The airplane-treadmill problem is a physics thought experiment that
has become a minor internet meme. It goes something like this:
"There is an airplane at rest on a treadmill the size of a runway
so that it is big enough for the airplane to take off. However, this is
a special treadmill that is designed to move as fast as the airplane's
wheels turn but in the opposite direction. So, as the airplane gets up to speed the wheels begin to
turn as the airplane moves forward but the treadmill also begins to
move just as fast as the wheels. Does the plane take off?"
The first known appearance of the meme was on a
Russian forum on April 8,
2003. Later it was
re-posted in
english to PhysOrg on July 19, 2005. From there the question
disseminated across the internet until it developed into a meme.
At first glance, the question seems to have an obvious answer: the
plane will not move. Unfortunately, that is the wrong answer and
here's why. For the purposes of simplicity I'm going to ignore the normal and gravitational forces as well as lift because they are perpendicular (and therefore irrelevant) to the range of motion which is important in the solution of this problem. When driving in a car there are three main forces; the
force of friction on the wheels, the air on the car, and the wheels
on the road. If we drew a force diagram of this it would look something
like this:
__________
/ | \ Fac
_____/___| \---------->
Fcr / \
<--------| __ __ |
\_/ \_________/ \_/ Frc
\__/ \__/------------->
The only force giving thrust to the car is Fcr; the force of the car on the road. So if you put a car on a treadmill like the airplane's, it would not move because Fcr and Frc
cancel each other out. Since the force of the air on the car is zero
when an object is at rest, there is no net force and therefore no
motion.
However, an airplane operates differently. Instead of pushing
against the ground to produce its thrust, an airplane pushes against
the air using a propeller or jet engine. The wheels are present not to
provide a means of propulsion but to reduce the friction between the
airplane and the ground, increasing net thrust. Again a force diagram:
__
/ | __
/ | / |
Fpa ___________/____|_________/ |
<----------/_| oooooooooooooooooooooooo |---------->
(____________\ |__________/ Fap
\ |
\ |
\__|
As you can see from this diagram, the only way that net force can be
zero is if the force of the thrust of the airplane is exactly equal to
the air resistance. Since we know airplanes can fly in the first
place, it's clear that this point doesn't happen before airplanes reach
their takeoff velocity. So, simply put, the airplane will take off of
the treadmill.
Now some of you may be saying "RedOmega you groovy dude, you
just ignored the friction of the wheels!" but the truth is that the
presence or absence of friction does not change the results of this
problem. As the plane accelerates, the wheels begin to turn. The
treadmill then speeds up to match the speed of the wheels but in doing
so the relative speed of the wheels increases causing another increase
in the speed of the treadmill much in the fashion of Zeno's Paradox.
What we have here is an infinite acceleration loop in the presence of
friction and unless the airplane has titanium-reinforced,
Teflon-coated, out of this world space wheels, the tires and eventually
the axles as well will melt into a pool of molten lava. Since the
airplane has no wheels now, the treadmill's speed drops to zero and the airplane takes off amidst a cloud of smoke with Samuel L. Jackson onboard.
Also all the frictional force is tangent to the point of rotation
so it goes directly into making the wheels spin, not slowing the plane
down.
Sources:
- http://waxy.org/2008/02/origins_of_the/
- Original Russian post: http://forum.ixbt.com/topic.cgi?id=64:417