So...about that plane on the treadmill...

On Tue, 12 Dec 2006 14:50:04 -0800, Nomen Nescio wrote
(in article ):

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From: "Brian"

With perfect frictionless bearings it will take 0 force. If the engine
is generating any thrust the airplane will move forward no matter what
the treadmill does.

As long as a magical massless wheel is attached to the magical frictionless
bearing.

There is no reason to assume that the treadmill will not fail first! So, not
having enough information to determine if the treadmill will stop running
before the wheels do, we cannot categorically say that the airplane will not
take off because the wheels will fail. The motor of the treadmill might
simply burn out first and the treadmill come to a stop.

On Mon, 11 Dec 2006 21:10:32 -0800, peter wrote
(in article om):

Peter Duniho wrote:
"peter" wrote in message
oups.com...
The problem is that as it is stated, the scenario is not one that could
ever be created with a real treadmill subject to normal engineering
constraints. [...]

You can interpret the question in that way of course. However, the intent
of the "puzzler" is clear, and the fact that it is poorly stated should not
interfere with making a reasonable, good faith effort to address the
intended question.

It's well and good to nitpick about physically impossible situations, but
rest assured if you started doing so in a true interactive situation in
which the person stating the puzzle had the opportunity to restate it, you
would quickly get past the nitpicking and get to the intended question.

It's a waste of time to do the nitpicking in the first place. It's easy
enough to infer what the interesting question really is.

My view was that it was exactly the infinite feedback mechanism that
made the problem as stated interesting. Otherwise it's trivial and
boring.


Heh, heh. So use a ski plane. Since the speed of the skis is "zero" under the
terms of the problem, the treadmill will remain motionless! Problem solved.

Christopher Campbell wrote:
On Wed, 13 Dec 2006 14:34:54 -0800, peter wrote
(in article . com):

Christopher Campbell wrote:

The other gotcha in this little puzzle is that it attempts to get you to
divide by zero.

Explain to us please where the statement of this problem ever involves
division by zero.
One can readily see where the statement implies a value of zero for air
speed since in the absence of wheel slip:

Treadmill speed = wheel speed (stated explicitly in the problem)
and
Air speed = wheel speed - treadmill speed (assuming calm air)
this directly implies that
Air speed = 0.

But I don't see where division by zero ever comes into play.
The stated problem does imply a runaway positive feedback in the
treadmill speed control. I.e. the moment the plane starts to roll
forward the control system would speed up the treadmill to match the
wheel speed. The motion of the treadmill would then speed up the wheel
rotation to a higher speed thus forcing the treadmill to move still
faster to catch up. The result would be an ever increasing treadmill
and wheel speed until something gives - most likely the tires (if we
ignore the technical difficulty of building the specified treadmill).

This is the old Achilles vs. the Tortoise conundrum that so
puzzled ancient Greek mathematicians. The puzzle was this: Achilles and a
Tortoise agree to have a race. Achilles agrees to let the Tortoise have a
head start of getting half way to the finish line. The starting gun sounds
and they are off! (Well, the Tortoise is, anyway.) The Tortoise reaches the
half-way mark and Achilles starts running. But by the time that Achilles
reaches the half-way mark, the Tortoise has moved forward. And by the time
that Achilles reaches the point where the Tortoise has moved to, the
Tortoise
has moved forward again, albeit not as far as before. Again Achilles reaches
the third point where the Tortoise was, but the Tortoise has moved forward
again. No matter how fast Achilles runs, he can never catch up with the
Tortoise. It was this sort of logic that led the Greeks to conclude that
everything was imaginary and that motion was impossible. They could not
solve
the problem because they did not have the number zero.

Zeno's Paradox. But I doubt if you could find any ancient Greeks who
actually concluded that motion was impossible. Even while puzzling
with Zeno over his problem, they continued to go to the markets to do
their shopping and to their respective work places.

And there's no need to have the concept of the number zero to solve
Zeno's paradox, just the idea of the convergence of some types of
infinite sums. I.e. if each successive run of Achilles is half as long
as the previous one (say he walks twice as fast as the tortoise) then
we have a sum for the total distance 'D' of the form:
D = x + x/2 + x/4 + x/8 +...
multiplying this by 2 gives:
2D = 2x + x + x/2 + x/4 + x/8 + ... = 2x + D
subtract D from both sides and we solve for the total distance Achilles
needs to walk:
D = 2x; i.e. twice the distance of the headstart he gives the tortoise.

The airplane-on-a-treadmill is just a restatement of the same problem. It
attempts to convince you that the airplane cannot move relative to an
outside
observer if the treadmill always moves at the same speed as the wheels. If
the wheels accelerate, then the treadmill accelerates, so the plane cannot
move, right? Wrong. The airplane does move, and it accelerates relative to
an
outside observer at the same rate as it would if the treadmill remained
stationary. The only thing that changes is that the wheels spin faster.

Sure, but airplane wheels have some maximum speed. Once the treadmill
gets up to that maximum speed the airplane wheels would fail and the
airplane is now sitting on a treadmill with a bunch of failed tires.
So the question becomes whether a plane can still take off after you
shoot out all the tires when it first begins its takeoff roll.

And yes, postulating a frictionless surface for the treadmill gets
around the problem and allows a normal takeoff. But the very term
treadmill implies a surface with reasonable friction, i.e. the tread.


Well, if you understand Zeno's paradox, then you understand enough that the
airplane will move forward on the treadmill. If the tires don't blow, it will
take off.

On the contrary, *if* the treadmill is able to perform as explicitly
stated in the problem; i.e. to always keep increasing speed so that it
is moving at the speed of the wheels but in the opposite direction,
then the plane won't be moving forward relative to the ground or still
air. The ability of a real treadmill to do that isn't relevant since
that performance is stipulated in the problem statement.

I will refer you to the book "Godel, Escher, Bach" for a discussion
of how the problem is created by an attempt to divide by zero.

You're the one making that claim - how about supporting it rather than
just handwaving about it being somewhere in a book?

If your only argument is that airplane tires will not stand the stress, then
you are placing a constraint on the problem that is not originally stated.
You are basically changing the question.

No, the problem doesn't say that there's anything out of the ordinary
about the plane, so characteristics that are common to all actual
planes will also apply to the one in the problem.

Some airplane tires might stand the stress; others might not. Tires are
highly variable in their design and intended purpose. You cannot flat-out
declare that all tires would fail.

The statement is true of all airplane tires that at some speed they
will fail.

In fact, why would not the treadmill break
down before the tires? The motor could overheat and stop the treadmill
entirely, or the treadmill surface could disintegrate, or it might be crushed
by the airplane. The airplane could be so heavy that the treadmill could not
turn at all. We cannot assume that the treadmill is any less immune to stress
than anything else stated in the problem.

But the problem explicitly states that the treadmill *will* keep up
with the speed of the airplane wheels. Unlike the airplane, the
treadmill is clearly a hypothetical construct that is only invented for
the purposes of posing the problem. I am not assuming anything about
it other than what is stated in the problem; i.e. that it is a
"tread"mill and therefore can be assumed to have a reasonable friction
surface, and that it is controlled and propelled in such a way as to
always keep up with the speed of the airplane wheels as they spin. If
the treadmill were to break and therefore no longer keep up with the
wheel speed that would be a direct contradiction of the explicit
problem statement.

So, lacking any further limitations
as stated in the problem, tires must be assumed to be capable of withstanding
the stress of the treadmill. Otherwise, why not throw in all other kinds of
variables not stated in the problem, like flap settings, wind, temperature,
density altitude, fuel on board, payload, visibility, clearance, and whether
it would violate FAA rules?

The problem makes no statement about these variables, nor about the
tire strength, for the very reason that these are not hypothetical
constructs like the treadmill, but common to everyday airplane
operations at every airport on a daily basis. The question is whether
a normal plane, operated with normal engines, normal controls, normal
tires, etc. can take off under the very abnormal condition of being on
a hypothetical treadmill with the given characteristics.

No, go with the problem as stated, and let us not make it a trick question by
assuming facts not presented to the audience.

I *am* going with the problem as stated. In particular where it
explicitly states that the treadmill will move backwards at the speed
of the airplane's wheels. Unless the tires are slipping, which rubber
tires shouldn't do appreciably on a 'treadmill' surface, then

Ground speed = wheel speed - treadmill speed
and since the problem stipulates that treadmill speed = wheel speed we
can easily solve this as:

Ground speed = wheel speed - wheel speed = 0.
If you're claiming that the ground speed (i.e. relative to the earth
and still air) is something other than zero, then it is you who is not
going with the explicit statement of the problem.

As for making it a trick question, I'd note that it is a puzzle
question, not an engineering design question. Puzzle questions are
generally intended to be trick questions in one or more ways - that's
what makes them interesting and provokes contradictory responses.

If it were given to me as an engineering question then I'd immediately
point out that the treadmill being requested can't possibly be built
since it requires instant acceleration of a massive structure and would
request that the project be modified to come up with something
feasible. But as a puzzle question having a gigantic treadmill that
can instantly accelerate to thousands of miles per hour is perfectly
legitimate - just don't ask me to build one.

On Wed, 13 Dec 2006 23:37:37 -0800, peter wrote:
On the contrary, *if* the treadmill is able to perform as explicitly
stated in the problem; i.e. to always keep increasing speed so that it
is moving at the speed of the wheels but in the opposite direction, then
the plane won't be moving forward relative to the ground or still air.
The ability of a real treadmill to do that isn't relevant since that
performance is stipulated in the problem statement.

Lock the brakes and do the takeoff... The treadmill will sense it as an
attempt of the wheels to go backwards and and start moving in the
direction that the plane wants to go anyway...

Grumman-581 wrote:
On Wed, 13 Dec 2006 23:37:37 -0800, peter wrote:
On the contrary, *if* the treadmill is able to perform as explicitly
stated in the problem; i.e. to always keep increasing speed so that it
is moving at the speed of the wheels but in the opposite direction, then
the plane won't be moving forward relative to the ground or still air.
The ability of a real treadmill to do that isn't relevant since that
performance is stipulated in the problem statement.

Lock the brakes and do the takeoff... The treadmill will sense it as an
attempt of the wheels to go backwards and and start moving in the
direction that the plane wants to go anyway...

If the brakes are locked then the wheel speed is zero and the treadmill
speed (as specified in the problem) must also immediately go to zero.

("peter" wrote)
If it were given to me as an engineering question then I'd immediately
point out that the treadmill being requested can't possibly be built
since it requires instant acceleration of a massive structure and would
request that the project be modified to come up with something
feasible. But as a puzzle question having a gigantic treadmill that
can instantly accelerate to thousands of miles per hour is perfectly
legitimate - just don't ask me to build one.


The treadmill need only be (approx) 6-ft wide x 8-ft long.
(If it needs to be any longer, your answer is wrong)

The object (the plane) isn't THAT heavy.

Our GIGANTIC treadmill only needs to average both accelerations - not have
instant acceleration. (Kind of like your home's thermostat keeping the room
at 68F. It has a 3 or 4 degree temp spread so it isn't "popping" on all the
time. BTDT)

When the plane roles forward two inches, the electric motors speed up
....until the 'curb feeler' sensors detect the axle has returned to point X.

Then, with basic computing, the treadmill readjusts its speed. It might be
only an inch for the "forward" or "back" tolerances ...or it might be a
foot.

It's a small Cessna/Piper/Cri-Cri we're dealing with, here. NOT an F-18 off
a carrier deck!

If you can't slap one of these puppies together in an afternoon... :-)


Montblack-to-the-drawing-board

Montblack wrote:
("peter" wrote)
If it were given to me as an engineering question then I'd immediately
point out that the treadmill being requested can't possibly be built
since it requires instant acceleration of a massive structure and would
request that the project be modified to come up with something
feasible. But as a puzzle question having a gigantic treadmill that
can instantly accelerate to thousands of miles per hour is perfectly
legitimate - just don't ask me to build one.


The treadmill need only be (approx) 6-ft wide x 8-ft long.
(If it needs to be any longer, your answer is wrong)

Unlike other responders here, I'm trying to go by what the problem
actually states, not what they think it should state instead. In
particular, the problem says "Imagine a plane is sitting on a massive
conveyor belt, as wide and as long as a runway." So it needs to be as
wide and long as a real runway to be in agreement with the problem
statement - that's much bigger than 6' x 8' - at least based on the
real runways I've come across. (But I commend you on your short field
landing and take off skills.)

The object (the plane) isn't THAT heavy.

Our GIGANTIC treadmill only needs to average both accelerations - not have
instant acceleration. (Kind of like your home's thermostat keeping the room
at 68F. It has a 3 or 4 degree temp spread so it isn't "popping" on all the
time. BTDT)

Your home heating system is in a negative feedback, well-controlled
situation as opposed to the treadmill which is in a positive feedback,
runaway and out-of-control situation. In the first case the action
taken in response to the stimulus (turning on the furnace when it gets
too cold) acts to reduce the stimulus. But in the second case,
speeding up the treadmill when the wheels speed up, only acts to make
the stimulus worse. So the faster the treadmill goes, the more it
pulls the wheels of the plane around and makes them spin even faster.
I've experienced that kind of positive feedback in miswired electronic
control circuits and it results in rapid escalation out of the physical
bounds of the devices - i.e. as soon as it was turned on there was a
sudden flash, a puff of smoke, and generation of lots of heat.

When the plane rolls forward two inches, the electric motors speed up
...until the 'curb feeler' sensors detect the axle has returned to point X.

But that won't happen easily since the plane is being pushed forward by
the thrust of its motor (a substantial force) and the only thing
pushing it back to point X is the slight frictional drag of the
spinning wheels. So until the treadmill reaches a really high speed
where that frictional drag becomes significant (probably when either
the wheel bearings start to overheat or the tire starts to fail) the
plane will keep moving forward and triggering the treadmill to go ever
faster.

Then, with basic computing, the treadmill readjusts its speed. It might be
only an inch for the "forward" or "back" tolerances ...or it might be a
foot.

Let's give it your maximum tolerance of a foot. That means that in the
time it takes your Cessna 150/whatever to move forward just one foot
the treadmill needs to speed up to the point where the wheel drag is
enough to equal the thrust from propellor. A 150 may not have a whole
lot of thrust, but it's still large compared to the drag of the wheels
turning at say 100 mph. I'd call an acceleration from 0 to 100mph in
the time it takes the plane to move a foot pretty impressive for a
treadmill the size of a runway - and that wouldn't even be enough since
the thrust is still larger than the 100 mph drag force. So the
treadmill has to go still faster until something in the plane's landing
gear (tires/bearings/etc.) breaks and results in a greater frictional
drag force which can counter the propellor thrust.

It's a small Cessna/Piper/Cri-Cri we're dealing with, here. NOT an F-18 off
a carrier deck!

If you can't slap one of these puppies together in an afternoon... :-)

I think you're seriously underestimating the difficulty of the design,
but you're welcome to prove me wrong with a working model.

In article .com,
"peter" wrote:


On the contrary, *if* the treadmill is able to perform as explicitly
stated in the problem; i.e. to always keep increasing speed so that it
is moving at the speed of the wheels but in the opposite direction,
then the plane won't be moving forward relative to the ground or still
air. The ability of a real treadmill to do that isn't relevant since
that performance is stipulated in the problem statement.

What keeps the wheels in contact with the treadmill when the treadmill
(and wheels) are going supersonic?

--
Bob Noel
Looking for a sig the
lawyers will hate

"Travis Marlatte" wrote in message
...
"Gig 601XL Builder" wrDOTgiaconaATcox.net wrote in message
What you saw was an aircraft that failed to achieve and or retain a
critical airspeed. Either the catapult failed or the engine failed or,
well any number of things. There is a reason carriers turn into the wind
to launch aircraft. There is also a reason that carriers can't launch
fixed wing aircraft while tied to the dock. Well they might be able to
but a lot of things have to be perfect.


Thanks. But it was a joke. I do question the word "can't" in your
explanation. I would believe "can't launch some fixed wing aircraft but
not as a general statement.


That's why I added the sentence that immediately follows the one you have a
problem with.

"Bob Noel" wrote in message
...
In article .com,
"peter" wrote:


On the contrary, *if* the treadmill is able to perform as explicitly
stated in the problem; i.e. to always keep increasing speed so that it
is moving at the speed of the wheels but in the opposite direction,
then the plane won't be moving forward relative to the ground or still
air. The ability of a real treadmill to do that isn't relevant since
that performance is stipulated in the problem statement.

What keeps the wheels in contact with the treadmill when the treadmill
(and wheels) are going supersonic?

--
Bob Noel
Looking for a sig the
lawyers will hate

In the original problem statement; nothing except gravity causes contact of
the wheels, or any other part of the airplane, to the treadmill at any
speed.

Peter