[QUOTE=Alexis Brooke]
The plane’s wheels will only rotate twice as fast if the plane is moving at twice the speed of the treadmill, or simply the speed it would move if the treadmill was not in motion.
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You’re getting confused by the reference point for relative velocity. In the quoted case, I’m talking about plane fuselage speed relative to the ground, as well as treadmill speed relative to the ground. The treadmill runs in one direction at the same speed (relative to ground) as the plane speed (relative to ground) does in the other–and thus the wheels must rotate twice as fast.
You could, of course, talk about plane fuselage speed relative to the treadmill, but that reduces the problem to one of four cases:
a) If the plane fuselage speed relative to the treadmill matches the treadmill speed relative to the ground in the opposite direction, then the plane doesn’t move at all relative to the ground (and turns into the same problem as wheel-speed-matching, covered by my above post).
b) If the plane fuselage speed relative to the treadmill matches the treadmill speed relative to the ground in the same direction, then the plane travels at twice the speed of the treadmill relative to the ground (and takes off with its wheels turning at half-speed).
c) If the plane fuselage speed relative to the treadmill matches the treadmill speed relative to the plane in the opposite direction, then that’s no restriction at all (the plane and treadmill can do anything at all and the condition will be satisfied).
d) If the plane fuselage speed relative to the treadmill matches the treadmill speed relative to the plane in the same direction, then that’s an over-constraint, and can only be satisfied if both speeds are zero.
[QUOTE=Alexis Brooke]
In order for the wheels to rotate, some force must be imparted upon the plane in the direction opposite the treadmill. Being tied to something in front of it will cause the plane to remain stationary. However, without the restraint, or thrust from the engines, the wheels will not rotate at all, and the plane will simply move backwards.
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Well, yes and no. First of all, I don’t think anyone anyone interpreted the problem as asking what would happen with the engines off (certainly I didn’t).
However, I suppose there’s nothing wrong with that assumption. Clearly, in this case the plane will move in the direction the treadmill moves. However, exactly what happens depends on the acceleration of the treadmill. Three things can happen:
a) If the treadmill acceleration is small (i.e., not enough to overcome friction in the wheel bearings), the plane will move at the same speed as the treadmill (with no wheel rotation).
b) If the treadmill acceleration is medium (i.e., big enough to overcome friction in the wheel bearings, but not large enough to overcome friction between the tire and treadmill), the wheels will begin to rotate with acceleration that depends on its mass and mass moment of inertia. However, some force is still transmitted to the plane, so the plane will also move, but at a speed much less than that of the treadmill.
c) If the treadmill acceleration is large (i.e., big enough to overcome friction between the tire and treadmill), the result is similar to case (b), but the force transmitted to the wheel is no longer a function of treadmill motion.
