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Plane on a Treadmill

OK, so it may not be as exciting as Snakes On A Plane, but here's the problem statement:

An airplane is positioned on a giant treadmill as long as a runway. The treadmill moves in the opposite direction and same speed as the airplane. Will the plane takeoff?

I can forgive the general public for not getting the correct answer. I just don't understand why pilots don't get it right.

Oh, the correct answer is the plane does take off in the same distance it would use without the treadmill. Why? Because the plane's wheels are free spinning. The airplane gets its motion from moving air - not ground - from front to back (using a propeller or turbine). It's the reaction of pushing against air - not the ground - that moves an airplane, so the treadmill can move as fast as it wants and the airplane will still move forward as it normally would.

I'm really surprised at the number of US Government-certified pilots who swear the airplane would remain stationary to an observer standing beside the treadmill.

Here's a short video clip of an experiment demonstrating the concept (using a skateboard in place of an airplane):
http://videos.streetfire.net/player.aspx?fileid=35E964D9-38DB-4EFD-BE8D-D6BA1A43A06B

Published Wednesday, December 13, 2006 8:51 PM by jtabor
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Comments

# re: Plane on a Treadmill

The problem is not so much in the failure of us pilots -- the problem lies entirely in the question.  How one interprets an ambiguity in the question leads inevitably to two different answers.

You are, of course, entirely correct that the airplane moves because of the thrust reaction with the air, not the ground, so the freely moving wheels or treadmill have no effect on its ability to move forward and thus takeoff.

But the question says that the treadmill will match the speed of the airplane.  What does that mean?

Some will interpret it to mean that the speed of the treadmill's surface will be equal to the ground speed of the plane.  If you make that interpretation, you get the answer you describe above, for the reasons you give above.

Some, however, will interpret "same speed as the airplane" to mean that the treadmill will be capable of assuming a speed that keeps the plane stationary.  No treadmill could do that, of course, and there's minimal friction with which to transfer that force (just the wheel bearing friction).  But the question doesn't let you consider whether it's possible -- it provides AS A GIVEN that the treadmill has this capability.  If you interpret that given as the capability to keep the plane stationary, then no relative wind, no lift, and no takeoff.

The skateboard video is a good example.  The paper could be pulled backward fast enough that the minimal force imparted through friction of the bearings would cumulate and equal the forward force of the fan.

B

Tuesday, January 16, 2007 7:30 PM by Brian

# re: Plane on a Treadmill

"Opposite direction and same speed" means just that. :)

There are two speeds the treadmill can match: Groundspeed and airspeed. You've already admitted matching the groundspeed of the airplane won't keep it from lifting off, but even matching the airspeed wouldn't keep it down. The wheels would simply turn faster.

The only way the treadmill can impart *any* energy to the airplane is through the wheels. If the wheels are free spinning, how can enough energy be transferred to the airplane to counter its forward thrust?

Tuesday, January 16, 2007 7:58 PM by jtabor

# re: Plane on a Treadmill

No, there is a third speed.  It's going to be really fast and theoretical only, but it's there.

And it's because "free spinning" isn't really free spinning.  As my physics prof used to say, friction always extracts its share.  :)

There is always some internal friction in the plane's wheel bearings.  Have you ever rolled to a stop without touching the brakes?  That's the internal friction of the wheel bearings.

For each revolution of the wheels, some amount of friction force is exerted in the opposite direction.  Let's call it "F."  There is some multiple of F that will equal the plane's thrust.  (xF=T)  When the treadmill moves fast enough that the wheel is going x, then the friction inside the wheel bearings will equal the plane's thrust.

This is oversimplifying the effect of friction.  (Friction forces change with speed, e.g., it takes more thrust to start a taxi than to continue it.)  And I am confident that the wheel hubs would melt long before xF=T, but it is theoretically possible.

And to my main point, because some (mis?)interpret this as a given, in their minds, the plane never moves.

Personally, I read the question the same as you.  The speed of the treadmill matches the "wheel speed" -- not some theoretical speed necessary to keep the plane stationary.  (If non-movement is a given of the question, it's not much of a question if you ask me!)  But I believe that this misunderstanding is at the root of all the internet "ink" that has been spilled on this topic.

Tuesday, January 16, 2007 8:42 PM by Brian

# re: Plane on a Treadmill

The question I posed specifically avoided wheel speed for this reason.

If the treadmill matched wheel speed as the airplane began moving forward, I'll grant the treadmill would attempt an infinite speed - which common sense tells us is well beyond the capability of any of the components of the wheels.

The airplane in my scenario would still take off.  :)

Wednesday, January 17, 2007 8:10 AM by jtabor
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