OK. I'm confused.. (not really)

Once again, since the obvious things didn't work...

If a plane is flying at 80 mph (close to stall speed of a Cessna), directly over a treadmill (as long as a runway, so there's enough space to do the test) that is currently running at 80 mph, the two objects are moving at the same speed, are they not?

Ok. So that's confirmed; the plane is flying over the treadmill at 80mph, as the treadmill is moving at 80mph.

Next step. The plane lowers his altitude *just* enough so that his tires are in now contact with the treadmill. The plane continues at 80mph forward. The treadmill continues at 80mph in reverse direction as the plane. BUT, you'll notice, the tires of the plane are now revolving at 160 mph equivalent.

So, multiple choice question:

The plane at this time is now moving at:

(A) 0 mph
(B) 80 mph
(C) 160 mph

If you guessed (A) or (C) you would be incorrect.

The correct answer is (B). Therefore the plane has forward movement at the same speed as the treadmill, in the opposite direction. The tires have an angular velocity of the plane's speed and the treadmill's speed, equivalent. But the plane only has 1 speed: 80 mph.

Angular velocity and velocity are NOT the same thing. They are a product of each other in the case of a wheel, but they are not a 1:1 relationship. They do not add or subtract directly to cancel each other out. Ever. Angular Velocity is the forward velocity divided by the circumference of the tire.

In this particular example, at 80 MPH, when you do the math, you'll find the tires will be spinning (angular velocity) (18.368) / (radius of tire) in radians per second. So if it's a 2' tall tire, the observed motion of the tire is that it's spinning at 36.7 radians per second. You don't observe the MPH equivalent the tires are spinning; it's a DEPENDENT property. The INDEPENDENT property is the net velocity (the sum of the airplane and the treadmill velocity).

In other words, the tires spin as a result of the horizontal motion of the airplane and the treadmill. It is a DEPENDENT motion. It is not physically measurable in "MPH", but is physically measured in RAD/S or DEGREE/S, and then converted using the radius of the tire to a MPH. You have to sort that out in your head, and be able to realize that angular velocity does not EVER get added or subtracted to motion velocity at any time; you can't act like they're the same thing, and add an independent to a dependent; math doesn't work that way, as it's a physical impossibility.

Whoops. Almost forgot. To finish it off, the very next second after proving the airplane will still continue forward at 80 mph, even if his tires are on a moving surface that's moving at negative 80 mph to him, the pilot pulls back on the stick, and viola: THE PLANE TAKES OFF...

If you REALLY want to think about it another way...Take this same example, only the treadmill is moving FORWARD at 80 mph. As the plane touches down onto the treadmill, the plane is moving forward 80 mph, the treadmill is moving forward at 80 mph, and the tires are rotating at a whopping 0 rev/s... The tires stand still, yet the plane is moving at 80 mph forward... And it still takes off. Example/Proof, yet again, that the angular velocity the tires move does NOT control the forward motion of a plane...

And one more, 'cause I just can't resist...Treadmill moving forward at 40 mph, plane moving forward at 80 mph. Plane touches down, and the wheels are only revolving at 40 mph equivalent, yet the plane is still moving at 80 mph... Oops. Lookie lookie...angular velocity of the tires still doesn't control the plane's movement...

But just in case I haven't annoyed you enough, yet..

Let's look at the case when the treadmill goes forward, and let's assume the argument of the "can't fly" bunch... If the plane moves at the same speed as the conveyor in the "backwards running conveyor", y'all have assumed the plane can't take off, 'cause the tires are moving at the same speed as the conveyor.

So, if the conveyor is moving forward at 80 mph, and the tires are moving forward at the same speed (80 mph equivalent), how fast is the airplane going?? If you said 80 mph, you'd be incorrect... the plane would be moving twice as fast as the conveyor, it'd be 160 mph!!

[insert "can't fly" protest]Argh!!! It CAN'T be moving at 160 mph, 'cause the tires are moving at the same speed as the conveyor!! But..........but I don't understand....[/insert "can't fly" protest]

Yep, that's right... the angular velocity of the tires isn't the same thing as the speed of the airplane... And since that's true for the conveyor moving forward, then it's true for the conveyor moving backward as well.

Guess what? The plane takes off.