The airplane-treadmill problem is stated as follows:
A plane is standing on a runway that can move (some sort of band conveyor).
The plane moves in one direction, while the conveyor moves in the opposite
direction. This conveyor has a control system that tracks the plane speed
and tunes the speed of the conveyor to be exactly the same (but in the
opposite direction). Can the plane take off?1
This interesting gedankenexperiment has drawn quite a bit of
discussion. This, in my opinion, is mainly because it is poorly worded.
Hence, I'll first try to get unambiguous descriptions of the problem, before I'll try to answer the question.
The whole problem with the wording is in the word "speed". In physics,
speed has no meaning if it isn't defined to a frame of reference, in other
words, an observer. There are two sensible definitions of speed possible: speed
with respect to the ground and speed with respect to the conveyor belt.
The first case is easiest, so I'll tackle that first.
If the speed of the airplane is defined with respect to the ground, the
conveyor belt will move in the opposite direction. Now, an airplane is
powered by its engines, either jet engines or one or
more propellers. These move the plane with respect to the air.
A movement with respect to the ground is in itself not very relevant for
the whole takeoff problem. The reason is that a plane lifts off when the speed of the air over its wings is high enough for the lift to be bigger than the weight of the airplane. The ground has little to do with that. Hence, the only thing that will happen is that the
engines will have to work a bit harder as the tires rotate at twice the
speed - for every meter the plane moves forward, the wheels must travel two
meters, one with respect to the Earth and one because the conveyor belt
moves in the opposite direction. In this scenario, the plane will simply
take off, as the friction in the wheels is rather negligible
compard to the drag induced by the air.
Now, another way of reading this is that we define speed as follows. The
speed of the plane is the speed with respect to the conveyor belt. The
speed of the conveyor belt is the speed with respect to us, an external
viewer, sitting on a lawn chair with a parasol (this parasol will be
important later),watching the whole spectacle. In this setup, the plane is
stationary with respect to the observer - for every meter the plane moves
forward on the belt, the belt moves back one meter, leaving at the exact
same spot. This probably isn't the most obvious way of reading the question,
but it is the most interesting one.
Before going to the cute theoretical answer to this paradox, it's perhaps
more fun to consider what will happen in real life. Imagine our plane is
passenger jet like a Boeing 767. This plane has a thrust of
about 270 kN 2. All this thrust is being counteracted by the
friction the wheels of the landing gear have with the conveyor belt
(whether this friction is between the surface of the belt and the wheel or
between the wheel and it axle is not too relevant at this point). There are,
apart from gravity and the normal force that prevents the plane from
sinking in the ground, no other forces acting on the airplane. This
does imply two things:
- The entire thrust has to be counteracted by the landing gear. I'm
not an airplane engineer, so I can't tell whether the landing gear can
hold 270 kN, or 27 tonnes, of force in that direction. Given the speed at
which an aircraft can brake at the runway, I think it might.
- The friction creates heat. Normally, the wheels of a landing gear
only brake for a very short amount of time, and even then, occasionally, a
tire pops. I wouldn't be surprised at all if the landing gear catches
fire. Of course, such a burning landing gear will eventually ignite the
plane and cause it to blow up. So, parts of it will take off - hence the
need for a parasol.
From the discussion above, it is clear that such an experiment will be
very difficult to conduct - the conveyor belt would need to run at a very
high speed, potentially a lot higher than the normal takeoff speed of an
airplane, to generate enough friction to counter the force of the
engines. However, let's assume we live in an Ideal World where this
is not a problem. Can we answer our paradox? It turns out that the answer
depends on the conveyor belt used.
As discussed, an airplane takes off when the lift generated by its wings is
larger than the weight of the airplane. Wings generate lift when air
blows past them. We have just decided that the plane remains fixed with
respect to the Earth. So, the only question remaining now is: Will it also
remain fixed with respect to the air? In other words, does the air
move?
There are two moving parts in our setup: the engines and the belt. The
engines do move air. If it's a jet, the engines will move the air from in
front of the wing to behind it, but the air will move through the engine,
and it will not hit the front of the wing. In case of a propeller -
driven aircraft, some air will hit the wings, but it will likely not be
enough to cause liftoff. In other words, the engines can't help us. Now,
let's look at our conveyor belt. Imagine I'm looking one millimeter above
the conveyor belt. It stands to reason the air at that point has the same
speed as the conveyor belt, as it is so close to it and the viscosity of
the air is enough to keep it at (almost) the same speed.
Imagine the plane is on a set of very small conveyor belts just under the
wheels, perhaps a few treadmills normally used for
exercises, only reinforced and powered by a very powerful engine. The air
very close to the treadmills will move at the speed of the treadmills; the
rest of the air will just have the speed of the ground, which is zero with
respect to the speed of the airplane. The plane won't take off.
Now, imagine the plane is on a gargantuan conveyor belt, one that has a
surface of several square kilometers. Now, the speed of the air above this
conveyor belt will be roughly equal to the speed of the conveyor belt - the
larger the conveyor belt, the smaller the difference. It is possible to
compute the minimum dimensions of the conveyor belt that is big enough for
this the speed difference to be small, but I think this is not a very easy
computation (the effect of the finite width of the conveyor belt are not
difficult, that of the finite length seems harder) and I don't have a
Computational Fluid Dynamics package handy here, so we'll assume that our
conveyor belt is big enough - a few kilometers by a few kilometers seems more
than big enough. In this case, the air will be blowing against the wings of
the plane, and it will be just like a normal takeoff. As a matter of fact,
if this is a really big conveyor belt, as big as the eye can see, the
passengers wouldn't be able to tell they were taking off from a treadmill and
not from a real runway (provided we paint nice runways on the rubber of
the treadmill, that is).
In summary, we can state that most of the confusion of this paradox is in
the poor wording, in particular the definition of speed. The outcome of
the experiment - takeoff, no takeoff, or an accident - depend on the setup. If
we say that the speed of the airplane is determined with respect to the
ground, then the problem is not very difficult, and the plane will take off.
If we define the speed with respect to the conveyor belt, it becomes more
interesting, and we have three possibilities. If the wheels become too hot,
an accident will happen. On a small conveyor belt - admittedly the most
logical setup - the plane won't take off, while on a large enough conveyor
belt, the plane will.
Sources:
- http://mouser.org/log/archives/2006/02/001003.html
- http://www.boeing.com/commercial/767family/pf/pf_200prod.html