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

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.