I’m tired of these motherf***ing planes on this motherf***ing treadmill.
The plane and treadmill problem is mentioned here, and here and here.
Basically, a jet plane is setting on a treadmill/conveyor belt. The treadmill is setup so it will match the exact speed of the plane in the opposite direction.
Will it take off?
My answer is yes. It would take off more or less normally. With the exception that the wheels will be spinning twice as fast as normal.
Why? F=MA.
The jet engines excert a huge force in one direction. The treadmill however, is only able to exert a very small force in the opposing direction.
Not convinced? Think of it this way. The plane is standing still on the treadmill with the engines off. You have a cable attached to the front of the plane. The treadmill starts up. How much force does it take to keep the plane from moving? Not that much. The wheels can only provide so much force, mostly just the rolling friction from the wheels.
Now try the opposite. The treadmill is off. You are holding the cable behind the plane. The jet engines start up and roll up to full throttle. How much force does it take to keep the plane in place. A frigging lot.
Unbalanced forces means a net force. Which means acceleration. Which means the plane moves. And takes off.
Well.
In my opinion, the problem is incomplete.
It does not tell you if the plane’s engines are running or not.
If the plane’s engines are not running, then the plane will not take off. That is because the wheels will be spinning but the air will not, and no lift will be produced against the wings.
If the plane’s engines are running, then they will produce thrust against the air, not against the ground/treadmill. At that point, the speed of the plane relative to the air will increase, and the plane will fall off the treadmill (and eventually take off).
There is a reason why in airplane modeling you are always taught that planes take off with the nose in the wind, not with the tail in the wind. It’s misleading because you think going in the wind will mean more speed for the plane – but that’s not relevant because lift comes from the speed of air against the wings not the speed of the plane against the ground.
The real treadmill in the aircraft industry is the wind tunnel – and then, yes, the plane’s relative speed to the ground is zero, but the plane will fly because the air moving in the wind tunnel will provide the necessary lift.