[quote=“newman,post:281,topic:37377"”]
but some part of me wants to tend this thing to infinity…
[/quote]
OK I’m here.
First of all, anyone who thinks that the plane might not take off should stop reading now. If you can’t wrap your head around the concept of the wheels/runway approximating a frictionless surface then you won’t get what I’m about to explain.
IF… The question reads: “The treadmill is equal to the wheel speed of the plane, but in the opposite direction” then yes the treadmill speed is asymptotic with infinity. Here’s why:
By nature of their physical link, wheel speed is dependent on ground speed (aka treadmill speed.) If the control of the treadmill is based on wheel speed, then you have an infinite loop. If this experiment is controlled by a computer program, the compiler will crash. Get it? Wheel speed is dependent on ground speed, but ground speed is dependent on wheel speed. :bloated:
So say you have the plane moving forward with the wheels moving at tangential velocity x, then the treadmill will have the velocity of -x. But wait, the plane is moving at a constant velocity regardless so wheel speed has become 2x. Plane speed - ground speed = x --x = 2x. But that means that treadmill speed is -2x to equal negative wheel speed. But that means that since the plane is still moving independently at it’s constant velocity, wheel speed has become -4x. But that means that treadmill = -4x, so wheel speed = -8x, and treadmill -8x, and wheel -16x, and treadmill -16x, and wheel -32x… On till infinity.
BUT, IF treadmill speed is dependent on plane speed (not wheel speed) then no problem. The plane moves independent of wheel/ground speed so no infinite loop. The plane moves dependent on relative air speed. Which is what this entire “riddle” depends on.
Get it?