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.