I think people may be misinterpreting your question slightly. It sounds like the question you’re asking envisions the plane remaining in place horizontally (i.e., doesn’t move forward) because of the treadmill. So the question you’re really asking is: would a plane’s engines still lift it into the air if it wasn’t moving across the ground?
So while it’s true that a plane on a treadmill will still fly *because it will ignore the treadmill*, I’m not sure that holds true if the plane isn’t moving forward.
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