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

"John T" wrote in message
...
"Darkwing" theducksmail"AT"yahoo.com wrote in message


This has NOT been adequately explained or there
would be no question about it. If the plane is not moving on the
treadmill but rather keeping up with the speed that the treadmill is
moving (yes planes DO have throttle controls) the thing is going to
takeoff with no air moving over the wings? NO WAY.

Assuming you're a pilot, I don't understand why you think no air would be
moving over the wings, but I'll give this one good "college try"...


Yes I am a pilot.


First, the question posed in the link by the OP of this thread is an
incorrect variation of the original. The original problem asks: "A plane
is standing on a giant treadmill. The plane moves in one direction, while
the treadmill moves in the opposite direction and at the same speed as the
plane. Can the plane take off?"

As has been explained, placing a car on the question's treadmill would
result in a stationary vehicle relative to the observer standing beside
the treadmill. The reason is the car derives its propulsion through the
wheels sitting on the treadmill and the speed of the car is measured by
how fast the wheels are turning. The faster the wheels turn, the "faster"
the car moves. However, this is only relative to the treadmill belt. To
the observer standing beside the treadmill, the car is motionless. If the
driver placed his hand out the window, he would feel no wind even though
his "speed" as indicated by the speedometer may be 100 miles per hour.

This is very similar to your example of running on the treadmill. You did
not feel a relative wind in your face because you were stationary relative
to the observer standing beside the treadmill. The reason you were
stationary is you generate your propulsion by moving your feet against the
ground (or belt, in this case) and the belt is moving in the opposite
direction and same speed of your "travel". Like the car, your speed is
measured by how fast your feet move from front to rear and they match the
speed of the belt to cancel out each other.

Now, replace the car and runner with an airplane. The airplane derives its
propulsion from its engine pushing air from front to back. None of this
energy is sent to the wheels to propel the airplane. The speed of the
airplane is measured by the flow of air past the airplane, not the turning
of its wheels. As the airplane's engine spools up to takeoff power, air is
forced from front to rear and the plane moves forward regardless how fast
its wheels are turning. The observer standing beside the treadmill would
notice the treadmill speed up, the airplane's wheels turn twice as fast as
normal, and the airplane move forward (not stationary).

Speed is relative and the key here is the means of propulsion. The
airplane's speed is measured by how fast the air is moving past it, not by
how fast its wheels are turning or how fast the ground is flashing by.
None of the airplane engine's energy is transmitted to the wheels to
generate speed. All of the airplane's propulsion is derived from moving
air (otherwise it would never stay in the air after takeoff). Since the
treadmill has very little effect on the air (and what little effect it
does have actually helps the airplane generate more lift), the airplane
will indeed takeoff in the same distance it normally would use without the
treadmill. However, the airplane wheels would be turning at twice their
normal speed at the time of takeoff.


Try this experiment:

Take a toy car and attach it to a string. Tie the other end of the string
to a small spring scale. Place the car on the treadmill belt and hold the
scale in front of the car while you turn on the treadmill. Observe nearly
zero (essentially 1G) force being exerted on the string/scale. Speed up
the treadmill (for simplicity, let's say you set it to a constant 10mph)
and you'll observe no significant difference in force exerted on the
string (the only additional force is the friction of the car's axles). Now
gently pull the string/scale forward. As long as you maintain a 1G force
on the string, the car will continue to accelerate.

Now, to the observer standing beside the treadmill, was the car stationary
or moving forward? It's speed was certainly not zero as the car most
definitely moved from rear to front of the belt. What was the speed of the
car relative to the "driver" sitting inside the toy? The wheels would be
turning faster than 10mph. If the "driver" were to put his hand out the
window, how fast would the air be moving? Much slower than his wheels
would say he's moving, but faster than the driver I mentioned at the
beginning of this post.

Replace the toy with the mythical airplane above, replace your arm with
the airplane's engine (and propeller, if appropriate), then replace the
string with the airplane engine mounts. You should now be able to
visualize why the airplane sitting on that giant treadmill would most
definitely takeoff.

If not, I wish you good luck and safe flight. You'll need it.

--
John T


Thank you for your reply. Here is my .02, it would seem that the plane never
actually moves in respect to the observer no matter how fast the treadmill
moves, the plane will just take off like it is hovering and then slowly
accelerate away?

I guess I'll have to set this up and try it, I do have a few RC planes
laying around and I have a treadmill so I guess I'll know one way or
another, unless Mythbusters beats me to the punch.

-------------------------------------------------------
DW

"Darkwing" theducksmailATyahoo.com wrote in message
...

"John T" wrote in message
...
"Darkwing" theducksmail"AT"yahoo.com wrote in message


First, the question posed in the link by the OP of this thread is an
incorrect variation of the original. The original problem asks: "A plane
is standing on a giant treadmill. The plane moves in one direction, while
the treadmill moves in the opposite direction and at the same speed as
the plane. Can the plane take off?"

As has been explained, placing a car on the question's treadmill would
result in a stationary vehicle relative to the observer standing beside
the treadmill. The reason is the car derives its propulsion through the
wheels sitting on the treadmill and the speed of the car is measured by
how fast the wheels are turning. The faster the wheels turn, the "faster"
the car moves. However, this is only relative to the treadmill belt. To
the observer standing beside the treadmill, the car is motionless. If the
driver placed his hand out the window, he would feel no wind even though
his "speed" as indicated by the speedometer may be 100 miles per hour.

Hmm. That presumes that "at the same speed as the plane" means "as fast as
necessary to cancel the forward motion." If you take your car analogy and
apply it to the plane, then the treadmill must try to run backwards as fast
as necessary to cancel forward motion - which is, Ah, let's just say
difficult.

To be consistent with your conclusions about the plane's motion, then the
car would also move. Using the object's motion as the defining parameter to
determine the treadmill speed, then a stable state can be reached with
either
1) a plane with forward motion X, treadmill with motion -X, wheels
spinning at 2X, thrust applied to achieve speed X
2) a car with forward motion X, treadmill with motion -X, wheels
spinning at 2X, thrust applied to achieve speed 2X

Accelerate either the plane or the car with X from 0 to, say, 65. The plane
will take off. The car will drive off the end of the treadmill.

John T


Thank you for your reply. Here is my .02, it would seem that the plane
never actually moves in respect to the observer no matter how fast the
treadmill moves, the plane will just take off like it is hovering and then
slowly accelerate away?

I guess I'll have to set this up and try it, I do have a few RC planes
laying around and I have a treadmill so I guess I'll know one way or
another, unless Mythbusters beats me to the punch.

-------------------------------------------------------
DW


DW,

None of the people that believe the plane will fly say that it will fly with
no forward motion. The claim is that the plane will accelerate to flying
speed in spite of the treadmill moving in the opposite direction.

Here is my .02, it would seem that the plane never
actually moves in respect to the observer no matter how fast the treadmill
moves, the plane will just take off like it is hovering and then slowly
accelerate away?

How about we make life easy for the moment and forget about the
treadmill. The airplane is magnetically suspended over the runway. No
part of the plane is touching the runway at all. It's all done with
electromagnets. (and for the nitpickers, let's ignore linear induction
motor effects).

The (otherwise ordinary) plane is magnetically suspended over the runway
with nothing touching the runway at all. Can the plane take off? How
does it do so?

Jose
--
"There are 3 secrets to the perfect landing. Unfortunately, nobody knows
what they are." - (mike).
for Email, make the obvious change in the address.

("Jose" wrote)
The (otherwise ordinary) plane is magnetically suspended over the runway
with nothing touching the runway at all. Can the plane take off?


Yes. But, ...Lord only knows what direction he's heading. g


Montblack

"Montblack" wrote in message
...
("Jose" wrote)
The (otherwise ordinary) plane is magnetically suspended over the runway with
nothing touching the runway at all. Can the plane take off?


Yes. But, ...Lord only knows what direction he's heading. g

The thrusters will take care of that, until the inertial dampeners take effect.
g
--
Jim in NC

("Darkwing" wrote)
Thank you for your reply. Here is my .02, it would seem that the plane
never actually moves in respect to the observer no matter how fast the
treadmill moves, the plane will just take off like it is hovering and then
slowly accelerate away?


Not unless the plane's "wheels" are coupled to the shaft of a gyro's rotor.

Try this one:

You're in a Class B airport terminal.
You're on roller-skates, Rollerblades, a skateboard... whatever.

You find yourself on an (evil) moving sidewalk - facing the wrong way.
The (evil) sidewalk ALWAYS matches your wheels' forward speed.

Someone moves a huge Hollywood 'film set' fan, in a few feet behind you.
They point the fan at your back and turn it on.

You hold open your jacket to make a sail (...like kids at the ice skating
rink have done for ages)

1. Will you get blown down to the far end of the moving sidewalk - your
destination?

2. Will you remain in the same spot - relative to the wall - no matter how
hard the giant fan blows?

3. Forgetting the fan, if you try pulling yourself forward using the
stationary handrails, will you in fact move forward? Or will the (evil)
moving sidewalk thwart your forward motion by speeding up? Or will your
upper body pull itself forward, while your feet remain behind ...(or
stationary, relative to the wall and the handrail)?

4. How is this the same as the airplane and the treadmill question? How is
it different?


Montblack

"Montblack" wrote in message
...
("Darkwing" wrote)
Thank you for your reply. Here is my .02, it would seem that the plane
never actually moves in respect to the observer no matter how fast the
treadmill moves, the plane will just take off like it is hovering and
then
slowly accelerate away?


Not unless the plane's "wheels" are coupled to the shaft of a gyro's
rotor.

Try this one:

You're in a Class B airport terminal.
You're on roller-skates, Rollerblades, a skateboard... whatever.

You find yourself on an (evil) moving sidewalk - facing the wrong way.
The (evil) sidewalk ALWAYS matches your wheels' forward speed.

Someone moves a huge Hollywood 'film set' fan, in a few feet behind you.
They point the fan at your back and turn it on.

You hold open your jacket to make a sail (...like kids at the ice skating
rink have done for ages)

1. Will you get blown down to the far end of the moving sidewalk - your
destination?

2. Will you remain in the same spot - relative to the wall - no matter how
hard the giant fan blows?

3. Forgetting the fan, if you try pulling yourself forward using the
stationary handrails, will you in fact move forward? Or will the (evil)
moving sidewalk thwart your forward motion by speeding up? Or will your
upper body pull itself forward, while your feet remain behind ...(or
stationary, relative to the wall and the handrail)?

4. How is this the same as the airplane and the treadmill question? How is
it different?


Montblack


It's basically the same question with the same ambiguities. The crux of most
of the hilarious debate is really over what defines the speed of the
treadmill. It seems like a more interesting puzzle if the treadmill (or evil
moving sidewalk) tries to match the forward speed of the object on the
wheels resulting in the wheels simply spinnning at twice the speed of the
forward movement of the object.

The other interpretation, which leads to an impossible solution, is that the
treadmill moves to counteract all forward motion - which results in a
treadmill accelerating to infinite speed (or until the wheels explode which
ever comes first).
--
-------------------------------
Travis
Lake N3094P
PWK

On Tue, 12 Dec 2006 21:23:56 -0800, Darkwing wrote
(in article ):


"John T" wrote in message
...
"Darkwing" theducksmail"AT"yahoo.com wrote in message


This has NOT been adequately explained or there
would be no question about it. If the plane is not moving on the
treadmill but rather keeping up with the speed that the treadmill is
moving (yes planes DO have throttle controls) the thing is going to
takeoff with no air moving over the wings? NO WAY.

Assuming you're a pilot, I don't understand why you think no air would be
moving over the wings, but I'll give this one good "college try"...


Yes I am a pilot.


First, the question posed in the link by the OP of this thread is an
incorrect variation of the original. The original problem asks: "A plane
is standing on a giant treadmill. The plane moves in one direction, while
the treadmill moves in the opposite direction and at the same speed as the
plane. Can the plane take off?"

As has been explained, placing a car on the question's treadmill would
result in a stationary vehicle relative to the observer standing beside
the treadmill. The reason is the car derives its propulsion through the
wheels sitting on the treadmill and the speed of the car is measured by
how fast the wheels are turning. The faster the wheels turn, the "faster"
the car moves. However, this is only relative to the treadmill belt. To
the observer standing beside the treadmill, the car is motionless. If the
driver placed his hand out the window, he would feel no wind even though
his "speed" as indicated by the speedometer may be 100 miles per hour.

This is very similar to your example of running on the treadmill. You did
not feel a relative wind in your face because you were stationary relative
to the observer standing beside the treadmill. The reason you were
stationary is you generate your propulsion by moving your feet against the
ground (or belt, in this case) and the belt is moving in the opposite
direction and same speed of your "travel". Like the car, your speed is
measured by how fast your feet move from front to rear and they match the
speed of the belt to cancel out each other.

Now, replace the car and runner with an airplane. The airplane derives its
propulsion from its engine pushing air from front to back. None of this
energy is sent to the wheels to propel the airplane. The speed of the
airplane is measured by the flow of air past the airplane, not the turning
of its wheels. As the airplane's engine spools up to takeoff power, air is
forced from front to rear and the plane moves forward regardless how fast
its wheels are turning. The observer standing beside the treadmill would
notice the treadmill speed up, the airplane's wheels turn twice as fast as
normal, and the airplane move forward (not stationary).

Speed is relative and the key here is the means of propulsion. The
airplane's speed is measured by how fast the air is moving past it, not by
how fast its wheels are turning or how fast the ground is flashing by.
None of the airplane engine's energy is transmitted to the wheels to
generate speed. All of the airplane's propulsion is derived from moving
air (otherwise it would never stay in the air after takeoff). Since the
treadmill has very little effect on the air (and what little effect it
does have actually helps the airplane generate more lift), the airplane
will indeed takeoff in the same distance it normally would use without the
treadmill. However, the airplane wheels would be turning at twice their
normal speed at the time of takeoff.


Try this experiment:

Take a toy car and attach it to a string. Tie the other end of the string
to a small spring scale. Place the car on the treadmill belt and hold the
scale in front of the car while you turn on the treadmill. Observe nearly
zero (essentially 1G) force being exerted on the string/scale. Speed up
the treadmill (for simplicity, let's say you set it to a constant 10mph)
and you'll observe no significant difference in force exerted on the
string (the only additional force is the friction of the car's axles). Now
gently pull the string/scale forward. As long as you maintain a 1G force
on the string, the car will continue to accelerate.

Now, to the observer standing beside the treadmill, was the car stationary
or moving forward? It's speed was certainly not zero as the car most
definitely moved from rear to front of the belt. What was the speed of the
car relative to the "driver" sitting inside the toy? The wheels would be
turning faster than 10mph. If the "driver" were to put his hand out the
window, how fast would the air be moving? Much slower than his wheels
would say he's moving, but faster than the driver I mentioned at the
beginning of this post.

Replace the toy with the mythical airplane above, replace your arm with
the airplane's engine (and propeller, if appropriate), then replace the
string with the airplane engine mounts. You should now be able to
visualize why the airplane sitting on that giant treadmill would most
definitely takeoff.

If not, I wish you good luck and safe flight. You'll need it.

--
John T


Thank you for your reply. Here is my .02, it would seem that the plane never
actually moves in respect to the observer no matter how fast the treadmill
moves, the plane will just take off like it is hovering and then slowly
accelerate away?

I guess I'll have to set this up and try it, I do have a few RC planes
laying around and I have a treadmill so I guess I'll know one way or
another, unless Mythbusters beats me to the punch.

-------------------------------------------------------
DW



You assumption is that the plane never moves relative to an observer. In
fact, the airplane will accelerate normally and run down the treadmill and
take off normally no matter how fast the treadmill is moving. The only thing
that would stop it is the wheels coming off. The treadmill cannot keep the
airplane from accelerating in this way. An observer standing next to the
treadmill will see the airplane moving down the treadmill and taking off,
just like it would from a normal runway.

Airplane engines cannot feel the wheels. They do not turn the wheels. So the
wheels will automatically spin fast enough to keep up with the accelerating
aircraft. This is easily demonstrated. All you need is a pair of roller
blades, a rope, and a treadmill. You stand on your roller blades on the
treadmill with a rope attached to the front. Measure the force needed to pull
yourself to the front of the treadmill with the treadmill off. Then try it at
different speed settings. It always requires about the same amount of force
to pull yourself forward. The only that changes is that your wheels spin
faster to compensate. The treadmill could be moving many times faster than
the airplane; it does not matter. The airplane will move down the runway and
take off normally.

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

The False Pythagorean theorem is similar; it postulates that the shortest
distance between two points is always a right angle, or in other words, the
hypotenuse of a right triangle is equal to the sum of the other two sides. It
is false on the face of it; we can see that this is obviously not true, but
nevertheless you can make a powerful argument that it is. If you have a right
triangle ABC where the hypotenuse is AC, you can measure the sum of AB and
BC. If you turn the hypotenuse into a series of steps, the rise of the steps
will always equal BC and the run of the steps will always equal AB. No matter
how large or small you make the steps, the rise will equal BC and the run
will equal AB. A straight line hypotenuse and be seen to be simply a series
of infinitely small steps; the sum of the rise of the infinite steps must be
BC and the sum of the run of the infinite steps must be AB. Therefore AC must
equal AB plus BC. Bzzzt. The Greeks could not solve that puzzle, either,
until Pythagoras was able to prove that it is the square of the hypotenuse
that equals the sum of the squares of the other two sides. But even then a
lot of people did not believe him.

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. None
of the thrust of the engines on an airplane is being used to overcome the
force of the treadmill because the wheels spin freely. It would be different
with an automobile. There the motor has to overcome the force of the
treadmill. But the only thing resisting the airplane is air, which remains
constant no matter how fast the treadmill is moving.

Think of it like this: raise the airplane a few inches off the treadmill, put
the engines to full power, and for grins leave the gear down and add a little
motor to keep the wheels spinning at the same speed as the treadmill. Have
the wheels accelerate at the same speed as they would on a normal takeoff,
and have the treadmill match speed with the wheels. Now the airplane is not
even touching the treadmill. Do you believe that the airplane will still
remain stationary with respect to an outside observer? Yet the treadmill's
relevance to the speed of the speed of the wheels is just the same as it was
before when the airplane actually sat on the treadmill. And the treadmill is
having exactly the same effect on the airplane's forward motion as it did
when the airplane was touching the treadmill, which is to say none.

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 plain 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.

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 plain 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. 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.

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.

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. 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. 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?

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