Exactly... and they didnt start by pulling fast enough allowing the airplane to accelerate forward.
Now... the tricky part is uncoupling that concept of pulling backwards as fast as the airplane would accelerate forward on solid ground. You have to think of the axial force exerted on the landing gear as the treadmill moves backwards and see how that force increases as the velocity of the treadmill increases.
the propeller at take off power will exert the same force for the most part at all times in the take off run.
now there is a certain velocity of the treadmill that will exert an equal and opposite force on the gear. I personally believe that speed would be much larger than that of the liftoff speed of the airplane due to the rotation of the tires (slippage) but that would have to be determined by empirical evidence gathered by actual testing.
So, as the car pulls the mat in one direction there are three possibilities.
One: he is pulling at a speed slower than that required to exert a force on the gear struts equal to the force of thrust created by the prop and the aircraft will accelerate forward in air and lift off. (this is what happened in the example as the car accelerated from rest. rest = speed less than that required)
Two: he is pulling at a CONSTANT speed required to create a backwards force on gear strut which will equal the force created by the prop at take off power and the airplane will remain stationary with zero airspeed.
three: he is pulling faster than that required thus causing a greater force on the gear strut than that force by the propeller and the aircraft will get pulled backwards.
All these assume a constant takeoff power setting creating a relatively constant force (thrust vector)
Again, the trick is not to visualize this as a car pulls in one direction accelerating at the same rate the airplane normally accelerates. This caused a smaller drag force than that by the prop and the airplane took off.
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A better way to visualize this is to start off with the airplane tied down at the wings and an endless treadmill.
apply take off power and the airplane will strain forward against the ropes.
move the treadmill at a speed which generates a force great enough to relax the tension in the ropes.
That is your equivalent treadmill speed. (notice this is about force and not speed. It would take an experiment to determine what speed this would be and it would be different for every airplane based on rated takeoff power)
Now, any speed less than this will always allow the plane to accelerate forward as thrust is greater than drag and the airplane will take off.
Any speed less than this and the airplane will be drug backwards as thrust is less than drag.
Again, if you simply accelerate (starting at rest) the treadmill backwards like the car did on mythbusters the plane will always accelerate regardless of wheel speed as the takeoff thrust is greater than the drag created by the treadmill.
It is NOT about aircraft speed. It is about the force created by the propeller and its relation to the force created by the drag of the treadmill.
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By the way... the more I think about this the more I wonder if it is even possible to pull the treadmill under the wheels fast enough to create a force large enough to equal the force created by the propeller. I wonder if the dang tires wouldn't fail first.
This post evolved after thinking through the free body diagram of the forces acting on this problem. they supersede my previous post where I used the words "matching the speed of the airplane."
After further thought.... I discovered that in my last post I didn't properly explain what I was thinking and how it actually was saying something the wrong way.