yes, that airplane treadmill problem again
and yes, it will absolutely take off.
The main confusion in this problem seems to be the idea of friction, so let's take a look at it. The formula friction (taken from wiki) is as follows:
f = μsFn
where f is the frictional force, μs is a constant relating to the surfaces (remember, contstant means that it never changes) and Fn is the normal force from the surface. Now, you might remember from high school physics that the normal force is perpendicular to the surface, so on the flat ground or on a treadmill Fn will be completely vertical due to gravity. Please excuse the following maths, although it is very simple.
Let's assume our plane is 500kg (in the range of ultralight aircraft) and μs = 0.5 (I just randomly picked this number, you should see in a minute that this doesn't matter). If we assume for the sake of simplicity that the only component influencing the normal force is gravity, we can work out the friction for a plane travelling at 1m/s:
μs = 0.5
Fn = m * g = 500 * 9.8 = 4 900N
therefore,
f = 0.5 * 500 * 9.8 = 2450N
So we have 2450N of force going against the direction the engine is trying to push the plane. Now let's calculate it if the plane is moving at 2m/s:
μs = 0.5
Fn = m * g = 500 * 9.8 = 4 900N
therefore,
f = 0.5 * 500 * 9.8 = 2450N
And 3m/s:
f = 0.5 * 500 * 9.8 = 2450N
4m/s:
f = 0.5 * 500 * 9.8 = 2450N
even ∞m/s
f = 0.5 * 500 * 9.8 = 2450N
In terms of kinetics, it does not matter if the plane is moving at 10m/s, the treadmill is moving at 10m/s or the treadmill and plane are moving each at 5m/s; the amount of friction will be exactly the same. Read this next bit very carefully: the frictional force for a flat surface does not depend on velocity. So if we have a plane whose engine can only excert exactly 2450N (plus air resistance) given any environment and any inputs then, and only then, we will have a stationary plane that will not take off.
However, you might be surprised when you take the plane off the treadmill and it doesn't lift off on a regular runway! This is because, similarly to the treadmill, the frictional force from the ground will never let the plane reach its take off velocity. So this hypothetical plane is actually not a plane; it is simply a stationary or slowly moving lump of metal.
If the plane engine and/or propeller has the capability of overcoming the friction in the wheels and the air resistance, then the plane will move in a direction opposite to the treadmill spin and it will take off. Obviously the air resistance of the plane does not depend on the speed of the treadmill and as we proved above, the friction on the wheels is not dependent on velocity. To put this another way: because vertical and horizontal forces are independent (they cannot affect eachother) the only impeding force that the runway [or treadmill] can exert on the plane is friction, which is always constant.
So, any plane that is capable of takeoff on a regular runway will be capable of takeoff on a treadmill. QED.
