1. Airplane on Treadmill

Problem:

Imagine an airplane on a treadmill. The treadmill’s speed is made to exactly match the lift-off speed of the airplane, and the treadmill is oriented such the airplane (moving at lift-off speed) would appear stationary with respect to the ground. Will the airplane be able to lift off?

Solution:

The airplane lifts off.

Background:

The problem that I am addressing today is well studied. Every few months someone posts the story on an internet forum or website, and people argue their perspectives on the issue. The problem grew to become so popular that it was featured on MythBusters [video]. The result is already known to many of you, but the reason why this result occurs is mostly unexplained. Most “explanations” of this problem stray too far from physics and instead rely on simplified intuitional models. I hope to present a solution which is rigorous in its physical derivation, but simple enough for the layman to understand.

The null hypothesis (that the airplane will not lift-off) is based on basic aerodynamic theory. If there is no airflow over the airplane’s wings then it cannot produce any lift. A stationary airplane will have no airflow, and, therefore, will not produce lift. All of this is true. The problem with the null hypothesis is the assumption that the airplane will remain stationary. Using basic physics, I will demonstrate that the airplane must necessarily accelerate forward.

Proof:

Before beginning the proof, a basic understanding of Newton’s laws of motion is required.

Newton’s 2^nd Law is that when an object is acted upon by a net force, the object will move with an acceleration which is related to the mass of the object and the magnitude of the force.

Newton’s 3^rd Law is that every force causes an equal and opposite force to occur. A good example of this is jumping. You push against the ground with your legs; the ground pushes you back which causes you to move upwards.

It would be illustrative to first talk about a car moving on a treadmill like the one mentioned in this problem. We will consider the system (the object that we are solving the equations of physics for) to be the car and the treadmill together. This is counterintuitive (the natural assumption is to make the car be the object), but it is an important and useful assumption for this problem. Let’s take a look at the forces involved in the problem. For the sake of this analysis, we need only look at forces in the horizontal direction.

The car moves by turning its wheels. The motion of the wheels creates a force on the treadmill (friction). By Newton’s 3^rd, the treadmill creates an opposing force on the car. The forces acting on the car and the treadmill are equal and opposite, so they will always cancel each other out. There is no net force. By Newton’s 2^nd, if there is no net force then there is no acceleration. The car remains stationary.

Now, solve the same system for the airplane. The airplane/treadmill is a more complex system than the car/treadmill because it has thrust in addition to frictional forces.

Just like the car, the friction generated by the airplane will cancel with the friction generated by the treadmill. The difference is the existence of the extra force, thrust. Apply Newton’s 2^nd. There is a net force (the thrust) and therefore there will be some net acceleration. The airplane will move! This video by neodocneodoc demonstrates my point.

Model Limitations:

There are practical limits to the amount of thrust an airplane’s engine can output. The simplified model above assumes that the airplane’s engine can produce enough thrust to accelerate to the speed needed by the airplane to lift off from the treadmill. In reality, there is a certain upper limit speed that the airplane can reach which depends on its engine type. This upper limit speed may not be enough to take off from a very fast moving treadmill.

Aerodynamic drag was ignored in this analysis because it is usually small compared to the other forces. This is a legitimate assumption for most situations, but if you drive a supersonic car then you will need to include drag effects. (^_^)

Next time: How airplanes produce lift.