Letters to the Editor
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"much fun to be had"
Best part first...
"Let us call the airspeed that is required to take off, "take off airspeed" aka "TOAS"."
"For fun replace the conveyor belt and wheel, with a river and floats, much fun to be had."
Good example.
Say the airplane had a "TOAS" of 60 MPH and with it's engine could only move on floats at 80 MPH in still water. If the river flowed at any speed over 21 MPH opposite the takeoff direction the plane could not take off.
This is the reason you take off into the wind. The reverse treadmill and the fast flowing river both act like a tailwind robbing the airplane of air speed. In both cases the reverse speed need not keep the plane stationary when viewed from the side but only need to keep it below "TOAS" to prevent takeoff.
Other notes...
"ex3: engine off, treadmill speed of TOAS backward, free wheels, no headwind = plane does not take off; ground speed is zero, wheel speed is TOAS"
No. The ground speed would not be "zero" it would be approaching negative "TOAS". The plane would be moving backwards. Not relevant to the problem.
"ex4: engine on, treadmill of TOAS backward, free wheels, no headwind = plane can take off; at moment of plane taking off ground speed is TOAS, wheel speed is twice TOAS."
With the magic reverse treadmill the plane's ground speed will always be zero. The length of the treadmill is not relevant since the from the side the plane appears stationary. Lots of noise and exhaust but no flight.
Unless the wings of the plane are moving THROUGH THE AIR at "TOAS" it WON'T take off. Ground or wheel speed is not relevant.
It is not magic. An airplane works in two modes. In the first mode the engines serve as a force to move the plane over the earth surface supported by wheels/floats/skis. The second mode is flight, where the engines provide force to move the wings through the air with sufficient speed to provide aerodynamic lift.
In the case of the reverse treadmill the plane is unable to transition from mode 1 to mode 2 because it can NEVER move through the air with sufficient speed to fly.
Imagine a wing with streamers hanging every 12" from the trailing edge. At perfect speed match on the magic reverse treadmill all the streamers outside any engine blast would hang down indicating the lack of movement through the air.