A plane is standing on a runway that can move (like a giant conveyor
belt). This conveyor has a control system that tracks the plane's
speed and tunes the speed of the conveyor to be exactly the same (but
in the opposite direction).

Will the plane be able to take off?

---

Clarification of Question by iflytri-ga on 05 Jan 2006 23:00 PST

This question tends to generate intense debate wherever asked, and yet
intuition tells me there can only be one right answer to this
relatively straightforward question: either the plane will move
forward and takeoff, or it will not.  I am looking for a simple yes/no
answer and a logical explanation to support it.

An airplane's lift is generated by air flow relative to the wings. The
speed of the aircraft relative to the runway is, in principle,
irrelevant.

   In fact, let's put it this way: As long as the plane's wheels are
free to roll, the conveyor belt can move either forwards or backwards,
at any speed, but the engines will exert thrust and the aircraft will
aquire airspeed, and will liftoff regardless the belt.

There is a discussion of this on the Straight Dope Message Board:

http://boards.straightdope.com/sdmb/showthread.php?t=348452

If the conveyer belt offsets the plane's speed, then the plane is
stationary to the ground and will achieve no more lift than if it was
just sitting still on the ground without a conveyor belt.

So unless there is a very strong headwind, it will not be able to
generate the lift necessary to take off.

The point is that the belt cannot play a role in the plane's
aquisition of speed due to engine thrust, as long as the wheels roll
freely. The rolling wheels decouple the plane from the belt's motion.
In fact, for idealised wheels which have no bearing friction, the
plane's inertia would prevent it from being carried by the belt even
in the absence of engine thrust. In such a scenario it is engine
thrust alone which propels the aircraft either forwards or backwards.

Qed100,

Read the question again.  The treadmill plays a crucial roll in the
plane's acquistion of speed as long as the plane is not airborne.  If
it is still on the ground and the wheels are rolling forward at a rate
to move the plane at, say 150 mph, the conveyor belt is pulling it
backwards at the same 150 mph.  So relative to the stationary earth,
the plane is going exactly nowhere.  This is like going up the steps
of the down escalator at the same rate of speed the steps are going
down - you get nowhere.  And if you are going nowhere and the wind is
not blowing, you will not generate any lift at all and therefore will
not be able to take off.

If wheels do not have any friction, you will slide on them, not roll. 
This is like a car that slides on the ice.  However, the clear
intention of the questioner is that the wheels have enough friction to
prevent sliding on the conveyor belt.  Therefore the plane is not
decoupled from the belt's motion.  And in the scenario described, will
not be able to take off.

I should clarify one point.  If the belt were accelerating, the plane
would roll backwards to some extent relative to the belt and to some
extent be carried along by the belt, but if the belt is moving at a
constant speed, the plane would not roll back at all.  It would be
carried by the belt.

Again, wheels need to have some friction with respect to the ground
(or in this case conveyor belt) in order to roll, not slide.  Unless
the problem specifically states otherwise, I think we have to assume
the wheels roll, not slide.

The plane is driven by the thrust of the engines acting on the air not
the ground.  It *will* accelerate forward as a result, and acquire
speed relative to the air and therefore take off.

The friction in the wheels is small enough to be irrelevant.

At take of, the runway will be spinning backwards at 2x the forward
motion of the plane.

The wheels/runway have nothing to do with acquiring airspeed. (for
instance imagine it on skids rather than wheels).

The runway can spin the wheels fast, but it can't hold the plane still
as it creates no force to do so.

No, the aircraft will not takeoff. As already mentioned lift is
generated by the velocity of the flow over the wings.

Let's look at it this way - if you are running on a treadmill, do you
feel any wind? If you wouldn't then the aircraft's wings wouldn't
either.

If the force transmitted via the conveyor belt via the wheels, which
presumably use the brakes, is the same as the thrust on the engines ..
then it will sit still.

Do you know what a 'Backwind' is ?
- that is an equivalent, and I was reliably assured that pilots are
not trained for it - as they are dead.

If you are looking at STOL then Ski Jumps worked for the Harrier, and
the idea of a full frontal blast from a static jet on an A/Carrier to
lift an aircraft might work
- oddly a reverse flush with an internal forward gust might do the trick
- I would call that 'Foghorn' - because it looks like it

"The plane is driven by the thrust of the engines acting on the air not
the ground.  It *will* accelerate forward as a result, and acquire
speed relative to the air and therefore take off."

This is the correct answer. Unlike a car, the wheels of a plane play
no role in accelerating the plane. All they do is support the weight
of the plane until the wings have enough lift to support it. It is
engine thrust acting on the air that propels the plane forward. If a
plane depended on thrust from its wheels to move, it would have a hard
time moving once it left the ground, wouldn't it?

QED is right.  If you really want to spend time on this question, read
the Straight Dope Message Board that Pinky linked.  And before anyone
starts talking about the plane's wheels or bearings causing problems,
that is also discussed there, with the thoughtful remark that if we
are contemplating the existance of such a conveyor belt, then we are
also accepting that the wheels and bearings to go along with that
exist.  That is to say (to Anse1001) that the free rolling wheels has
no effect on the plane's forward exceleration.  As someone posted on
Straight Dope, if the plane were pulled by cable, it would move
forward regardless of what the conveyor belt was doing.  Same
situation with the plane's propellor or jet engines that work against
the ambient air.  The conveyor belt only makes the wheels turn faster.
Need I go on?

Wow. Someone on the Straight Dope message board said they hope this
question is a joke, or we're going to need to start taking away voting
rights -- I have to say I agree. An object's movement relative to the
ground (whether real or on a treadmill) has NOTHING to do with flying!
The only thing that matters is the amount of air passing through the
wings. If anything, a plane could theoretically fly in a windtunnel
while stationary, but treadmills have absolutely nothing to do with
flying.

Poet and Ted131, you both copied, word for word, the same comment off
the internet.  The comment is:

The plane is driven by the thrust of the engines acting on the air not
the ground.  It *will* accelerate forward as a result, and acquire
speed relative to the air and therefore take off.

This comment is wrong.  It completely ignores the friction of the
wheels on the ground.

There are two kinds of friction that are generated by wheels.  The
first is the friction from the ball bearings.  This type of friction
is small and irrelevant to the question.  The second is the friction
of the wheels on the ground, or in this case, the conveyor belt.  This
friction needs to be great enough to make the wheels grip the road
under a variety of conditions.  This friction is not so great (nor
insignificant) when the wheels roll, but it is dramatically greater
when the wheels skid.

Once the airplane?s wheels start to roll, the conveyor belt will match
their speed, effectively keeping the plane in place.  The rotational
speed of the wheels and the speed of the conveyor belt will both
increase together until the maximum rotational speed of the wheels is
reached.  As long as the wheels roll without skidding, the plane will
go exactly nowhere.  For clarity, let?s assume the maximum rotational
speed of the wheels is 300mph.  Once that speed is reached, if the
airplane?s engine still has additional power, the wheels will start to
slip.  The wheel?s rotational speed, and therefore the conveyor belt?s
speed, will both max out at 300mph.  Let?s assume the plane needs to
achieve airspeed of 100mph to take off.  The engine, then, would have
to have enough power to make the plane skid on its wheels down the
runway at 100mph.  How much power is that?  If the plane were not on a
treadmill, it is equivalent to locking the wheels so they cannot
rotate, and accelerating the plane on skidding wheels to 100mph.  That
would take a LOT of power.  And also quickly shred the tires.

"As long as the wheels roll without skidding, the plane will
go exactly nowhere."

   That's a bunch of crap. It violates the conservation of momentum.
It effectively says that rockets cannot propel themselves through
space. And yet they do.

Anse1001, 
You've shown more savvy on other questions.  You are describing what
would happen with a car on the conveyor belt; the situation of the car
trying to advance by the motor turning the wheels.
The quoted comment is correct.  The 100% friction between the conveyor
belt and the tires only spins the tires faster as the plane moves
ahead.
Give up, you're wrong.  I screwed up a couple of days ago on another
question and calculated noon to be east instead of west of Greenwich
Mean Time 15:30.  :-/
Myoarin

Qed001 etal,

Regarding conservation of momentum:  A rocket going thru space only
has the rocket and the exhaust.  A plane on the ground has the plane,
the exhaust and the wheels in contact with the ground.  You overlooked
the last item.  Therefore it is a false analogy.

Let's look at this a little differently for clarity, by looking at the
movement of the wheel and the conveyor belt sequentially.  If the
wheel has a 12 inch circumference, with one rotation it will move
forward 12 inches.  If the conveyor belt tracks with the wheel it will
move backward 12 inches.  Since the wheel is on the conveyor belt it
will also move back 12 inches.  So the wheel ends up in the same place
as it started.  Whether this takes one second to complete, 1/10th of a
second, or 1/100th of a second, the result is the same.  The wheel
ends up in the same place it started.  In reality the wheel and
conveyor belt move simultaneously.  But if they each make the same
amount of movement as they did when they moved sequentially, the
result will be the same.  And the wheel will go nowhere.

They only way the wheel will move is if there is slippage.  If the
wheel makes one rotation and slips one inch forward during that
rotation it will move 13 inches forward.  It will move forward 12
inches based on the rotation of the wheel and one inch because of
slippage.  The conveyor belt will only track with the rotation, not
the slippage.  So it will only move back 12 inches.  So you have a net
forward movement of 1 inch.  All due to slippage.

None of those who have disagreed with me have examined how the
conveyor belt will interact with the rotation of the wheel at all.  At
best, you've just waved your hand at it.  If you are right, you should
be able to explain in detail how the wheel and conveyor belt interact
with respect to the wheel's rotation and slippage and show me where I
am wrong.  If you can't do that you should agree with me.

"Regarding conservation of momentum:  A rocket going thru space only
has the rocket and the exhaust.  A plane on the ground has the plane,
the exhaust and the wheels in contact with the ground.  You overlooked
the last item.  Therefore it is a false analogy."

   Wrong. What we see is that the entire role of freely turning wheels
is to eliminate the role of the runway as a source of drag, reducing
the dynamics of the system to an engine & its exhaust. The treadmill
can run at any rate, forwards or backwards, dragging the wheels with
it. All the moving belt does is exert torque on a wheel. The
frictional force does work on the wheel. But since the wheel turns, no
net work is done on the rest of the airframe. There is no drag. The
aircraft will be propelled through the air, and it will aquire lift.

Qed001,

Have you ever seen an airplane towed at an airport?  The vehicle
towing a large plane needs to have a fair amount of power to move the
plane with the its freely turning wheels.  The reason is friction.

Iflytri --

Suggested Google search strategy:
downwind takeoff airplane

In particular, see the Avweb article that addresses your question directly:
"It's The Medium, Manfred," (Durden, Nov. 27, 2005)
http://www.avweb.com/news/columns/191034-1.html

The answer is that it's "all about the airspeed," as the Rick Durden
correctly notes.

It's an important concept for pilots, particularly at takeoff and
landing.  Seemingly small tailwind effects can lengthen both.  It's
strong enough that some commercial carriers have policies forbidding
departures with a tailwind.

Best regards,

Omnivorous-GA

Ansel001

1.  Could you provide me the link to what I am supposed to have
plagiarised?  I might be wrong, but I don't recall copying it.

2.  You're wrong about the plane.

Poet

"Have you ever seen an airplane towed at an airport?  The vehicle
towing a large plane needs to have a fair amount of power to move the
plane with the its freely turning wheels.  The reason is friction."

   Yes, there is some friction in the wheel bearings. But if the
wheels were missing from the planes, I doubt that those runway tow
vehicles could manage to tow anything.

   It's a fact that the role of wheels is to elminate friction between
road & payload. That there is *some* friction in a well designed
bearing is irrelevant. The resistance is tolerably small, and aircraft
engines routinely overcome the friction in the wheel bearings every
day. I have seen aircaft both being towed and plowing forwards over
the runway under their own power.

Poet,

Upon closer examination, your comment is worded a little differently
so it must be your own.  However, Ted131 did copy you.

However, I respectfully disagree with your conclusion.

You might want to consider the implication of one of your own comments.

The wheels/runway have nothing to do with acquiring airspeed. (for
instance imagine it on skids rather than wheels).

The point I raised is that it is only when the wheels start to slip
(like skids would) that you get any forward motion of the plane.  To
the extent the wheels roll rather than skid, the conveyor belt exactly
counter acts the forward progress of the wheels.  Switching from
wheels to skids makes all the difference.  You won't acquire airspeed
(assuming there is no wind) until the plane starts to move down the
conveyor belt.  And that will only happen when the wheels start to
skid.

You made another statement I don't understand.

At take of, the runway will be spinning backwards at 2x the forward
motion of the plane.

Please explain how you arrived at this conclusion.  I respectfully
disagree with this statement.

Iflytri,
You are right with your clarification.
There must be a way to get everyone off this treadmill.   I find it so
strange that so many people can't/don't want to understand that
airplane propulsion has nothing to do with the plane's wheels.  They
just support it on the ground, spinning freely, regardless of the
conveyor belt.
Would it help to point out that if the conveyor belt were running in
the other direction, and the wheel bearings had zero friction, that
the plane would just sit there with its wheels spinning?  Only when
the plane's engines are started, would it begin to move.
Myoarin

Myoarin,

You made the comment:

Would it help to point out that if the conveyor belt were running in
the other direction, and the wheel bearings had zero friction, that
the plane would just sit there with its wheels spinning?  Only when
the plane's engines are started, would it begin to move.

You can do an experiment to check this out.  Put on your roller
blades, go to the gym and hop on the treadmill.  Turn it on and just
stand there without holding on.  Do the wheels begin to freely spin so
that you remain stationary, or does the treadmill carry you backwards
off the back end?  I guarantee you, you will go off the back end.

"You can do an experiment to check this out.  Put on your roller
blades, go to the gym and hop on the treadmill.  Turn it on and just
stand there without holding on.  Do the wheels begin to freely spin so
that you remain stationary, or does the treadmill carry you backwards
off the back end?  I guarantee you, you will go off the back end."

   Sure. There's some friction in the bearings. But airplanes do
overcome this friction several thousand times each day worldwide. An
airplane on a giant treadmill would be capable of the same. It would
aquire airspeed, and would aquire lift.

Qed100,

You seem to have a blind spot on this question.  You assume the
desired answer as your starting point.

You said:

Sure. There's some friction in the bearings. But airplanes do
overcome this friction several thousand times each day worldwide. An
airplane on a giant treadmill would be capable of the same. It would
aquire airspeed, and would aquire lift.
 
You seem to think that just because planes can take off on stationary
runways, they can take off just as easily on a treadmill that moves
backwards just as fast as the wheels of the planes rotate forward. 
Read your comment above and see what I mean.  It contains no reasoning
at all.  And by the way, it is the friction between the wheels and the
treadmill that is important, not the friction in the bearings.

I'm sorry, but you've provided no more rigorous analysis than anyone
else in this thread.

   And it cannot be the friction between the wheel surface & the belt
surface. Read again what I've said regarding the advantage delivered
by wheel technology. The wheel/road friction is rendered moot.

Qed100,

I have read what you said and it's nonsense.  I have in the various
posts, addressed and refuted all your statements.  You have simply
ignored mine and waved your hands and restated your "analysis" in
which you assume the desired answer as your starting point.

This being the case, I see no point in addressing any additional
comments you make on this thread.

It's way too easy to denigrate an argument with pre-emptive codewords
such as "handwaving", "It's not even wrong" and "You haven't been
published in a reputable peer-reviewed journal. I see this sort of
crap all the time on Usenet, but it doesn't demonstrate your argument.

   The friction between a runway & the perimeter surface of a wheel is
not a source of drag against the progress of the aircraft parallel to
the runway. Only friction in the wheel bearing can do this. If the
engine exerts thrust which doesn't exceed the bearing friction, then
indeed the plane won't go anywhere. If the engine thrust exceeds the
bearing friction, then the plane *will* move forwards. It will
accelerate as long as the thrust exceeds all sources of drag. Even if
drag accumulates with speed, such as with fluid resistance for
example, the plane will cease aquiring more speed, but will
nevertheless continue to move forwards with constant speed;
acceleration due to thrust will be balanced by counter-acceleration
due to drag. Terminal velocity will have been reached.

   The only question is thrust vs drag in the wheel bearing. It cannot
be (given a fixed runway) that bearing friction rises with wheel RPM
such that it always maintains terminal velocity of less than takeoff
airspeed; it's an empirical fact that airplanes achieve takeoff
airspeed all the time. It is true that on a treadmill the runway speed
can exceed the airspeed. The wheel RPM can be greater on a moving
treadmill than on a fixed runway. Does the wheel bearing drag increase
with RPM such that the belt can be moved fast enough to create bearing
drag sufficient to keep terminal velocity below takeoff airspeed? If
it does, then be my guest. Demonstrate.

in a 1937 study the coefficint of friction for several airplane wheels
was measured on different surfaces. I think it's ok to assume that
airplane wheels are better but definately not worse than the wheels
back then.

The worst results they saw on concrete were
.035


Now let's take a 747. It weighs roughly 850,000 pounds at max takeoff
weight. That means you can calculate it's rolling resistance.

850,000 * .035 or 29750 pounds. (Worst case with tires built in 37)

Now that same 747 has 4 engines that produce around 58000 pounds of
thrust each. That means that there are 232000 pounds of thrust
counteracting that 29750 pounds of friction.

That means that only 12% of the total thrust is going towards
overcoming the friction again in this worse case scenero of 1937
technology wheels on a really big plane where the rolling friction
numbers will be the highest.

You can do the calculation for any plane's weight and thrust you want.

Check this out for an explination of friction
http://hyperphysics.phy-astr.gsu.edu/hbase/frict2.html

If the plane can overcome it's initial static coefficient of friction
it can overcome it's kenetic coefficient of friction.

I vote no, the plane will not take off. My reasoning is that as soon
as the plane's wheels leave the treadmill and actually starts to MOVE,
the plane will hit the wall in front of the treadmill and crash to the
floor.

"I vote no, the plane will not take off. My reasoning is that as soon
as the plane's wheels leave the treadmill and actually starts to MOVE,
the plane will hit the wall in front of the treadmill and crash to the
floor."

   This is the most sensible "no" argument I've read so far. Thanks! ;)

That is the difference between a Researcher and the rest!  :)

I would say that the type of enigne in the aircraft makes a huge
difference. If we look at a wing on its own then we must have a
relative airflow before the wing produces lift. If there is no
airflow, there is no lift and hence the wing wouldn't take off (Lift =
1/2 x rho x Vsquared x S) where rho is air density, v is True Airspeed
(TAS) and S is the wing area. If TAS = 0 then Lift = 0). It is lift we
reqire to counter weight (thrust counters drag). No lift means the
aircraft won't take off.

On take off, the Harrier aircraft uses the engine nozzles to produce
upwards motion. However, this is pure thrust as there is no relative
airflow over its wings.

So, in the scenarion you've asked about, if the conveyor could counter
any forward motion and there was nil wind then the wing would produce
no lift. Say, for example, it was a propellor driven aircraft.
Propellors produce airflow over the wings (nothing to do with the
conveyor). This lift must oppose the weight of the aircraft and on a
normal aircraft would not be sufficent enough to lift the aircraft off
the ground. During light aircraft run-up checks we put the brakes on
and apply nearly full power. Very little relative airflow so we don't
lift off. It is only when we release the brakes and allow the aircraft
to accelerate that the aircraft can take off. In this scenario, the
treadmill would be countering any acceleration so the aircraft would
remain on the ground.

iflytri-ga,
Re your clarification:  you HAVE gotten your answer if you read
Qed100's, Poet's and my comments.
I thought the example of roller blades on a treadmill would be
convincing, but roller blade wheels are actually not as "free
wheeling" as they could be, and some fitness treadmills are sloped
backwards, so the example is apparently inadequate in some people's
eyes.

I will assume that you do understand the problem and recognize the
correct answer since you refer to the debate it causes.  That is more
flattering than assuming the contrary.  The fact that commenters here
and on other sites refuse to accept that the plane's wheels have
nothing to do with the plane's forward propulsion from its engines is
their mistake.
Cheers, Myoarin

Alright...i read a little over half the posts before i couldnt take it
anymore.  The plane will fly plain and simple.  The example of the
cable attached to it and pulling it along is correct...  Here is the
reasons why.
First of all, it ISNT a car.  It doesnt propel itself at the wheels. A
car has a drivetrain that spins the wheels and requires the friction
of ground to push itself forward.  A plane on the other hand, pushes
against the air to move forward.  A plane doesnt need wheels... thats
why it can take off on the snow with skis or on water with floats.. 
Wheels are there to provide less resistance for the plane.  It would
act the same as with cement blocks...just would have more friction
counteracting it.

second... to solve the problem.. simply make a free body diagram. 
according to newtons laws... Force = mass * acceleration.  SO if the
net of all the forces is positive...it is accelerating and will
continue to till it has sufficient wind over the wings to create lift.
 There are four main forces acting on it.  Thrust (to the right) Drag
(to the left) the normal force (up) and gravity (down).  So if thrust
is greater than drag... it will accelerate.  the drag forces are wind
resistance (would be the same as on a normal runway).  Bearing
resistance (too minimal to matter in airplane grade bearings...  and
rolling resistance.

Most people tend to be hung up on this rolling resistance thing... 
rolling resistance = coefficient of rolling friction * weight.  As you
can see...this equation has nothing to do with speed.  There may be a
slight increase, as the extra heat may increase the CRF... but very
minimally.

Thus the plane will accelerate and continue too untill it creates
lift.  If someone still doubts that... if you ever want to convince
someone....simply show how the drag force is equal to or greter than
the thrust force...

Intense debate is an understatement, here's a thread where this very
discussion is  up to 44 pages:
http://forums.flightinfo.com/showthread.php?t=66860

I think someone should contact MythBusters and ler Jamie and Adam get
to the bottom of it. At the very least, they will consider it,
reference this excerpt from the Transcript of Jamie and Adam's Nov.
10, 2004, Online Chat...

..."Q) Joe: Can I submit a myth to bust?
http://dsc.discovery.com/fansites/mythbusters/chat/transcripts/04nov10/04nov10_09.html

Answer by Jamie: Absolutely.

Answer by Adam: Post it on the Web site. We read every single posting.
While we often don't have nearly as much time as we'd like to post
replies, we do read every email that comes to us..." (more at the
link)

If you can somehow design and execute the experiment, AND get it on
video tape, you might get on the show:
http://dsc.discovery.com/fansites/mythbusters/application/application.html


Personally, I like this explanation of why the plane won't fly:
http://forum.physorg.com/index.php?showtopic=2417&st=1155&#entry42611

Cynthia, you were right the first time, the plane hits a wall, like
this question does.  ;)

First I would like to say that reading this has been very enjoyable
for me.  I love the way that people got offended, upset, and
belligerent all over a theoretical runway.  As far as I can tell the
plane would take off, but I'm only an Aerospace Engineer, I could be
wrong.  My laymen's justification is that the problem is very
mathematically similar to sailing a boat up a river with swift
current.  It is possible to sail a boat up a river with no motor, why
you say?  Why because the thrust generated by the air on the sail is
enough to over come the  drag of the boat in the water.  That's really
all I have to contribute...the rest of the math stuff has already been
said, I especially liked the reference to 1937 airplane wheels...that
guy was on the right track.  Also the guy who noted the runway moving
at 2x the speed of the plane could be wrong say the it could be more,
likely it would be much more because once the plane starts moving the
runway will rapidly accelerate without result causing great
frustration for the runway and all confused onlookers. Have a blessed
day everyone..

matt

ucabednego,

   Well said.

Yes, quite, although now I am beginning to feel that it is a little
unfair to those stuck on the treadmill to dissolution them.  ;)

They do all of the time.  The "treadmill" or moving runway is called
an aircraft carrier.  And they move much faster than the treadmills at
the YMCA.

And you can land on a moving surface too --
http://science.howstuffworks.com/aircraft-carrier5.htm

I just saw this question posed on Cecil Adams' "The Straight Dope."
His answer was that the plane would take off. However I and all of my
co-workers think it would not take off due to the problem of lack of
airflow across the wings to create lift. It's been entertaining
reading all the comments here, and there have been several people who
agree with my position. However, I have not seen any reasonable
refutation to this position. Again, simply put, the plane is not
moving relative to the surrounding air. Without airflow over the wings
there will be no lift. Without lift the plane cannot take off. Seems
pretty simple to me, and again, I have not seen anyone counter this
particular argument.

The only ways I can see that my position is incorrect is if 1) there
is sufficient airflow generated over the wings by the plane's
propulsion system (either prop or jet). 2) there is something else
besides lift from the wings that allows a plane to leave the ground.
As far as I know, the latter is pretty solid, planes fly due to lift
created by airflow across the surface of the wings. Number 1 I'm not
cartain about, at least for propellers. However, jets are a different
matter, they propel the plane forward through the principle of equal
and oposite reaction, the plane moving in the opposite direction to
the jet's thrust. Therefore, a jet engine, say all the way in the back
of the plane, not evem near the wings, shouldn't create any airflow
over the wings, so agin, no lift.

I particularly like the comparison to a person running on a treadmill.
Yo do not generate a headwind by doing so, so if you were a plane,
again, no airflow, no lift.

If someone has a reasonable counter-argument to this, I'd like to hear it!

lupulin,

   I'm sorry, but if that's what you got out of this thread, then I'm
willing to say that you haven't been paying attention to the whole
thread. Skates on a treadmill is definitely not equivalent to an
aircraft generating its own thrust.

There is an unstated assumption here, the invalidity of which gives
the answer to the question.  I think the reason that folks such as
lupulin and his co-workers don't understand the correct answer is that
their unstated assumption is never challenged.  The unstated
assumption is that the airplane on the treadmill, or conveyor belt,
will remain stationary in relationship to the conveyor belt.  This is
not so.  The airplane's thrust is not expended against the belt, but
against the surrounding air.  Thus, treadmill or not, the plane will
move forward and eventually achieve enough airspeed for take off.  Of
course, if the conveyor belt is short, the plane will fall off the end
before having achieved enough airspeed to take off, so a passenger jet
would need a very, very long conveyor belt!  This means that it would
also make no difference if the conveyor belt was running in the same
direction as the airplane--the plane would still have to develop the
same amount of thrust against the air to take off.  On the other hand,
if one could somehow make the surrounding air move at the same speed
as the thrust of the engines or propellers, then the airplane would
stand still on the belt (or even on solid ground).

johasco,

   Very well said. Thanks.

How about this?
Apply the brakes on the plane and get the conveyor rolling to its max
speed of 100kph.  The wind-speed over the wing is now -100kph.  Take
the brakes off and accelerate the plane to its maximum velocity of
100kph.  The wind-speed goes from -100kph to zero.  There is no lift,
despite the jet or prop 'pushing at the air'.      I think this is in
the spirit of the question(s).

Just to keep score:  9 say it flies, 5 say it doesn't.  Cynthia,
Pinkfreud and two commenters posted non-commital comments, as did
Omnivorous, but he then seemed to agree that it flies, so I counted
him in the 9.
Qed and Anse posted almost half the comments.

I was a little reassured that the majority think the plane flies, but
still wonder what the length of the discussion here and even more so
on the other sites says about high school physics education.

Cheers, Myoarin

"How about this?
Apply the brakes on the plane and get the conveyor rolling to its max
speed of 100kph.  The wind-speed over the wing is now -100kph.  Take
the brakes off and accelerate the plane to its maximum velocity of
100kph.  The wind-speed goes from -100kph to zero.  There is no lift,
despite the jet or prop 'pushing at the air'.      I think this is in
the spirit of the question(s)."

   This isn't really relevant. It's agreeable that if the plane is
dragged backwards at -100 m/h, then accelerates itself forwards a
total of +100 m/h, then of course at that point there's no net airflow
over the foils, and no lift. This doesn't in principle prevent the
aircraft from continuing to accelerate right past that point 'til
there is airflow and lift.

   Interestingly, if the wings' angle of attack were situated
properly, being dragged backwards through air at 100 m/h would
actually generate a lifting force under the wings.

5 days without a comment--you all should be ashamed.  Here's comment #50:

Ansel001 is right:  most of the rolling resistance of a typical
(rubber) wheel on a typical surface comes not from the bearing, but
from wheel/road interface.  As the wheel rolls, the tire deforms. 
There is always a flat patch pressed against the ground.  There is
precious little deformation of the steel bearings.  The deformation
takes mechanical energy and transforms it into heat: friction. 
Roller-blade, bicycle, car, 747, doesn't matter.  The main source of
rolling resistance in all (assuming we're not talking about air drag)
is from the movement of the tire on the road.

One thing to be clear about is the reference frame relative to which
we measure the velocities.  I assume we use the airport reference
frame.  That is, both the airplane speed and the conveyor speed are
measured relative to the control tower.  In this case, I think it's
pretty clear that the plane can take off (sorry Ansel).  But actually
it would have a (slightly) harder time than than a plane which is not
on a conveyor.  This is because of the rotational inertia of the
wheels.  The wheels of the plane on the conveyor are spinning twice as
fast at takeoff as are the wheels of the other plane.  In order to
spin the wheels up to the higher speed, an extra force, over and above
that required to overcome the rolling resistance, is needed between
the wheels and the conveyor belt.  That extra force tries to hold the
plane back, and so its acceleration will be a little less, and it will
take a bit longer to become airborne.

I just took a look at the Straight Dope link posted by Pinkfreud.  The
question there is worded a little differently.  Instead of matching
the speed of the PLANE, the conveyor matches the speed of the WHEELS. 
This seems to be the definition Ansel has been using.  If instead of
watching how fast the plane is moving past the control tower, you look
at a spedometer attached to a wheel, and match that speed with the
conveyor speed, the plane won't move, and it certainly won't take off.
 As Ansel says, if the wheels don't slip, and the conveyor moves
backwards as fast as they spin forwards, then regardless of anything
you might say about engines, thrust, etc, the plane obviously goes
nowhere.  In this scenario, the thrust from the engines goes 100% into
spinning the wheels faster and faster.  In a very short time, they
will be going so fast that the centrifugal force will tear them apart,
and then the plane won't roll anymore, and will be carried backward
with the conveyor.  If you ignore limitations on the strengths of
materials, and imagine that the wheels would never break, eventually
the speed of their perimeters (and the speed of the conveyor belt)
would approach the speed of light, still without the plane ever having
moved.

However, I think this is the answer to a different question, because
to me "the plane's speed" means the speed of the center of mass of the
airplane, not the rotation speed of the wheels.

Rracecarr,
Clever of you to catch that difference in the definitions of the
problem, but I don't see that there is any difference in the solution.
If the plane can take off at 200 mph, when it starts moving, the
conveyor belt will move at the same speed in the other direction, but
since we are assuming no material failure and minimal bearing
friction, as before, the wheels will just spin faster, and the
conveyor belt will still unable to keep the plane from advancing and
eventually taking off.

Tire failure is, incidentally, a real problem, as someone mentioned. 
Heard some guys discussing taking off from high altitude airfields,
where the thinner air requires a higher speed to achieve liftoff. 
Apparently the max. speed is marked on all tires.  But if we are
assuming the existence of the conveyor belt that can run at such
speeds  (a much more likely failure), then the tires can't be part of
the problem.

Hi Myoarin--

If 
1) there is no tire slippage, and 
2) the speed of the conveyor belt is exactly the same as the speed at
which the wheels are spinning,

then the plane remains stationary.  This is not a statement that has
anything to do with the specific nature of airplanes--it is just a
mathematical fact.  So the problem in your previous post is the phrase
"when it starts moving" -- it never does.

Forget about the plane for a minute, and just think about one of the
wheels.  Let's say it has a circumference of 1 meter, and is spinning
at 10 revolutions per second.  If the conveyor belt weren't moving,
the wheel would roll along at 10 m/s.  However, the conveyor belt is
moving at 10 m/s in the opposite direction, and the wheel just spins
in place.

Hi Rracecarr,
I am a little disappointed with your analysis.  
What makes the plane move is the engines acting on the ambient air. 
The fact that it begins to move at all has nothing to do with its form
of contact with the ground  - beyond overcoming negligable wheel
bearing friction.  It starts creeping forward, and the belt creeps
backwards, making the wheels turn faster (but still slowly at this
stage, but hardly braking the advance of the plane).  Since the wheels
can rotate freely, the plane continues to accelerate, as does the
belt, the wheels spinning even faster, much faster than in the other
definition of the problem, I admit, and maybe at takeoff at some
extremely high speed.

As someone has long sinced pointed out, the plane's propulsion by its
engines can be compared with its being pulled by a cable parallel to
the ground  - or your pushing a roller skate forward on a treadmill or
conveyor belt.  You can do it, despite the counter motion of the belt.

Since we are assuming no practical problems (bearings' freezing up,
tires' flying to pieces, the existence of the conveyor belt), let's
take it to the extreme and assume absolutely no friction from the
wheel bearings.
It this case, the belt could be flapping along the runway before the
plane even starts it engines and the plane would still stand still  -
no bearing friction, no overcoming of the inertia of the motionless
plane.  The engines start, the plane moves forwards, regardless of
what the wheels and belt are doing.
Indeed, by assuming zero friction in this hypothetical situation, they
no longer are part of the equation.  If the belt were moving in the
direction of takeoff, with zero bearings' friction, it would also have
no effect on the plane, just allow the wheels to turn more slowly  -
or even backwards  - even at takeoff - if it were moving forward at
more than 200 mph.

Am I convincing anyone?

All hypothetical, of course, but the problem by definition is.

Cheers, Myoarin

Hi again Myoarin,

I'm sorry that you're disappointed.  But I really think you wouldn't
be if you would take two minutes to consider what I wrote.  It is very
simple, and it is correct.  If the plane is rolling forward relative
to the conveyor at velocity V, and the conveyor is moving at -V
(relative to the airport), then the velocity of the plane (relative to
the airport) is the sum of the two.  What is V - V?  How can you keep
arguing that it's not zero?

The cable analogy is wrong.  It is fine (and helpful for some people I
think) for the original problem.  But in the case where the speed of
the wheels (or alternatively the speed of the plane relative to the
conveyor belt) is used to set the speed of the conveyor belt, it
isn't.  However you try to move the airplane, whether with its engines
or a cable, IT WILL NOT BUDGE.  However much forward force you apply
with the cable or engines, the exact same amount of force will be
applied in the opposite direction by the conveyor belt on the tires. 
That force will act to accelerate the spin of the wheels.  And forget
about bearing friction (and wheel/surface friction too).  Things like
that don't belong in thought problems like this.  The force the
conveyor belt applies to the wheels is not acting against friction. 
It is acting against the inertia of the wheels.  It accelerates their
spin.  This is the factor that is ignored by the people who say that
the the wheels "decouple the plane from the runway".

You might ask: "what if I put the plane on the conveyor belt, and then
pull it along at 10 meters per second with a cable?"  Well, that is a
logical impossiblity.  You cannot have the wheel speed match the
conveyor speed, no wheel slippage, and still move at 10 meters per
second.  It is like asking what happens if you're tied to an immovable
wall with an unbreakable rope, and you start to walk away from the
wall at 10 m/s.  You can't even start to answer, because the question
itself is impossible.  All the conditions of the question cannot
possibly hold true.  In the airplane question, this problem arises if
you specify that the wheels are massless, because it doesn't take any
force to spin a massless wheel, and so the conveyor belt cannot apply
a backward force to the plane.  In this case, the question is
logically inconsistent.  Something has to give.  Either the wheels
must slip, or the belt must fail to perform as specified.  It is
pointless to debate what would actually happen, because the question
itself is wrong, and everyone can have their own opinion about which
part of it should be changed.

This has once again gotten to be a longwinded comment.  Sorry.  Here
is the essential point for the 3rd and last (I promise) time: 
Specifying that the velocity of the conveyor RELATIVE TO THE AIRPORT
is equal and opposite to the velocity of the plane RELATIVE TO THE
CONVEYOR, is EXACTLY THE SAME as specifying that the velocity of the
plane is zero.  The answer is in the question.

I disagree, but 'nuf's enough.  :-)

Anyone who thinks that the airplane will take off is, i'm sorry to
say, a complete idiot. Or ignorant. Anyone with a basic knowledge of
the principles of lift will know that lift is created as an airstream
passes by an airfoil or aerofoil and is deflected downward. The force
created by this deflection of the air creates an equal and opposite
UPWARD force according to Newton's third law of motion. Because a
specific shape of wing will deflect a FLOW of air downwards at a
certain force, an equal and opposite force will act on the airplane,
giving it lift and allowing it to fly. For this to happen, the
airstream needs to be of a sufficient speed - this is the reason
planes have to move forward on a runway, or more accurately, move
forwards through the AIR. If the plane was on a treadmill in a
windtunnel providing a fast airstream it would take off - however,
from the wording of the question the plane is in a normal "windless"
(or close to windless) situation. It has NOTHING to do with friction
in the wheels or any of that crap. An airplane needs AIR moving ACROSS
THE WINGS to gain LIFT in order to take off. A notable exception is
the harrier jumpjet; but a vertical take-off in this case occurs due
to a significant downward force provided by engines, not an airstream
moving over the wings. If the plane is not moving through the air (or
the air moving past the plane) at sufficient speed it WILL NOT TAKE
OFF! There is NO airstream if the plane is running on a treadmill,
even if the engines are running. The engines DO NOT push air over the
wings; their purpose is to move the plane fast enough through the air
to anable LIFT to be generated. The engines can run as "fast" as they
want but if the plane is on a treadmill it won't move!!!!  Why is this
so hard for so many people to understand?

"Why is this so hard for so many people to understand?"

   Regardless any arguments regarding wheel inertia, etc., it's hard
to see why you can't see: an aircraft with airbreathing motors doesn't
even ostensibly generate lift by blowing air over the foils directly
with the motors. The motors generate horizontal thrust, which propels
the craft forward THROUGH THE AIR, thus generating the lift force. The
ONLY question in this regard is if the forward acceleration, due to
engine thrust, will be negated by the wheels spinning passively on the
treadmill. Answer: it won't.

Right.

QED, if we are so brilliant, why can't we convince anyone?  ;-)

Maybe we're not so brilliant. >:)

Yes, it will take off.  However not because of forward movement.  As
with many hypothetical questions we must assume some things not stated
in the question.  Let's assume this is a magical converor belt that
can run fast enough to keep the aircraft relatively stationary with
respect to the ground next to the converor belt.  We must also assume
that the aircraft tires/wheels will not fly apart (which they likely
would).  Further, let's assume the intent of the question is that this
converor belt has the ability to counter the forward motion of the
aircraft.  Since the aircraft will tend to move, not as a result of
it's wheels roating, but as a result of engine thrust, the only way
this magic converor belt can "stop" the aircraft from moving forward
is to accelearte so fast that the aircraft engine's energy is all
consumed by wheel friction and the initeria of spinning up the
aircrafts wheels.  Very quickly the converor belt will be moving at
impossible speeds, but this is a hypothetical question and if we
assume it is possible for a converor belt to hold this aircraft in
place we have to assume it can move this fast.

So, very quickly after the aircraft attempts to move forward, the
conveyor belt is spinning under the aircraft at thousands of feet per
second.  Friction between the air at the surface of the converor belt
and the belt will cause the air to move with respect to the ground
next to the belt.  At some point the magic belt will be moving fast
enough to generate enough airflow over the wings that the aircraft
will lift off the belt, again assuming the belt and wheels/tires are
magic and do not fly apart.

So, yes the aircraft will lift off the belt.  What happens next is the
subject of another debate.

Well, that is a new twist, but the problem states that the belt only
moves as fast as the plane.

doug has a point.  Myoarin, I think maybe you still haven't understood
that there is more than 1 possibility for the definition of the
'speed' of the plane.

no

There are several assumptions made here which render this question
unsolvable as currently stated.

Let's solve two of the possibilities.

1: Pure fantasy. The airplane engine and power of treadmill are
infinite and bearing frictions are 0. The wheel rotation has no effect
on the plane and it takes off.

2. Real world. The treadmill exerts a force on the plane via the
friction of the bearings in the wheel axle. This friction increases
with rotational velocity, as long as the treadmill engine is capable
of spinning the wheel at high enough speeds to counteract the thrust
generated by the plane engines the plane will remain in one spot and
not be able to take off. If the treadmill cannot maintain the speeds
necessary the plane will begin to accelerate after the treadmill has
reached it's max speed, if the plane engine is strong enough to
achieve enough speed to generate the required lift on its wings it
will take off.

This scenario is easy to simulate with a toy object on a treadmill.
Imagine if the toy has very good bearings and you can easily hold it
on the treadmill, the wheels will spin up to the treadmill max speed
and you will be able to push the toy forward. In this case the thrust
generated overcomes the bearing friction and flight would be achieved.
Now imagine the toy has really terrible bearings and is very hard to
make the wheels move. At the same speeds the treadmill is capable of
transmitting a much greater force to the toy and if the friction on
the bearings is enough you will not be able to push the toy forward.

This is somewhat of a "trick" question. Not because it is phrased in a
deliberately tricky way, but because people tend to have trouble
thinking about the operation of other vehicles apart from cars which
they know so well.

The heart of the confusion is simply these two important facts:
   * cars propel themselves by pushing against the ground via friction
   * airplanes propel themselves by pushing against the air
If you can let go of how cars operate and think about what an airplane
does, you'll be able to see the problem clearly.

One good way of tackling this problem is to find a good analogy. But
the analogy must be a valid one else you'll just get more confused.
For example, someone posted the analogy of running on a treadmill. Why
is that a bad analogy? Because one runs by pushing against the ground
via friction between their shoe and the ground. This is how a car
propels itself! It is not how an airplane propels itself, by pushing
against the air. Bad analogy.

Let's use this analogy. Instead of looking at the airplane, let's back
up and go into the airport. Suppose you're walking down to your gate
and pulling your carry-on bag behind you. It's a nice new bag with low
friction wheels. No problem! Up ahead you see one of those moving
walkways. You don't see anyone coming, so you decide to do a little
experiment. You go over to the walkway that is moving TOWARDS you and
place your bag on it. Meanwhile, you step off to the side of the
walkway, and still holding on to the handle of your bag, you continue
to walk along. In fact, you intentionally walk along at the same speed
that the moving walkway is going, just in the opposite direction.
Question: does the bag move or does it remain stationary as you keep
walking? Obviously it moves with you. So why does your bag move
forward when you are walking at the same speed of the conveyor going
in the opposite direction?

The answer to that question is also the answer to the
airplane-conveyor question. To complete the analogy, the pull of your
arm is analogous to the force of the airplane engines. The bag's
wheels are analogous to the airplane tires. Do the nice low-friction
wheels on your bag on the conveyor pull against you anymore than they
do when you're just pulling your bag along normally?  No, they don't.
They are free-wheeling, after all. Meanwhile, you're pulling the bag
with the same force in both cases. So in both cases, the bag keeps
moving forward. Likewise with the airplane, the pull of the engines
doesn't change nor does the force on the airplane imparted by the
tires change no matter what the ground is doing underneath the tires.
You have the same force imbalance in either case, and since Force =
mass x accceleration, you have the same acceleration. Remember, we are
talking airplane engines which push against the AIR, not the ground.
The acceleration is with respect to the AIR, thus the airplane
develops a speed relative to the air and can eventually take off.

That's a long winded analogy. Here's a quicker solution. Engineers
learn to draw Free Body Diagrams to understand such problems. A FBD is
just a block diagram which illustrates the forces acting upon an
object. The net force can be calculated from all the contributing
forces. If that net force is not zero, the object accelerates in the
direction of that force. Let's draw a FBD now. Represent the airplane
by a simple rectangle (the shape doesn't matter). Indicate the force
of the engines pulling on the plane with a large arrow, labeled "F_e"
with e for engines. What force are the tires imparting to the plane?
Remember, they spin freely except for bearing friction and rolling
friction, which produces forces that are quite tiny compared to the
engines. Represent those forces with a small arrow, labeled F_t where
t means tire. What other forces are operating? There is a drag force
too. But remember this drag force is also small compared to the
propulsion of the engines at least at takeoff speeds. Label that F_d.

      FREE BODY DIAGRAM

   F_d |----------|          F_e
   <---| Plane    |------------->    
       |----------|
     <-- F_t

So what do we have here? We haven't put numerical quantities on these
forces, but instead have just been talking in terms of large and
small. That's okay for the purposes of this illustration. It's enough
to know that F_e is going to be much greater than F_d + F_t pulling in
the opposite direction. If that's the case, we have an unbalanced
force on the plane. Therefore, it accelerates. And it accelerates with
respect to the AIR, since F_e is produces by the engines pushing
against the air.

To sum up: Yes, the airplane takes off. The motion of the surface
underneath the freely-spinning tires is irrelevant to the acceleration
of the aircraft since the tires cannot impart any force to the
aircraft (aside from the aforementioned very small rolling and bearing
friction).

I hope this discussion has helped somewhat and not muddied the waters any further.

John Strong
Ph.D., biochemical engineering
M.S., chemical engineering
B.S., mechanical engineering

Ah, somebody who knows what he's talking about.  Great analogy, not
that it will necessarily convince the pedestrians with only their feet
to move them.
Thanks.

Jstrong--

Obviously the plane can take off if you measure the speed of the plane
relative to the fixed ground, and match it with the conveyor speed,
also measured relative to the fixed ground.  But if you measure the
speed of the plane relative to the conveyor belt (by using a
spedometer attached to the plane's wheels, say), and match that speed
with the conveyor belt, the velocity of the plane must always be zero.
 It will never move forward.

Also you're leaving a force out of your diagram--the backward force
exerted by the conveyor belt on the tires that goes into accelerating
the spin of the wheels.  This force is unimportant in the first case
(speed measured relative to fixed ground) but not in the second.

Nomen est omen:  Rracecarr, all four wheels on the ground and accelerating.  :)

Wow - how funny - this seems so simple to me, yet I am sure everyone
feels the same way about their answer...  The problem is, everyone
forgets it is a hypothetical.



Lets say you put a skateboard on a treadmill and turn it on to the
fastest spead possible as quickly as possible - it will shoot off the
back of the treadmill due to the friction of the tires/bearings....

Lets say you attach a string between the skateboard and the front of
the treadmill to a device to measure the force....  If you start the
treamill ever so slowly, the skateboard will not pull so hard.... if
you start quickly there will be more force against the string.  This
one is hard to underststand but the ACCELERATION in the bearings does
cause friction

This is the key to the whole puzzle.  Do you accept that acceleration
of the treadmill does cause more friction.  Its a tiny amount but the
amount is there.

So if you accept that the treadmill can increase the ammount of
friction, can it do so infinitly?  The force of the engine is huge- I
don't think it would be possible to concieve of a treadmill fast
enough to hold back a plane - because the resistance provided by the
bearings and tire friction is so minute...

BUT THIS IS HYPOTHETICAL.  If you accept that

1) the treadmill creates friction in the bearing.
2) the friction or force backwards is increased the faster the
treadmill accellerates
3) that the force the engines can exert is fixed.
4) the force the treadmill can exert is NOT fixed because it can spead
up infinitely fast to an infinite speed

Then you see that the plane can never take off.

The problem is that the force of the engine would require astronomical
amounts of friction from the bearings and this really isn't possible. 
BUT in hypothetical world we can continue acceleration backwards at
impossible speeds.

So it really comes down to - does the force exerted by the treadmill
increase the faster you accelerate?  If you say yes.  And you agree
that there is nothing to stop the system (its hypothetical) from
speeding up, then the plane CAN NOT TAKE OFF.

Let's try to come up with a final answer that everyone can agree on...
sorry if it's all been said, i  haven't read all the posts.

quoted from a post by egon_spangler-ga about half way down this page...

"in a 1937 study the coefficint of friction for several airplane wheels
was measured on different surfaces. I think it's ok to assume that
airplane wheels are better but definately not worse than the wheels
back then.

The worst results they saw on concrete were
.035
....
Now that same 747 has 4 engines that produce around 58000 pounds of
thrust each. That means that there are 232000 pounds of thrust
counteracting that 29750 pounds of friction.
...
If the plane can overcome it's initial static coefficient of friction
it can overcome it's kenetic coefficient of friction."

So....232,000lbs thrust - 29,750lbs friction = 202,250 lbs total force
propelling the plane forward.
*The important thing to note here is that friction comes from the
bearings in the wheels, and bearing frcition does not depend on speed.
Here is an quote from another forum
Excerpt from SKF Interactive bearing catalog for rolling element bearings:

"Under certain conditions (bearing load P » 0,1 C, good lubrication,
normal operating conditions) the frictional moment can be calculated
with sufficient accuracy using a constant coefficient of friction µ
from the following equation

M = 0,5 µ F d

where 
M  = frictional moment, Nmm 
µ = coefficient of friction for bearing (Table 1) 
F = bearing load, N 
d  = bearing bore diameter, mm"
(http://www.eng-tips.com/viewthread.cfm?qid=113896&page=5)

So, speed is irrelevant for calculating the friction drag on the
plane.  Only the weight of the plane matters, which is constant.

Therefore, the friction force in the plane is always approximately
29,750lbs in the above example.  It doens't matter how fast the
conveyor belt is moving, it will never (theoretically) prodcue much
more drag on the aircraft...the theoretically part comes in when the
bearings start to spin faster than their working limit and fail, lock
up, and then there'll be lots of friction.

So, you can spin that conveyer belt all you want, the 29,750lbs of
friction don't do much to stop the 232,000lbs of thrust.  The plane
will still have plenty of thrust with the conveyor belt spinning to
take off.  In fact, since the thrust has to overcome the bearing
friction force even on a non-moving runway, the added friction with
the backward-moving runway is small.

The plane would take off, unless the bearings lock up, even
considering bearing friction.

one addition to the last thing I just wrote...

moving the wheels faster does in fact create more "friction force."
Rather, it creates some force which opposes the forward motion of the
aircraft, not actually friction.
This force comes from the inertia of the wheels themselves.  It takes
some energy to accelerate the wheels, energy which shows itself in the
form of an additional drag force on the aircraft.
Therefore, if you continue to accelerate the conveyer belt, then yes
you can continue to add drag.  However, as soon as the conveyer belt
stopped accellerating and returned to constant velocity, the drag
force would die back down to the bearing friction force, no matter how
fast the belt is moving at this point.  Since keeping the aircraft
stationary under these conditions would require a conveyer belt which
could continue to accelerate indefinately, this scenario is
impractical.  If this was possible, however, the yes the plane would
remain stationary and no it would not take off.  In reality, this
wouldnt work nad the plane would take off.

Plane can not take off unless there is a strong headwind :)

The catergorical answer is YES. 

The fundamental point to appreciate is that the power from a plane for
forward momentum is not through its wheels and the wheels are free to
spin. The other point to appreciate is that the questions states that
the treadmill is tracking the planes speed and not the other way
around. (See scenario B)

I think people's yes/no answers depend largely on them ignoring these
two points. To clarify, imagine the treadmill runway but with a fixed
control tower at the start.
 
Scenario A - Lets assume that the plane tracks the treadmill's speed:
The treatmill starts moving at whichever ludicrously high speed you
care to mention and the plane's engines are running. For the plane to
remain stationary relative to the control tower, the engines would
only need to operate at a power to overcome the friction on the
free-spinning wheels - as previously quoted by egon_spangler, about
12% for a 747 with the wrong wheels on a concrete treadmill (a whole
new question, I feel?). In this instance, will the plane take off?

Answer: Absolutely not as there is no airspeed/lift (see everyone
elses comments for further explanation)

Scenario B - the treadmill tracks the planes speed: The "speed" of the
plane would be it's speed relative to the control tower. If the plane
accelerates to 100mph, the treadmill would accelerate to 100mph, and
the net effect would cause the wheels to spin at 200mph. However, the
plane would still be moving forward at 100mph. (For the plane to
remain stationary relative to the control tower, there would have to
be no speed, and so the treadmill would not move either.) Will the
plane take off?

Answer: Absolutely yes. Once it achieves the necessary speed relative
to the control tower, the airspeed would be sufficient to create the
lift. Granted the plane will have to overcome more friction on the
wheels than it would on a static runway as they are moving at twice
the speed, but the engines would easily overcome this.

Different conditions would cause the effects that others are arguing
towards, but none apply to this question:

If the treadmill was replaced by an windtunnel and the speed of the
windtunnel increased to match the thrust of the engines or vice versa,
then at the necessary speed, the plane could takeoff and fly whilst
remaining stationary relative to the control tower.

Alternatively, if you replaced the plane with a car with wings where
its forward momentum on the ground is through the wheels, then the
treadmill tracking the speed would increase until the car reached it's
top speed. The car would remain stationary relative to the tower and
it would never take off due to the lack of airspeed. Aside from the
treadmill question, I do recognise that a winged car with no means of
propulsion once in the air would be pretty rubbish.

In conclusion, given the conditions of the question, the plane will
take off everytime.

If I'd read John Strongs answer before I'd posted my own, then I think
I wouldn't have bothered!

I accept Rracecarr's point that there are two interpretations of
"speed": relative to the control tower and relative to the conveyor
belt. However, as he clearly states, to answer the question based on
the latter interpretation is a logical impossibility "You can't even
start to answer, because the question itself is impossible.  All the
conditions of the question cannot
possibly hold true." Would this not suggest then, that there can only
be one logical interpretation of "speed". Anything else makes the
question redundant, which I am sure isn't the point.

First, you have to assume a few things. 

1. The conveyor belt can react quick enough to the plane's speed so
their speed stays exactly the same.
2. The engine or propeller that powers the plane can only cause the
plane to go forward (or backward) but not create any lift. So the only
lift generated will be from the wings.
3. The conveyor belt's speed is linked to the speed of the plane
relative to the ground and therefore also to the air around it. It is
not linked to the turning speed of the wheels.
4. The wheels of the plane does not power or cause the plane to go forward.
5. The wheels can turn freely and without friction, but only in
reaction to the movement of the plane or the conveyor belt.

Let's start with the plane standing still. The conveyor belt is not
moving because the plane is not moving.
As the engines start to cause the plane to go forward, the conveyor
belt starts to move at the same speed of the plane.
But because the wheels are free to move, the conveyor belt has no
effect on the plane, only the wheels, so the plane continues to go
forward.
The faster the plane goes forward, the faster the conveyor belt moves
backward, but it's only affecting the wheels, not the plane.

The plane will still take off, but the wheels will turn at twice the
speed because of the conveyor belt. Because the wheels are free to
move and does not affect the speed of the plane, the conveyor belt has
no effect on the speed of the plane.

If the plane was powered by it's wheels, like a car, and the conveyor
belt speed was linked to the speed of the wheels. Then the plane will
not move.

The people who claim that the wheel will move backwards are
overlooking something very important...


Put a ball on a piece of paper, now pull the piece of paper forward
fast. What happens? The ball spins in a relatively same space?
Hmmmm... The fact is the WHEEL which will is modeled by the BALL will
start SPINNING corresponding to the motion of the runway...the only
way the net movement of the wheel will exceed 0 is either if the wheel
spins faster than the runway, or the runway faster than the wheel...

The wheel created friction in TWO ways...one is against the runway,
the other is against the plane! Friction against the runway transfers
motion from the wheel to the runway and from the runway to the
wheel...So if the runway moves, the wheel turns but due to friction
connecting the wheel to the plane, the wheel doesnt transfer all of
the energy because some is lost to friction, so the plane unless it
were being pushed forward would move BACKWARDS if the runway was just
moving by itself.

Now let's assume that the plane has no friction connecting the wheels
to the plane. Lets say they have magnetic bearings that just float the
plane due to magnatism. The wheels still have normal friction to the
runway, but now when we turn on the runway the wheels spin, but the
energy isnt transfered to the plane...so the plane doesnt move no
matter how fast we spin the wheels...

So now...let's assume that when the plane turns on its engines,
forward thrust is created, this creates a force forward and the plane
will start moving forward. This forward motion pushed forward on the
bearings of the plane, which push forward on the wheels (without
friction, but with magnetism) but since the wheels are touching the
runway friction prevents them from just sliding forward (There is a
forward force on them remember?). They must ROLL to move forward in
reaction to the force...

When the wheels roll they exert a force on the runway in the opposite
direction of their movement (just as you do on the earth when you walk
across it) but when you move push agains the earth, the earth pushes
back and you move. In this case the runway does NOT push back but
absorbs that force as it turns.

So when the force is absorbed (kind of like you trying to jump on a
floor that falls through right when you apply force to it, you will
stay in the same place as the floor falls under you, and then gravity
will take over and you will fall too)

The plane will not be able to provide enough force to the wheels to
make them move as all of the force will be cancelled out by the moving
runway.

BUT the plane's engine do NOT only apply force to the WHEELS!!! There
is a force pushing on the rest of the plane, and unless this plane is
in a wind tunnel that has air blowing on it to create enough
resistance to counteract the force of the cockpit pushing against the
air, the REST OF THE AIRPLANE will move forward.

So what happens then?

The wheels stay in one place, the rest moves forward? The plane begins
to rotate around the stationary point(s) (this being the wheels) and
this means the plane starts pointing towards the ground.

Think of it this way, you are running on a treadmill as fast as you
can, your feet are literally being swept under you as fast as you can
place another one on there. There is no way they are moving forward
anymore. Now I come up and push your head forward. What happens? Do
you move forward? NO you cant! Your head travels faster than your
feet, but since you are connected your head doesnt travel straight
forward but is pulled by the rest of your body like a ball on a tether
and travels in an ARC, downward and then you fall.

Same thing with the airplane, the wheels cant move, the rest of the
plane doesnt have any forces to prevent it from moving (aside air
friction) so the plane starts moving like a ball tethered to a pole
(the pole being the wheels) IT starts rotating towards the earth,
ultimately around the front wheel as when it tilts slightly the back
wheels will come off the ground and then only the fast spinning front
wheel will keep it from moving. So inevitably the plane rotates until
it is pointing down into the ground enough that the tip of the nose
catches the runway and thats the end of that flight.

Then of course the other scenario is that this plane can generat
STATIC lift by the engines. IF the engines can move enough air from
the front of the plane to the back of the plane while the wheels do
not move, then the air flowing under the wings could generate the lift
it needs to get the wheels off the ground.

Another thing to consider is that in our universe by our laws of
physics nothing can travel at the speed of light so assuming that the
engine can generate infinite energy is unrealistic, and then to assume
the runway could rotate at infinite speeds is unrealistic too.

This question does not have enough information to be reasonably
answered. IF the engine and the runway can only go so fast, then the
scenario I described will occur and the plane will tip over and crash.
IF they wheels and the runway can move infinitely fast than they will
reach the speed of light almost instantly and physicists arent smart
enough to figure out what happens yet, some say that its impossible to
accelerate to a speed faster than that of light so... then the wheels
and the runway would be spinning at the speed of light creating
massive gravitational forces and just things that this question wasn't
meant to ask.

IF they COULD spin faster than the speed of light they would end up
traveling back in time... Then what? the plane wouldnt have any wheels
and runway to get in its way and take off? HAHA (no because the mass
of an object moving near the speed of light would crush the plane, so
the plane would be crushed by the gravity of the runway and the wheels
spinning so fast).


There ya go... THAT is what would really happen.

Very interesting. I am going with the notion that it doesn't take off.

If the conveyor is moving backwards at the speed the plane is
thrusting forwards, it will esensially be at a stand still (much like
the aformentioned escalator theory).

Even if you suddenly stopped the conveyor moving, whilst the plane is
still thrusting, the plane would still need to cover a certain
distance to gain the velocity needed to have lift.

Also, another thing that corssed my mind. If you had the conveyor
matching the thrust speed of the plane, and the conveyor was dropped
from underneath the plane, the plane would fall with the conveyor.

I suppose its the same principal that comes with having a fly in your
car flying around, yet you are doing 100mph.

I have no qualifications, and was a school drop out at 16, so if my
answer seems a bit primitive, that's why. But i have a good
understanding, and the common sense to see that the plane would not
take off.

Ok, I've just changed my mind lol

Thinking about it, when a plane thrusts, it thrusts against the
atmosphere, not through the wheels on the belt.

lol I can see why this is a bhig debate, There's not really any solid
way of proving which is right or wrong unless its actually tried.

How about thinking of it the other way? If you sat the plane on the
conveyor belt, but diddn't switch it on......then you thrust the plane
to take off.....would the belt move in the opposite direction of the
plane?

Like if you were on a free running tredmill, if you exert a force to
start sprinting with your legs, the belt will move the opposite way.

Most confusing, and i'd love to know the answer :)

Hmmm...for an experiment we simply need a device large enough moving
in the opposite direction under the airplane.  Let's use planet earth.
 Will an airplane, pointed against the rotational direction of the
earth, have difficulty taking off?  No.  BTW the earth would be
spinning approx 1070 mi/hr against the plane if at the equator.

  If you believe the drag is great enough to prevent the plane from
taking off then the same would apply when the plane is landing. As
soon as the wheels touch down then plane should come to a halt on it's
own but this is not true and major brakes must be applied because
there is not enough friction to stop without them.