Okay, let’s do this. The web tubes have been buzzing about this little brain teaser for years. The even did a bit about it on Mythbusters. If you put an airplane on a treadmill, can it take off? Well, it turns out there are several different answers depending on precisely how the question is worded. But, IMHO, the vast majority of these answers have missed the point entirely.
You might think it’s a puzzle designed to see if the student knows how airplanes work. Something like this:
TEACHER: Imagine an airplane which accelerates down a mile-long runway until it reaches takeoff speed, which is 100 mph. What would happen if you put the airplane on a treadmill instead of using the runway?
STUDENT: Is the treadmill capable of turning faster than 100 mph?
TEACHER: Yes. It can turn any speed you want.
STUDENT: If you turn on the airplane’s engines and spin up the treadmill at the same time, when it reaches 100 mph the plane should take off, right?
TEACHER: (smugly) Ah, you failed to realize that the speed of the wheels relative to the surface is totally irrelevant. Airplanes do not push against the ground with their wheels the way cars do. They create lift because of their wings rushing through the air. It doesn’t matter how fast the wheels are turning. If there’s no air flowing over the wings, there’s no lift, and the airplane can’t take off.
STUDENT: (humbly) oh great teacher, thank you for explaining the difference between how cars work and how airplanes work.
But then along comes a smart ass.
SMART ASS: But wait! If the treadmill is turning 100 mph and the plane’s wheels are also turning 100 mph and the plane is sitting still, why can’t the pilot simply throttle up the engines a bit more until the plane starts to move down the treadmill? And then when the plane’s wings are moving 100 mph relative to the air, the plane will take off! As you said, it doesn’t matter how fast the wheels are turning.
TEACHER: (flustered) Well, that won’t work because the treadmill operator can just speed up the treadmill at the same time the plane is throttling up its engines. The treadmill operator can prevent the plane from moving.
Did you notice that the question has changed? A minute ago, we wanted to know if it’s possible to make the airplane take off. Now, we are discussing whether it’s possible to prevent the airplane from taking off. Those are different questions! But our smart ass isn’t done yet…
SMART ASS: How can the treadmill prevent the plane from moving? The only force it can exert on the airplane is the rolling resistance of the wheels. This force is caused by friction and it does not increase as the speed increases! If the plane’s engines are capable of producing more thrust than the friction of the wheels (and they must, or else the plane was never capable of taxiing down the runway in the first place) then the airplane can move! And if it can move, it can accelerate. Once it reaches 100 mph relative to the air, it will take off, regardless of how fast the treadmill is turning at that time.
TEACHER: (utterly defeated) I bow to your superior intellect.
THE VOICE OF REASON: Excuse me for butting in, but how long would the treadmill need to be?
SMART ASS: The treadmill would need to be just as long as the runway was.
THE VOICE OF REASON: So, if we needed a 1 mile runway, now we need a 1 mile treadmill? How is that better?
SMART ASS: uh…
THE VOICE OF REASON: Why did we want to build the treadmill in the first place?
TEACHER: The runway was inconveniently long. We wanted to launch our airplane in a shorter distance, say from the roof of an apartment building. Let’s call it 150 feet.
THE VOICE OF REASON: Okay, if you take an airplane which needs a 1 mile runway to get up to 100 mph in order to take off, and you put that airplane onto a 150 foot treadmill which is capable of turning whatever speed you want, what will happen?
SMART ASS: The airplane will move down the treadmill, fall off the end and crash.
THE VOICE OF REASON: So, if your goal is to eliminate the runway, having a treadmill doesn’t help, does it?
TEACHER: (triumphant) Yes! That’s right! The treadmill is useless. If you want to launch the airplane in only 150 feet, you don’t need a treadmill; you need a catapult.
STUDENT: I think this explains why they don’t build aircraft carriers with treadmills on them.