PFE009: Airplane on a Treadmill

This was a request (for those wondering about how topics are picked) so if you have another topic you'd like to hear about, please ask and, if I think it's interesting, I will do my best to address it.

Also, I decided to do a post on friction first so things like slipping would make sense.

The "airplane on a treadmill" problem seems to be one of those internet things that just got way out of control when it hit critical mass. To address this, Randal Munroe of xkcd fame put this together which is actually pretty good.

My explanation is going to (hopefully) be a little simpler. But before I address the problem, I'd like to talk about airplanes.

Let's first think about what has to happen for an airplane to fly and remember all of those failed attempts.

If we assume an airplane is already in the air and figure out how to keep it in the air, we can do the same thing, just more so, to get it to take off. So there's an airplane in the air, and mother earth is kind of pissed and keeps trying to make it crash into the ground. So the plane has to provide some sort of lift force. So why not point those big loud engines down, that's a lot of lift right? While this would provide a metric ton (read: way more than enough) lift, it would be nearly impossible to control. Moreover, going forward wouldn't be very easy. I've actually just sort of described how a helicopter works.

Your regular airplane is what we call "fixed wing" (none of those pesky rotors) and flies basically by going forwards really fast.

The wings are shaped in this really neat fashion called an airfoil which, for the purpose of this problem, we can just say that they provide lift when air flows over them. So planes need to be moving fast, relative to the air, in order to take off. As an example, it's easier to take off into the wind than with it.

Onto the airplane on the treadmill problem itself. It should be noted that the main problem here has to do with misconceptions about the problem's formulation. In the most basic sense, the question is:

If you put an airplane on a treadmill, will it take off?

As discussed above, the only way for an airplane to take off, is if lots of air is moving over the wings.

First, we assume that the engines are off, but the treadmill is running. In this (boring case), the plane lumbers forward. You might imagine that the plane would stay still and the wheel would spin, but there is some friction as the wheels turn (imagine pulling a rug out from under a cart - it will largely result in spinny wheels, but depending on the speed of the yank and the stickiness of the wheels, it should go forward some) so the plane would lumber forward and (given a crazy long runway) eventually take off.

Alright, more interesting is the problem of when you turn the jets on. The first thing to note here is that, with the jets on, the plane will move forwards no matter what. You can get all confused about velocities on the bottom of the wheel, and the treadmill, but regardless of what is happening with the wheels and the treadmill, the plane is going to move forward relative to the air (and relative to the ground, we're assuming a calm day here) and will take off.

If that was so easy then what's the confusion all about?

In a typical wording of the problem, the author will try to be clever and explain that the treadmill is matching the speed of the wheels. This tends to lead to pages of confusion talking about reference frames and speeds everywhere on the plane. Planes aren't powered by their wheels. So they can be slipping, spinning extra fast, and the treadmill can be doing whatever it wants, the plane is still going to move forward.

If somebody asks you just admit that you probably don't know what's going on with the wheels in the way they worded it, but point out that the engines will always push the plane forward in the air and it will take off.

That's an airplane on a treadmill.