Can the Jet Take Off from a Conveyor Belt? Yes and No

The above physics problem has caused a lot of controversy and heated debates. You might wonder why. After all, didn't Myth Busters settle the question? Here's their video clip:

The above video certainly nearly settles the following question: "Can a plane take off from a conveyor belt if the plane and conveyor belt are going equal and opposite speeds?" Well, not if both the plane and belt speeds equal zero. LOL! Obviously the answer is also no if the plane taxis below the minimum take-off speed. In any case, the Myth Buster question is not the one before the court. The question is, "Can the 747 take off if the belt is designed to exactly match the speed of the wheels?" This question is subtly different from the Myth Buster question. It focuses on the wheel speed rather than the plane speed. As we shall see below, focusing on the wheel speed can change your answer to "Will the Jet Fly?"

To solve the problem we should first examine why the plane flies in the Myth Buster video. Below is a free body diagram:

The plane flies because the thrust force (or propeller force) is greater than the friction forces. But what happens if the thrust force exactly equals the friction forces? The plane stays on the conveyor belt. Suppose the friction forces are greater than the thrust force? The plane slides backwards. The video below provides an excellent demonstration:

Now, with all the above in mind, let's examine the question again: "Can the 747 take off if the belt is designed to exactly match the speed of the wheels?" To answer the question we should think of a scenario where the wheel speed matches the conveyor belt speed. How about when the thrust force equals the friction forces? That would keep the jet in the same spot on the conveyor belt. We can imagine the belt driving the wheels (like gears) and their respective velocities matching. Below is the relevant math:

At equation 3 we have the gear formula that shows if the conveyor belt turns at a velocity v, the wheels also turn at velocity v. At equation 4 we see that these equal and opposite velocities net zero. This corresponds and is consistent with equation 5 where the net force is zero. So we can conclude that when the thrust force is equal to the friction forces, The wheels of the jet have the same speed or velocity as the conveyor belt. The jet obviously fails to take off under this condition. Note the jet is staying in place. It's displacement velocity is zero or less than the belt velocity. The jet can only take off when the wheel speed exceeds the belt speed; i.e., when the thrust force overcomes the friction forces--when its displacement velocity is greater than zero.

So, can the 747 take off if the belt is designed to exactly match the speed of the wheels? No.

Update: If Myth Busters had done their experiment with a car, the car's speedometer would have shown the car going twice as fast as the conveyor belt. Car speed is measured from the rotation of the wheels. The plane, by contrast, measures its speed with air speed. So even though the plane's wheel speed is twice that of the belt's, the pilot thinks the plane's speed exactly matches the belt's . Given the fact the wheel speed is twice as fast, it is no surprise the plane takes off.

Update: How about a little Einstein's equivalence principle? Imagine the jet on the conveyor belt with its engines shut off. The conveyor belt accelerates until it overcomes any friction. The jet stays where is, its wheels spinning at the same rate as the conveyor belt. It kind of reminds me of the old tablecloth trick, where you yank the table cloth from under the dishes.