So lets say I know my little RC plane takes off at exactly 25mph. So I crank
up the treadmill to step up to 25mph so I can keep the RC plane up on the
treadmill, the plane is completely stationary in regards to anyone standing
next to the plane but when the treadmill hits 25mph and my little RC plane
is staying even with the treadmill you are telling me I can pull back on the
elevator stick and the plane will take off? I don't think so.

Ok, now you've changed the problem. The way you state it here, your RC
plane will not take off, because it has no airspeed. However, in =your=
case, the treadmill speed is the independent condition, and the pilot is
adjusting his throttle to compensate. The pilot is ensuring that
airspeed is zero (assuming no wind). It will require about as much
power as it does to taxi, because all he's doing is overcoming wheel
friction.

There will be excess power available to the pilot, which he could use to
move forward faster than the treadmill is moving backwards, acquiring
airspeed and thus taking off. (but the pilot won't do that due to the
constraints of your problem, which says he won't do that).

In the =original= problem, the =pilot= is the independent variable. He
can do what he wants, and it's the treadmill that is tasked with
"keeping up" (whatever that means). Let's consider the following three
cases:

1: The treadmill is frictionless. In this case, we can ignore it. The
problem is the same as taking off from a maglev rail. The plane's
engines will push against the air, the plane will move forward, whatever
the treadmill does will not affect the forward motion of the airplane,
and the airplane will take off.

2: The treadmill and the plane's wheels are very sticky. The plane's
engines will push against the air, the plane will attempt to move
forward, pulling the treadmill bed with it. IF this is permitted, the
plane will gain airspeed and will take off, perhaps with the treadmill
dangling below its wheels. This may be a problem on landing. If the
treadmill bed is -not- permitted to move, then (by hypothesis) the
plane, being glued thereto, will not move. It's a static test stand,
not an airplane. But in that case all the king's horses couldn't move
the plane either.

5: The test is announced on usenet first. Everybody comes out to see
the test stand, and the resulting bluster of hot air allows the airplane
to easily reach a high enough airspeed even while standing still. The
airplane takes off before the test is even begun.

Jose
--
"There are 3 secrets to the perfect landing. Unfortunately, nobody knows
what they are." - (mike).
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