to those who still believe the plane will not fly
Yesterday, I proved that friction does not depend on velocity. If that hasn't changed you from a 'no-fly', then hopefully this post will be the final nail in that theory's coffin. I will hopefully cover all of the interpretations of the problem. A quick explanation before we start: Force diagrams.
Force diagrams are used by physicists, mathematicians and engineers to determine the total force on an object (although we call them vector diagrams). They are extremely simple to understand but if you cannot remember how to read them or have never seen them before, here is an example:
In a force diagram, the two important things we need to look at are the direction of the arrow and the length of the arrow. So in this example we have force 1 going right and force 2 going left. This is the direction in which the force acts on the object. The length of the arrow represents how strong the force is; looking at the example, we can see that both forces are equally strong (ignoring direction for the moment). So if we arrange them side by side, we get something like this:
In this case, we say that the forces "cancel out" or in more mathematical terms, that the net force (total force) is 0. With a net force of 0, we know that there will be no force on the object and it will not accelerate in any direction. It is important to note that the direction was responsible for this cancelling; if both forces were pointing in the same direction, we would have a very large net force!
Statement of the problem: If we place a plane on an infinitely powerful treadmill (which spins opposite to the takeoff direction) and match the speed of the aircraft during takeoff with the speed of the treadmill, will the aeroplane take off?
Interpretation 1: We match the takeoff velocity of the craft to the speed of the treadmill.
Why it will still fly: As we proved yesterday, the amount of friction does not depend on velocity. So let's take a look at a force diagram for this scenario:
As we can see in this force diagram, the force generated by the aircraft will overcome friction from the treadmill and air resistance. If we cannot overcome the friction (a longer red line), the pilot can just increase the power of the engine (make the blue arrow longer) and then the difference between the forces will become even greater, causing the aircraft to speed up faster. This is because the jet engine/propeller generates thrust with the air, not with the wheels.
Interpretation 2: We spin the treadmill so that the relative velocity of the plane to the ground is 0. To put this in a less wordy way: we spin the treadmill so that if you were standing next to the treadmill, the plane would appear not to move.
Why it will still fly: It won't. "What?!", you say; didn't I tell you that the plane will always fly?
The problem with this interpretation is that it can never occur. Impossible. Not with the strongest of treadmills and the most ideal of components. Remember the only two forces that can stop the plane taking off are friction and air resistance. We proved yesterday that friction is constant because vertical and horizontal forces are independent and it should be clear that the treadmill cannot affect the air resistance of the plane. To keep an object at a constant velocity (0m/s), the net force on the object must be 0. So to keep the plane at a relative velocity of 0, the force diagram will have to look like this:
As you can see in the diagram, the grey arrow represents a force that is stopping the plane from taking off. The problem here is that there is no possible way that the treadmill can generate the grey force. No argument about it; for the plane to stay still, there needs to be some magical force coming out of nowhere which is stopping it from moving. Apologies to those who believe in magic, but the treadmill cannot, in any way, shape or form force the plane to remain stationary.
The trick is that when the problem is described to you, this is often implied as an assumption. When you go to solve the problem your brain shortcuts and automatically takes that assumption as valid, which essentially leaves you to solve a problem that has no solution.
Therefore, under all possible, realistic conditions the plane will take off from the treadmill.
One last note: There is actually one case in which the relative velocity (with respect to the ground) of the plane will be zero. This is when the force generated by the engine and propeller is exactly equal to the force stopping it from moving. You should be an expert at force diagrams by now, so here is one more:
If these conditions are met, the plane will indeed sit still. However, the only way a plane could ever do this is if the pilot was intentionally driving the plane slowly. If the engine is only ever powerful enough to match the retarding force it wouldn't take off on the treadmill, but it also would not take off on a regular runway! The friction on the treadmill will be very similar to that of the runway, so your plane isn't actually a plane!
