|The case of the plane on a conveyor belt has been extensively discussed in the online world, and still doesn't have a definite conclusion. Or rather, there are two opposing sides which believe wholeheartedly in their explanation, and these sides will never agree. The question is this:
A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?I've read a thousand people's opinions about this and heard every comparison ranging from a skateboard on a treadmill to a weightless car on a sheet of paper. Taking into account some major assumptions (a plane on a conveyor belt is actually plausible; frictionless wheels, bearings, conveyor belt; no wind; ideal/instantaneous control system), my take on it is this: The plane won't take off [Edit: I changed my mind]. Here's my reasoning:
1. In order for a plane to take off, it needs to have air passing over its wings at a certain speed. This air can come from the plane moving down the runway, or it can come from wind. Theoretically, a plane can take off while sitting completely still, as long as there is a significant amount of headwind. However, since there's no wind in this example, the plane must be moving forward at a considerable velocity.
2. A plane's wheels are "dumb". In other words, they're only there to reduce friction. A plane could just as easily have no wheels and just rest on its belly on the runway. It could still take off because its motion is produced by thrust from its engines or the movement of air from its propellers. The wheels will spin when the plane is in motion and in contact with a surface.
3. As many people have pointed out, the question's wording can be confusing.
a. If "plane speed" means "angular velocity of the wheels with respect to a stationary/ground observer", the conveyor belt would spin at an infinitely increasing rate, which is logically impossible. For example, if the wheels started to spin at 100 rpm, the conveyor would ramp up and spin at 100 rpm. But this would cause the wheels to actually be spinning at 200 rpm from the point of view of a stationary observer because the ground is no longer stationary but is moving at 100 rpm in the opposite direction. This would force the conveyor belt to spin at 200 rpm, 400 rpm, 800 rpm, etc., ad infinitum. So in conclusion, the plane wouldn't take off because it wouldn't move from its original location.
b. If "plane speed" means "angular velocity of the wheels with respect to the conveyor belt", the velocity of the conveyor belt would always equal the velocity of the wheels, no matter what. This means that the plane wouldn't move, no matter what. If the wheels started to spin at 100 rpm, the conveyor would ramp up and spin at 100 rpm. From the point of view of the conveyor belt, the wheels would still be spinning at 100 rpm even though they're actually spinning at 200 rpm from the point of view of a stationary observer. No matter what speed the wheels spun, the conveyor would always be spinning at the same speed as the wheels. This would prevent any forward motion of the plane.
c. If "plane speed" means "linear/horizontal velocity of the plane with respect to a stationary/ground observer", the plane speed would always be zero because the conveyor belt would always cancel out any forward motion of the plane. For example, if the plane started moving at 100 mph to the right (with motion derived from thrust), the conveyor belt would immediately begin moving at 100 mph to the left. Although the wheels would be spinning at an incredibly high rate (the wheel diameter doesn't equal the conveyor belt diameter, so the conveyor belt speed of 100 mph would translate to a wheel speed of something like 100,000 rpm [total guess, but the concept is there]), the plane would not change position from the point of view of a stationary observer. If it started at point A, it would stay at point A. This is the same result as part b.
d. If "plane speed" means "linear/horizontal velocity of the plane with respect to the conveyor belt", it's the same as part a. The conveyor belt would spin at an infinitely increasing rate.
Part of the reason this whole thing gets me so riled up is the attitude of the people who think they're right. Cecil Adams said, "Everything clear now? Maybe not. But believe this: The plane takes off." Thanks for your mediocre and confusing explanation, followed by an unqualified, unproven conclusion. Michael Buffington said, "Jason's Case of the Plane and Conveyor Belt riddle is confusing very smart people, so I thought I might explain it." Thanks, Michael. Obviously you know everything and everybody else knows nothing. Without you, we'd be nowhere.
This is my explanation. I put a lot of thought into it, and I even lost some sleep over it last night. I sort of think I'm right, but I'd be willing to be proven wrong if somebody has a good explanation. I'd also love to see this on MythBusters. #science