The Treadmill Problem
Last night I was skimming through the latest postings on BoingBoing, and came upon this one. I had heard of this “question” before, but never gave it much thought. Here’s the gist of it:
Imagine a plane is sitting on a massive conveyor belt, as wide and as long as a runway. The conveyer belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off?
At first, I thought this was the stupidest question I’d ever heard of. I had initially imagined the problem as the plane was sitting on a treadmill just big enough for the wheels to sit on, and these people thought that by spinning up the wheels the plane would takeoff. However, the problem actually states that the treadmill is the exact dimensions of a standard runway.
So, I think of it like this: Planes fly because of “lift”. In order to get lift, according to Bernoulli’s Principle, you must have sufficient airflow over the wings so that an area of low pressure is created above the wing (due to the air having to travel a longer distance than below the wing, so it moves faster) and an area of high pressure is created below the wing. When this pressure differential is great enough to overcome the weight of the plane, the wings produce lift and take the plane aloft.
Understanding that, if the plane is sitting on a treadmill that is able to adjust it’s speed (apparently instantly) to match the rotational speed of the wheels, can the engines still move the plane forwards, and at great enough speed to get the required airflow over the wings to generate lift? My answer: Not being a physics expert, I’m not sure. Here are my two theories for both sides of the argument.
1) If the treadmill is exactly matching the rotational speed of the wheels at any given instant, for every X amount of distance the wheel moves “forward”, the treadmill is moving it “backward”. Thus, no actual forward movement of the wheel means no forward movement of the airplane and no airflow over the wings. The plane remains motionless and does not takeoff.
2) The relationship between the wheels and the surface of the treadmill makes no difference. Since the wheels are able to spin freely with the treadmill, the plane is essentially “floating” on top of the treadmill-runway much like a hovercraft. Therefore, the engines are able to push the airplane forward at a sufficient speed to get enough airflow over the wings to produce lift. The plane takes off.
I have no clue if I’m right or if my hypotheses are correctly reasoned. But that’s my take on the whole thing. Any physics majors out there?
December 15th, 2006 at 7:14 am
Starting from your no.2 hypothesis…
As the plane accellerates, the wheels start to revolve.
The treadmill speeds up to match the wheel’s rotation.
However, as the plane is already moving forward, this simply increases the speed that the wheels rotate.
So the treadmill speeds up again.
Which makes the wheels turn faster.
Which makes the treadmill speed up again.
This goes on in a loop until one of three things happen….
1. The treadmill can’t go any faster, or breaks
2. The wheel bearings fail, or the tyres explode
3. The plane takes off before 1 or 2 happen.