From the “den of topics that have been written about to death, yet are still interesting to the decreasingly small percentage of the world that haven’t heard about the topic still, but will soon“^TM comes the Airplane on a Treadmill Problem.

The problem as listed is more or less:

Imagine an airplane is sitting on a conveyor belt, as wide and long as a normal runway. The conveyor belt is designed to exactly match the forward speed of the wheels, except the conveyor belt moves moving in the opposite direction. Can the plane take off?

First some basic principles of flight, lift, and thrust:

• A plane takes off when it has sufficient “lift”.
• “Lift” is created by the speed that air passes under and over a properly angled wing.
• Lift occurs when the plane achieves a certain thrust relative to the air.
• Thrust comes from the engines of the plane.
• Lift and thrust does not depend on the speed of the plane relative to the runway.
• Lift and thrust also does not depend on the speed of which the wheels are turning.
• It is true that a sufficiently strong gust of wind could cause an airplane to achieve lift, and as a result, “fly”. Albeit in an uncontrolled manner.

And now some basic principles of how planes fly:

• Planes do not have motors for their wheels. This is why they are tugged around by those weird machines across runways. Thrust does not come from the plane’s wheels.
• Planes move via their engine thrust, which push backwards on the air. Thanks to Good ol’ Newton and his Third Law of Motion, this will result in an equal and opposite force (opposite being forward) which sends the plane flying in the proper direction.
• Jet engines do not traditionally fly by pushing air downwards, with the exception of vertical takeoff. Airplanes achieve upward speed via lift.
• Also, Jet engines do not produce wind themselves, but rather use the prexisting air around them. For the most part, all engines do is push the plane forward.
• The speed of the the runway (in treadmill form or not) and the speed of the air are independent. So even if the runway is moving really fast (in treadmill form), it will not affect the wind-based mechanics of an airplane.

The trick in the problem is that the method that the conveyor belt is being used to “exactly match the forward speed of the wheels” is not defined:

1. The treadmill could be moving backwards at the same speed the wheels of the plane are moving forward. This would be true of a treadmill with no motor.
2. The treadmill could be moving backward at the same speed as the plane is moving forward relative to the ground. (If the plane moves 5mph forward relative to the ground, the treadmill is set to move 5mph backward.)
3. The treadmill could be moving backward at the same speed as the plane is moving forward relative to the treadmill. (If the plane moves 5mph forward relative to the ground, the treadmill is set to move 5mph backward, which would create a plane moving 10mph relative to the treadmill.)

Treadmill with No Motor

The way this works is as the plane accelerates, it moves forward at an increasing speed. The treadmill, pushed by the thrust, moves backwards at an increasing speed. The plane takes off fine. As we noted before, the airplane does not move via it’s wheels, therefore the plane will still achieve adequate thrust from the power of the engines. Those things are producing a quarter of a million pounds of thrust all time, regardless of whether the plane is moving forward or not.

Treadmill Moves the Opposite of the Plane’s Speed Relative to the Ground

The plane still takes off fine. Again, the airplane is not moving via the wheels. The additional backwards motion of the treadmill will make the wheels spin twice as fast, but the engines will still achieve thrust.

Treadmill Moves the Opposite of the Plane’s Speed Relative to the Treadmill

This is where things start to blow up. This interpretation is the only interpretation that would create for a perfectly stationary plane. Except there’s a big problem. Assume the airplane accelerates to an arbitrary five kilometers per hour forward.

The treadmill then reacts and starts moving five kilometers per hour backward. This means that the speed of the airplane relative to the treadmill, or the difference between the two speeds, is ten kilometers per hour. Thus the treadmill reacts and starts moving ten kilometers per hour backwards. Of course this means the speed of the airplane relative to the treadmill is now 15km/hr. This makes for an infinite cycle.

This also means the wheels blow off. Which means the plane falls down. Which means it’s not going anywhere. …Except backward off the treadmill.

So…

If you subscribe to interpretation #1 or #2, the plane can take off. If you subscribe to interpretation #3, it does not. This is where the argument comes from. Both answers are technically correct.

Except, of course, when you take this into reality with a real treadmill, because then all the millions of other factors (axle friction, ground friction, air resistance, wind, imperfections in matching treadmill speed, etc.) come in and cause havoc.

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