Boing Boing was talking about an airplane on a treadmill problem this morning (See also David Pogue’s NYTimes blog and a thread at The Straight Dope). The problem is as follows:
Imagine a plane is sitting on a massive conveyor belt, as wide and as long as a runway, and intends to take off. The conveyer belt is designed to exactly match the speed of the wheels at any given time, moving in the opposite direction of rotation.
Can the plane take off?
I’ve been thinking about this all day (I also read through the various responses to the question, and they were totally unhelpful), and I’ve come to the conclusion that the answer involves a flaming plane wreck. But what kind of flaming plane wreck, I’m not sure. I did minor in physics after all.
A couple of things to start with: I’m assuming that the plane is just a regular, stock plane that exists in the world today. No superscience planes with unobtanium ball bearings or that use their wheels for thrust. The conveyor belt can be superscience, if you like, but I prefer solutions that don’t require superscience conveyor belts (only slightly less than I prefer solutions with fiery crashes).
To start with, I noticed that a lot of people in the various threads want to assume no friction. This is lame, and leads to the boring conclusions. Either the wheels and conveyor belt spin up to infinity as soon as thrust is applied (and then the plane cruises along and takes off), or the pilot applies the brakes and throttles up and the plane glides along the unmoving surface of the conveyor belt until it takes off.[edit: I just remembered that without friction the airplane’s wheels don’t spin at all (friction is what makes them spin). So the treadmill doesn’t either. So ignore the bit about infinite speeds. It’s easy to make mistakes when you are thinking about impossible things…]
The rest of my answers assume friction.
The most realistic answer is that the plane can’t take off, because the only way to satisfy the condition that the conveyor belt and the wheels move at the same speed is to have neither one move at all. This is because in the real world, there are bound to be tiny variations between wheel and conveyor belt speed. You can engineer to certain tolerances, but perfect is pretty much out of the question. Again, a boring answer.
So lets say that negligible differences between conveyor speed and wheel speed are allowed. Assume that the test has been done many times before, and the friction calculations have been made, so that the conveyor knows how fast the wheels have to spin to compensate for any amount of thrust from the engine (and thus keep the wheels and conveyor moving at the same speed), and the conveyor has been engineered to detect how much thrust is being applied, perhaps even before it is applied (you could do this by having computer program apply the thrust, or with a camera in the cockpit monitoring the pilot; there is a mechanical delay between when they push the throttle up and when the engines actually speed up).
It should be obvious that the conveyor belt is going to have to move very fast to generate enough frictional forces to overcome the thrust forces and keep the plane in one place (and thus keep the wheels and the conveyor moving at the same speeds). Without sitting down and doing the calculations, I imagine something like this: The pilot throttles up a teeny bit, the conveyor goes to a million miles an hour, and either rips the wheels completely off, or causes the bearings (and the rest of the wheel?) to almost instantly turn into white hot molten slag. But the plane might not crash to the ground immediately. At those speeds, the conveyor belt will drag a lot of air along with it. So maybe the plane “takes off” when it is thrown (sans landing gear) high into the air by the hurricane force winds generated by your giant, superscience conveyor belt. If we’re lucky, maybe the wings will get ripped off too. Then it all comes crashing to the ground.
That was fun, and I figured I had it solved until I realized that if the plane doesn’t have wheels, the conveyor belt can’t move, so it would never get to a million miles an hour (or whatever speed). In fact, the wheels are going to have an upper speed limit that is probably pretty low. Beyond that, they’ll blow or rip off or the bearings will seize up or have some other kind of failure. This means that the conveyor belt doesn’t need to be a superscience conveyor at all, because it doesn’t ever need to move faster that the max speed for the wheels. You could figure out how fast by testing the wheels beforehand. (heck, the manufacturers have probably already done that work for you)
So in that case, the scenario goes like this: Pilot applies just enough thrust to overcome the static friction on the wheels and get them moving. Wheels and conveyor belt start moving faster and faster until they reach the max speed for the wheels. At this point, an equilibrium is reached, and the pilot will probably have to throttle back to avoid having the wheels fail. If he doesn’t, the wheels quickly fail, the conveyor belt either stops or keeps moving at the last speed of the wheels, and the plane probably can’t take off.
Now I am bored and will wander off, but in the meantime, feel free to offer your criticism…