The Airplane/Treadmill Problem: Corrections

Since my last post on the airplane/treadmill problem, I've had some extremely interesting conversations about it with a few friends. I'd like to make some clarifications and qualify a few of my earlier statements. I apologize, this looks like it will be another long post, but it also looks like now we're getting to the real meat of the problem.

"Eustace Bright" asked me if this has ever been tested. I'm sorry to say I never gave a very good picture of what would happen if you tried to test it and why. I tend to favor theory over practice, but theory divorced from practice quickly becomes aimless rambling. So, with that in mind, let me try to start from a real-world perspective this time.

The short answer is that some aspects of the problem would be extremely hard to test. Now for the full answer...

The Wings and Engines

I said before that with "powerful enough" engines, the plane would take off regardless of how the engines themselves work. I understated the point a little bit that for some types of engine "powerful enough" might mean "like the warp drive from the Enterprise". With big propellers and a very light plane, there might be a chance it could take off from a stationary position. I really have no idea how powerful of a rocket or jet engine it would take to create enough airflow, if it's even possible. Also keep in mind that more powerful engines tend to be bigger and heavier. If the engines are attached to the plane, they're going to have to meet a certain power-to-weight ratio.

In other words, it may be possible to build a plane that could lift off in place (with only horizontal thrust), but if you pick a plane at random, you could be reasonably certain that it wouldn't pass the test.

The Wheels

Remember that a quirk in the problem description requires the plane to be stationary without necessarily providing the forces to hold the plane stationary. This creates a weird kind of reverse causality where you can try to find the answer and find that you've lost the question. The upshot is that if you can show that no such force can hold the plane stationary, you thereby prove that the problem has no exact solution.

I had listed two candidate forces to hold the plane stationary, friction and rotational inertia (the resistance to spinning faster or slower) in the wheels. The friction I'm referring to is in the wheel bearings. As far as rotational inertia, I think I need to clarify what I meant just a bit with another quick example:

This time, imagine a ball on a treadmill. When you turn the treadmill on, the ball moves back a bit before it gets rolling. Once the ball's spin gets up to speed, its momentum alone keeps it spinning, like a ball spinning in mid-air except supported by the treadmill. When you stop the treadmill, the ball will "coast" forwards quite a ways before friction finally brings it to a stop.

I had suggested that those two forces would be enough to keep the plane stationary, but then I discovered a flaw in my reasoning: neither of those forces is proportional to the speed of the treadmill.

The sliding friction in the wheels is a constant force, regardless of speed. Unless the wheels have some serious wobble or mechanical problem that makes them start to lock up at higher speeds, we'll only ever get a fixed-strength force from the friction. If the engines exceed that force (which is extremely likely), there's nothing we can do with the conveyor belt to compensate.

The rotational inertia is a little more promising, but not much: its associated force won't be proportional to the speed of the belt, but it will be proportional to the acceleration of the belt. Momentum and inertia are related to velocity whereas force is related to acceleration. Remember that once my hypothetical ball was up to speed, it stopped moving backwards without any additional forces having to act on it (not even friction). That means with the conveyor belt spinning at a constant speed, the wheels will offer no real resistance to a constant force from the engines.

My conclusion on the "exact match" constraint between the belt and wheels is that it very well may be impossible to balance the forces and create a stationary plane to meet the requirements in any practical scenario.

Postscript: An Apology

Last time I mentioned Mythbusters and some YouTube videos that claimed to solve the airplane/treadmill problem, but didn't seem to have anything to do with the real issues. They demonstrated exactly that the plane wasn't stationary, so they couldn't have been answering the question exactly, which had specified that the plane must be stationary. Well, that was true, but they were trying to answer the follow-up question of whether any such real-world forces could hold the plane back against the force of the engines, which turns out to be a tough part of the problem. Since the friction is a constant force, they've definitively ruled out friction as a candidate in any realistic scenario.

That said, I still have some ideas to kick around before I'll concede that the problem can't be saved. It'll be a longshot, though.

Edit: I did some more thinking and ruled some ideas out. I think I'll leave the whole airplane/treadmill thing alone for a while. It's more fun to discuss it in person, anyway.

The Airplane/Treadmill Problem: My "Solution"

A few days ago I posted a description of the Airplane/Treadmill Problem, which is basically a trick question of physics and logic:

Imagine a 747 is sitting on a conveyor belt, as wide and long as a runway. The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off?

Now I'm going to share my thoughts on what's wrong with the question and how I would go about "answering" the question at face value.

Since there's a lot of ground to cover, I'll give you a quick outline of my plan of attack:

• The problem description implies that the plane can't move, but doesn't say why.
• With powerful enough engines, the plane will lift off.
• Any reading of the problem description that makes sense of the premises leaves the plane stationary and the air flowing past the wings fast enough for an in-place liftoff.
  

There are several angles to look at the problem from, but there's one observation that's relevant to all of them: the problem defines the plane as stationary but doesn't necessarily provide a mechanism for holding the plane in place. We're told that no matter what else happens, the wheels don't slip and the conveyor belt speed exactly cancels out the wheel speed. By the principles of geometry, not physics, this boils down to an obscure way of saying that the plane doesn't move. That means that any statement of the problem that requires the plane to move forward or backward entails a contradiction, and can be read as "imagine a plane is moving and also not moving".

(Note: I read the problem to say the treadmill moves backwards at the "speedometer speed" of the wheels. A lot of people seem to read the problem as matching the air speed of the plane. I believe that's why on Mythbusters and several YouTube videos, the plane is clearly not stationary when it lifts off. If you solve the problem using air speed instead of wheel speed, the answer is trivial and turns out to just require the wheels to spin twice as fast as they would on solid ground.)

One other thing is certain. With powerful enough engines, the plane can always achieve liftoff, whether it's being held stationary, pushed forward, pulled backward, or completely uninhibited by the conveyor belt. I can prove it using only Newton's third law (actions and reactions) and some back-of-the-envelope aerodynamics:

1. The wing's shape is designed so that if enough air flows past it, it will pull upwards enough to lift the plane. It doesn't matter whether the plane moves through the air or the air moves past the plane (like in a wind tunnel).
   
2. Any engine (turbine, rocket, etc.) achieves acceleration by forcing something in the other direction (air flow, rocket exhaust, etc.). If enough of anything is forced in one direction, the balancing effect of air pressure will cause some breeze to follow it.
3. With enough energy ("enough" being the operative word), that air is going to move, and with enough airflow, that plane will go up.
   

That said, I doubt such engines can be made at any time in the near future, given that more powerful engines will tend to make the plane weigh more.

So, the plane must be stationary for the question not to be bogus, and the plane will lift off in any non-bogus but idealized) statement of the problem. But there's still one question left to answer: what's stopping the plane from moving forward? To answer that question, let's forget about the plane engines and consider a slightly different problem:

Imagine a wagon is sitting on a conveyor belt as wide as a wagon trail. The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction. You stand in front of the conveyor belt holding a rope that's attached to the wagon. If you give the rope a tug, what stops the wagon from moving?

In other words, since the wagon can't move a fraction of an inch without making either the universe or our heads explode, who or what are you playing tug-o-war with?

Here's where we bring back all of the little pieces of physics we ignored. If we neglect all of the technicalities of real-world physics and declare that the wheels have absolutely zero resistance to motion, then the motion of the conveyor belt is completely unrelated to the wagon and your rope. It could be tracking 10,000 mph in either direction, and you simply wouldn't feel it from the rope. That means that the wagon would necessarily move, and the problem description breaks down. Since that won't work, I'll relax the constraints a bit and see how I can link the motion of the conveyor belt to an equal and opposite tug on the wagon. The question is whether there is some way to interpret the problem so that there is still a single correct answer, or whether every reading turns out to be bogus.

The simplest way to make the problem work would be to say the wheels don't roll perfectly, that there's some friction in them. That way, if you get the conveyor belt moving fast enough, it will match your tug with its own.

Another option is to give the wheels some mass, which eats up some of the energy in getting the wheels to spin. As an example of this effect, think of how you can lift a yo-yo by the string alone before it gets spinning very fast. The same idea would allow the conveyor belt to pull on the wagon.

The theory of relativity might give a third option for balancing the system, but I'll spare you the details.

The Airplane/Treadmill Problem

It's time for a physics/logic problem. This problem has been circulating for a while, and there have been a couple of interesting arguments and an ocean of stupid ones.

Imagine a 747 is sitting on a conveyor belt, as wide and long as a runway. The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off?

The problem is actually sort of bogus. Figuring out exactly what's bogus about it and how to explain it is the real challenge. There are several ways to solve the problem just by saying "physics doesn't work that way", but since they're technical explanations and they're not satisfying answers, I'll go ahead and rule real-world physics out: the wheels are massless and frictionless, and all physics discovered after the year 1900 is off-limits.

I have my own thoughts on how to "solve" the riddle, but I'll give you a chance to look it over and ponder it for a while before I post my "solution".

Illustrated Philosophy: A Different Angle on Subjectivism

There seems to be a lot of confusion about subjectivism and its implications, so rather than expound arguments against objective values, I thought this time I'd take a step back and get a birds-eye view of objective vs. subjective worldviews.

First let's see how things look to a subjectivist:

Subjective Outlook

In this model, there's an objective truth that underlies all shared experience (meaning things that almost all humans have in common, such as biological humanity, our location in the universe, and especially the things considered painful or pleasurable). The body of human knowledge floats on top of this layer of experience, so it's also indirectly supported by truth. If the details of human experience were significantly modified, any bit of knowledge might be invalidated or become nonsensical. Different people can agree on qualities of human experience, such as moral values, because of a shared vocabulary and shared experience, but calling these shared values objective, universal truth is only accurate in a weak sense, as a form of conversational shorthand.

Now let's look at a world of objective values:

Objective Outlook

In this case, some knowledge is supported directly by truth, without being filtered through human experience. Everyone can agree on objective values, because they aren't supported only by the details of human experience, and they have universal significance outside of human experience. This implies that humans have a special kind of direct access to truth, such as divine revelation or an internal moral compass.