Airplane-Treadmill Problem Solution - Nihilist
Tuesday, December 12th, 2006This is not what all the talking heads are saying, but I’m almost positive this would be the end result of this mental exercise. For the uninitiated, look at BoingBoing:
David Pogue at the NYT has presented this classic airplane on a giant treadmill problem, and people are arguing about whether or not the plane would take off or not. Here’s the problem:
“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? “I say no, because the plane will not move relative the the ground and air, and thus, very little air will flow over the wings. However, other people are convinced that since the wheels of a plane are free spinning, and not powered by the engines, and the engines provide thrust against the air, that somehow that makes a difference and air will flow over the wing.”
I say yes. Let’s assume the friction in the wheel bearings is negligible. Putting a plane on a treadmill is like putting it on an icy lake. When you fire up the jets, the plane is going to shoot down the lake and take off just like it would on a runway.
The most important issue is that even if the the friction is negligiblein most situations, it does exist. Therefore it is important in this case. The detail being missed by most people is the way the question is set up: the treadmill is set to match the speed of the wheels, not the land speed (actual or normal) of the aircraft. So, in order to answer this question you have to get technical. Let us assume that there is enough tread on the wheels so that the friction of the wheels does matter (meaning that when the plane goes forward, the wheels move with it). This is to defeat the “icy lake” hypothesis which is based on the fact that one an icy lake, the wheels don’t turn at the land speed, mainly because the plane is in effect in a long slide. Here’s the solution:
- The engines start up creating a forward force in the upper half of the plane body, because the land speed is not yet big enough to create lift and the engines are parallel to the ground, this force pushes down onto the wheels with the help of our friend gravity.
- The wheels move as the plane’s body begins to be acted upon that force.
- The treadmill matched the speed of the wheels movement, creating a backwards force that initially matches the forward force.
- 2 and 3 get into a bit of a Cold War era armament situation: because the forward force is not generated by the wheels, the wheels spin free, meaning their speed is increased beyond what it would take in a standard situation to take off. The treadmill speed matches the wheel speed. Now, remember that in order for the plane to move forward and the wheels to meet the friction needed in the axiom (because the treadmill is a surface that would give traction, there is friction) the wheels would need to spin faster than the treadmill for forward momentum. Therefore the backward force generated by the treadmill would get significantly higher. So rinse and repeat until 5.
- The treadmill is designed (by nature of the problem) to be able to handle infinitely high speeds. The wheels, being true physical objects have a failure speed. The wheels fail, the plane makes physical contact with the treadmill, and is thrown back into the airport. Everyone dies.
The end result before death would be the forward force remaining relatively constant and the backward force being several orders of magnitude stronger. Which of course would beg the question, why don’t we use this magical treadmill to launch these planes into the air. If it generates so much force, short flights wouldn’t even need engines…
EDIT: I suppose that I should directly address Mark’s point. The wheels in a normal situation act as the icy lake, they remove the friction from the ground for the plane as a whole. What happens with the treadmill, is that by moving at would be around an infinite speed (hence the implausibility of this problem) it would in fact create a whole lot of friction for the plane, making it the exact opposite of the frozen lake. Oh and before anyone attacks my idea as overly simplistic, I forgot to give a few assumptions:
- 1-5 occur almost instantly.
- The wheels won’t slip. If they do, the plane will take off or a wheel will blow, killing everyone. If it takes off, I wouldn’t want to be on the plane. We still need that wheel when we land.
- If you ignore the practical limits of airplanes, this question becomes totally meaningless, as both the wheel and the treadmill will have broken the speed of light and saved Lois Lane from an earthquake. Therefore, they both heroes and should be left alone for their retirement.
- Either the tire will blow or the bearings will break: the plane will never take off, only crash and be flung back into the airport.