but in the opposite direction. So the wheels at any time would be spinning at twice their normal rate, while the plane accelerates to takeoff speed.

Additionally, if you'd like you can imagine an airplane taking off from a frictionless flat surface*. Even if the surface begins to move, the airplane will remain at the same velocity, and can take off in any direction, even directly against the motion of the surface.

*I'd better not use the word "plane" for this too.

Edited by cubinator

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25 minutes ago, Snark said:

It's like saying "I have a car that runs fast enough to make <arbitrary thing unrelated to car's speed> happen."

Oh so it's like the TV show Flash. Now I understand completely.

"But I don't know how to make bruschetta!"

"But I've calculated that if you can run at mach 18.27 exactly for 1.8 miles exactly, the knowledge will get pulled into your brain from a parallel dimension where you're a chef!"

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19 minutes ago, StrandedonEarth said:

I have trouble believing a conveyor belt can accelerate fast enough for friction from the aircraft's wheel bearings to counteract the thrust.

Correct, because it can't.  Sliding friction does not increase with speed.

19 minutes ago, StrandedonEarth said:

If that was the case, the bearings would quickly seize. At which point, the conveyor belt would hurl the aircraft backwards.

No, it wouldn't "hurl" the aircraft backwards.  It would simply gradually accelerate it backwards.  The amount of acceleration would be μg - a, where μ is the coefficient of friction between wheels and conveyor belt, g is gravity, and a is the acceleration the plane can get from its engines.  Sliding friction of rubber on surfaces will vary based on the surface, but is typically somewhere in the range 0.5 to 1 or thereabouts.  So take that, subtract acceleration from engines, and the backwards acceleration will be significantly less than 1g.  Dramatic, sure, but not fireworks-spectacular.

19 minutes ago, StrandedonEarth said:

The sudden acceleration would probably cause the plane to pitch forward

No, because it would essentially be the same set of forces that the plane experiences when it's landing on a normal runway and turns on the brakes.  Which is why the rear wheels are only slightly behind the CoM, but the front gear is way out in front of the plane.  So that's actually the designed behavior and it's not going to pitch down by any appreciable amount.

19 minutes ago, StrandedonEarth said:

giving the wings a positive AoA with respect to the direction of travel.  The aircraft would then take off backwards .

Well, the AoA thing won't happen.  And unless there's a long strip of conveyor belt behind the plane, it won't get anywhere near takeoff speed going backwards that way.  And even if it could approach those speeds... pretty sure that it would be horribly aerodynamically unstable in that configuration and would likely spin around.  Whatever happens next would be unpleasant, entertaining, and likely fatal, but I'm fairly confident it would not involve "takeoff" in any meaningful sense of the word.

19 minutes ago, StrandedonEarth said:

Special case: floatplanes fighting against the current will have difficulty taking off, as water offers much more resistance than wheels,

And that's different in more ways than one.  It's not just that water has more resistance-- it's that it's a fundamentally different kind of resistance.  Resistance to fluid flow (i.e. air, water) scales with speed.  Sliding friction does not.  Floatplanes have to drag their floats through the water, which gets harder and harder the faster you go (until you start hydropaning, as you say).  Whereas sliding friction (such as wheels rolling over a surface, or skids) doesn't change appreciably with speed.

5 minutes ago, 5thHorseman said:

Oh so it's like the TV show Flash. Now I understand completely.

"But I don't know how to make bruschetta!"

"But I've calculated that if you can run at mach 18.27 exactly for 1.8 miles exactly, the knowledge will get pulled into your brain from a parallel dimension where you're a chef!"

Pretty much, yeah.

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11 minutes ago, cubinator said:

As I understand it, the treadmill in the traditional problem is not magical, rather it is set to whatever acceleration the plane achieves, but in the opposite direction.

That's the problem, though.  Thrust is going to be constantly accelerating the plane from 0 to some positive value (X).  The treadmill must then accelerate from whatever speed it's at (call it Y) to X + Y.  This brings the plane to a stationary position again.  The plane now accelerates to X, and the treadmill must accelerate again to 2X + Y.  Then 3X + Y, 4X + Y, etc...

The treadmill is going to very quickly end up moving at unrealistic speeds, and will need to continue to accelerate to hold the plane stationary.  Physics can't help, so it has to be magic.

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35 minutes ago, Snark said:

Specifically﻿,

1. what acceleration can the plane manage from its engines ?

2. what speed can the wheels spin before they burst ?

3.  what's the rolling friction of the wheels ?

4.  what's the takeoff airspeed of the ﻿plane ?

5. what's the max acceleration that the conveyor belt can do ?﻿

Yeah.

But here's a thought :

A treadmill the width of a runway and the clear area around it that doesn't accelerate too quickly would just count as "ground" wrt the plane.

The only "difference" will be that there's a relative wind speed wrt the ground (and henceforth the plane).

Therefore, this condition is just the same as a tailwind.

And FAA and ICAO doesn't like you to takeoff/landing in a tailwind.

Facing the other way in a tailwind is a headwind :

So no matter how wrongly quickly the treadmill goes, you can always turn around and fly off.

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1 minute ago, razark said:

Thrust is going to be constantly accelerating the plane from 0 to some positive value (X).  The treadmill must then accelerate from whatever speed it's at (call it Y) to X + Y.

Yes.

1 minute ago, razark said:

This brings the plane to a stationary position again.

No it doesn't. The speed of the treadmill has no bearing on the speed of the airplane, because can assume the wheels have negligible friction, so the only effect of the treadmill's acceleration will be that the wheels spin faster as the plane moves, until it takes off and the treadmill is left wondering what it did wrong.

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22 minutes ago, 5thHorseman said:

Oh so it's like the TV show Flash. Now I understand completely.

"But I don't know how to make bruschetta!"

"But I've calculated that if you can run at mach 18.27 exactly for 1.8 miles exactly, the knowledge will get pulled into your brain from a parallel dimension where you're a chef!"

How could somebody not know how to make bruschetta? You just slice some bread, toast it, and spread some toppings onto it.

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5 minutes ago, cubinator said:

No it doesn't. The speed of the treadmill has no bearing on the speed of the airplane, because can assume the wheels have negligible friction, so the only effect of the treadmill's acceleration will be that the wheels spin faster as the plane moves, until it takes off and the treadmill is left wondering what it did wrong.

But the wheels are always spinning at the speed the treadmill is moving backwards, so they can't actually move forward, therefore the plane can't move forward and gain lift.  The way the question is set up requires a number of situations that can't actually exist, and therefore it has no connection to reality.  It's an absurdity, therefore the only answer is "magic".

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21 minutes ago, Snark said:

No, it wouldn't "hurl" the aircraft backwards.  It would simply gradually accelerate it backwards.  The amount of acceleration would be μg - a, where μ is the coefficient of friction between wheels and conveyor belt, g is gravity, and a is the acceleration the plane can get from its engines.  Sliding friction of rubber on surfaces will vary based on the surface, but is typically somewhere in the range 0.5 to 1 or thereabouts.  So take that, subtract acceleration from engines, and the backwards acceleration will be significantly less than 1g.  Dramatic, sure, but not fireworks-spectacular.

36 minutes ago, StrandedonEarth said:

The sudden acceleration would probably cause the plane to pitch forward

No, because it would essentially be the same set of forces that the plane experiences when it's landing on a normal runway and turns on the brakes.  Which is why the rear wheels are only slightly behind the CoM, but the front gear is way out in front of the plane.  So that's actually the designed behavior and it's not going to pitch down by any appreciable amount.﻿

36 minutes ago, StrandedonEarth said:

giving the wings a positive AoA with respect to the direction of travel.  The aircraft would then take off backwards .

Well, the AoA thing won't happen.  And unless there's a long strip of conveyor belt behind the plane, it won't get anywhere near takeoff speed going backwards that way.  And even if it could approach those speeds... pretty sure that it would be horribly aerodynamically unstable in that configuration and would likely spin around.  Whatever happens next would be unpleasant, entertaining, and likely fatal, but I'm fairly confident it would not involve "takeoff" in any meaningful sense of the word.

Well, since we're dealing with a magic conveyor belt here, much of physics was apparently thrown out the window anyways. So my post was somewhat tongue-in-cheek anyways. I could argue that the nose wheel would probably seize first (only one compared to two main gears, yaddyadda) Or maybe it's a taildragger, hmm? Pitchdown behavior would also depend on the control surfaces, and sudden deceleration could throw the pilot against the yoke, also forcing a pitchdown.

And nowhere did I say or imply that a backwards takeoff would be stable

I don't think there's enough popcorn in the world for this thread...

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19 minutes ago, razark said:

Thrust is going to be constantly accelerating the plane from 0 to some positive value (X).

Okay.

19 minutes ago, razark said:

The treadmill must then accelerate from whatever speed it's at (call it Y) to X + Y.

Okay.

19 minutes ago, razark said:

This brings the plane to a stationary position again.﻿

Wrong.  It just makes the wheels spin faster.  Doesn't affect the plane's motion.  @cubinator had it absolutely correct.

4 minutes ago, razark said:

But the wheels are always spinning at the speed the treadmill is moving backwards

No, they're spinning at the backwards speed of the conveyor belt plus the forward speed of the plane.

Conveyor belt accelerates a bit faster?  Wheels spin faster.  Doesn't make the plane move backwards.

Plane accelerates forwards some?  Wheels spin faster.

The conveyor belt can affect the spinning speed of the wheels by going slower or faster-- but it can't affect the speed of the plane.

15 minutes ago, YNM said:

A treadmill the width of a runway and the clear area around it that doesn't accelerate too quickly would just count as "ground" wrt the plane.

The only "difference" will be that there's a relative wind speed wrt the ground (and henceforth the plane).

Therefore, this condition is just the same as a tailwind.

Actually, no it isn't the same.

Planes don't care about ground speed... but they care very much indeed about airspeed.

"Plane on backwards-moving conveyor belt" isn't the same as "plane with tail wind".

Rather, it's the same as "plane that's just on a regular runway".

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15 minutes ago, Snark said:

Actually﻿﻿, no it isn't the same.﻿

It is, it's stationary wrt the crap it sits on either way. Same as a tailwind.

(tailwind here being the wind blowing from tail to nose, not just direction of travel.)

Edited by YNM

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5 minutes ago, Snark said:

Wrong.  It just makes the wheels spin faster.

5 minutes ago, Snark said:

No, they're spinning at the backwards speed of the conveyor belt plus the forward speed of the plane.

But the way the question is defined, the treadmill is always moving at the same speed as the wheels.  The wheels, by definition, cannot be moving faster than the belt.  Therefore, the plane must be stationary.  You've explained exactly why this is not the case, and I understand it.  (See here)  Therefore, the only way that fits the definition of the question is to ignore the physics and throw in some undefined, nonexistent force.  Or, "magic".

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2 minutes ago, razark said:

But﻿ the way the question is defined, the treadmill is always moving at the ﻿same speed as the wheels.﻿

Then turn the plane around and takeoff.

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21 minutes ago, YNM said:

It is, it's stationary wrt the crap it sits on either way.

No.  The plane is stationary with respect to the ground.

• Case A:  It's sitting on a conveyor belt, stationary with respect to the ground and the air (until the plane's engines move it forward).
• Case B:  It's sitting on the ground, stationary with respect to the ground and the air (until the plane's engines move it forward).

At no point does the plane move backwards.  No tailwind.

15 minutes ago, razark said:

But the way the question is defined, the treadmill is always moving at the same speed as the wheels.

Right.  Which is physically impossible-- the treadmill can't do that, because the plane moves forward.  A self-contradictory definition isn't a real definition, and a problem statement that involves physical infinities isn't a real problem.

You might as well say:  "The treadmill moves so fast that chocolate chip cookies rain from the sky.  Can the plane take off?"  And then get into arguments about whether engines can survive having cookies sucked through them.  It's a word-play, not an actual physical question.

In terms of physical reality, nothing that the treadmill does can affect the plane's motion in any significant way, and trying to word the question to make it matter simply invalidates the question.

Of course, if we say "but we're just making up a misleading linguistic conundrum and nothing here has anything to do with physical reality", then sure, we can say anything we want about what will happen because it's just flapping our gums.  Something involving rainbows and unicorns, perhaps.

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14 minutes ago, razark said:

But the way the question is defined, the treadmill is always moving at the same speed as the wheels.  The wheels, by definition, cannot be moving faster than the belt.  Therefore, the plane must be stationary.  You've explained exactly why this is not the case, and I understand it.  (See here)  Therefore, the only way that fits the definition of the question is to ignore the physics and throw in some undefined, nonexistent force.  Or, "magic".

What happens when the unstoppable cannonball hits the immovable post? Well, obviously, the question is ill-posed because if there is an unstoppable cannonball then there can't be an immovable post, or vice versa.

Similarly, all you have shown here is that the definition of the problem is impossible.

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10 minutes ago, Snark said:

The﻿ plane is ﻿stationary with respect to the ground.﻿

I wonder how baggage works then.

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9 minutes ago, mikegarrison said:

Similarly, all you have shown here is that the definition of the problem is impossible.

Well, yeah.  That was the point in bringing up the absurdity and calling it "magic".

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22 minutes ago, razark said:

Well, yeah.  That was the point in bringing up the absurdity and calling it "magic".

It ain't magic, but both yours and @Snark's stance are preposterous.

- The plane will start to move with the treadmill.

- The plane will be able to takeoff unles the takeoff "ground" speed is larger than VLO.

- ICAO and FAA vehemently opposes the idea of taking off in a tailwind situation.

So Tokyo Drift around and takeoff.

Edited by YNM

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1 hour ago, YNM said:

It is, it's stationary wrt the crap it sits on either way. Same as a tailwind.

(tailwind here being the wind blowing from tail to nose, not just direction of travel.)

No, because while a tailwind is exerting a force on the whole plane, the treadmill is only exerting a force on the wheels, which are always spinning in place relative to the plane.

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I was wrong earlier- the plane actually can take off, on a perfect treadmill.  It will go backward if they turn off their engines.

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28 minutes ago, cubinator said:

No, because while a tailwind is exerting a force on the whole plane, the treadmill is only exerting a force on the wheels, which are always spinning in place relative to the plane.

No leakage ?

So, flywheel ? (geddit ?)

19 minutes ago, DAL59 said:

I was wrong earlier- the plane actually can take off, on a perfect treadmill.  It will go backward if they turn off their engines.

Finally ! Someone with their common sense wired up.

Edited by YNM

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2 hours ago, mikegarrison said:

How could somebody not know how to make bruschetta? You just slice some bread, toast it, and spread some toppings onto it.

OMG YOU'RE THE FLASH!?

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@Snark's explanation here is excellent, though I would like to add that in practice I imagine the bearings in the wheels would seize up at a certain speed. and definitely break apart like a skateboard wheel in a water jet

of course nobody has explained why the plane is in this situation.

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1 hour ago, YNM said:

It ain't magic, but both yours and @Snark's stance are preposterous.

Well, sure, if by "preposterous" you mean "correct".

Seriously, though, if you think there's something wrong with my explanation, what in particular do you think is wrong about it?  Which particular assertion?  (As opposed to just saying "preposterous" without any explanation.)

1 hour ago, YNM said:

The plane will start to move with the treadmill.﻿

Can you explain?  If you mean "the plane will very slowly start to drift backwards, if it doesn't use its engines at all"... then yes, but that's exactly what I said in my "preposterous" explanation that you appear to disagree with, so I'll assume you don't mean this.

If, on the other hand, you mean "the plane will literally move with the treadmill", as in "the fact that it's on a treadmill will impart a significant motion to the plane" ... then what do you base that assertion on?  Objects move when there's a force on them, and to substantially accelerate a large object with a lot of mass needs a lot of force.  You only have a very little force between the treadmill and the plane (along the direction of motion), since the plane is sitting on wheels and is therefore close to frictionless.  Because of those wheels, the motion of the treadmill imparts very little force to the plane, and therefore doesn't accelerate it much.

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1 hour ago, Snark said:

If﻿ you﻿ mean "the plane will very slowly start to drift backwards, if it doesn't use its engines at all"... then yes﻿...

Yeah. That's what I mean.

1 hour ago, Snark said:

"the﻿﻿﻿﻿ plane will literally move with the treadmill", as in "the fact th﻿at﻿ it's on a treadmill will impart a significant motion to the plane"﻿﻿

Yes. If the engine is not on.

It's not even mutually exclusive. I can love a cow and still eat it's beef happily.

1 hour ago, cineboxandrew said:

in﻿ practice I imagine the bearings in the wheels would seize up at a certain ﻿speed. ﻿

It's called V-speeds.

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