Plane on a Conveyor Belt: My Guess
I didn't know about this controversy until a day or so ago.
I had a friend who used to like to get groups of people who didn't know much about the actual physics to try to figure out this kind of physics problem.
I'm with the faction who says that the Mythbusters screwed up.
In brief:
Seems to me that the question is equivalent to: Can an aircraft that is stationary relative to the ground take off? (And: not with a very substantial headwind.)
The Mythbusters think their experiment showed that it can.
But that plane wasn't stationary relative to the ground--as you can see in the video if you look at the cones.
So, as others have said: they probably weren't pulling the canvas (on which the plane was resting) fast enough to keep the plane stationary relative to the ground.
Anyway.
Can a plane that's stationary relative to the ground (and, again, without a significant headwind) take off?
My guess:
Maybe, but probably not given the kinds of planes they're actually interested in.
Many planes can take off like that, of course.
E.g. VSTOL aircraft like the AV-8B Harrier, the F-35B...and helicopters. These are tacitly ruled out. The Harrier and Fat Amy-B reorient their nozzles to direct thrust downward. So that's out of the spirit of the thing. As are helicopters.
More relevantly: Some bush-hoppers used e.g. in Alaska can take off and land almost vertically given a stiff headwind...and with very small runways otherwise. See e.g.:
Anyway, I say: the right kind of plane--e.g. with phenomenal power--could take off while stationary relative to the ground.
But there's no general answer to the problem.
If you want to know whether a plane can take off from a conveyor belt that's keeping it stationary relative to the ground, just have it try to take off with the brakes locked--or the wheels chocked. Most won't be able to. But some might. If you offered a billion dollar prize for making such a bird, I can semi-guarantee that you'd get one.
But the kinds of planes normally said to be at issue--747s, Piper Cubs--no.

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