# Airplanes, conveyor belts, mythbusters

## by PhilipJ on 28 January 2008

After the airplane and conveyor belt internet hysteria, it looks like the guys on Mythbusters will do the experiment. January 30th, 9pm.

For anyone who missed it the first time around, the problem is such:

“A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?”

**Update** — physics is saved! (the plane takes off.)

Exciting. Place your bets now!

What gets planes airborne is airspeed, as lift relies mostly on downwards airflow (Newton’s law of action and reaction) and to a lesser extent the Venturi effect on the airfoil cross-section. If the plane is standing still with respect to the air neither will operate, although if the plane moves through the air, then irrespective of the conveyer, it will take off.

My first thought was like John’s, but I now believe he is wrong. The conveyor belt can’t change the plane’s speed relative to the air or ground. Unlike a runner on a treadmill, or a car on the road, the thrust for a plane comes from its jet engines, not it’s wheels. The plane is pushing against the air, not the road, and that is what accelerates it. The conveyor belt can make the wheels rotate faster, but the effect on the plane’s speed relative to the air will be small.

Imagine pushing a rolling pin against a conveyor belt. The speed of the belt might make the pin rotate faster, but it’s speed relative to the ground —- and air —- only depends on how hard you push —- i.e., the thrust.

I say the plane takes off.

Sorry, John. I didn’t read all the way to the end of your comment. I guess you covered both possibilities, so you can’t be wrong ,,,

no way the plane takes off.

Of course the plane takes off. The angular velocity of the wheels will be twice that of a normal take off, but otherwise everything is the same.

I’m not bragging, but my I.Q. fortunately surpasses 9 points.

Anyone who knows anything about physics should understand the concept of lift.

(it’s what makes planes fly).

I was blown away by two “so-called-myths” recently.

Ordinarily a great show, they actually tested the tread-mill-plane “myth” and the freeze your tongue to a pole “myth”.

I’m from the North and freezing your tongue to a sub-zero pole has absolutely been proven.

If you are from San Diego, and have no concept of physics, enjoy the outcome of these “myths”.

I still tip my hat to Adam and Jamie for shutting up you virginal/semi-retarded science geeks.

Jeff, thanks for freezing your tongue to a pole for science.

In case it wasn’t clear from Jeff’s helpful comment, the plane takes off! The semi-retarded science geeks were correct.

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Remember that the people who get this wrong are still qualified to vote for the next President of the United States.

It was especially worrisome that the pilot they hired didn’t think he would be able to take off. You would hope that pilots at least would understand how planes work.

This myth was tested improperly in my book.

If the conveyer belt goes backwards at 25 MPH and the plane on the conveyer belt speeds up to 50 MPH, the planes relative speed to the air around it is still 25 MPH which is all that matters… This is clear as the plane still passed the road cones while it was on the ground, meaning the plane was traveling forward faster than the treadmill was going backwards…

The real myth should have been if the conveyer belt moved back at the exact same speed the plane was moving forward (at all times), would it take off?

That was sort of the whole point. A conveyor cannot keep the plane from accelerating. It can make the wheels spin faster, but the engine and propeller are acting against the air, not the ground. That’s the only interesting thing about this myth. The way it is worded makes you believe an impossible premise, that is that the conveyor can keep the plane stationary with respect to the ground and air.

[Sorry— the thrust doesn’t actually come from “pressing against the atmosphere”, it results only from conservation of momentum, in this case the air with mass m_air pushed out by the engine with velocity v_air such that

m_air X v_air = P = m_airplane X v_airplane, with dP/dt = F. This doesn’t actually change the result I believe]

An airplane with mass M sits on a conveyor belt. The airplane’s jet engine is turned OFF. The conveyor belt starts moving backward at a constant velocity V_conveyor. (note that all velocities will be quoted relative to the ground referential)

Which direction does the plane go? Backward, with velocity V_conveyor.

Intuitively it is reasonable to expect that the plane will have to overcome this backward velocity before taking off.

At time T=0, the jet engine is turned on, generating a thrust, F by pressing against (we all agree) the atmosphere. As a result, the plane experiences a forward acceleration A = F/M. Note that the conveyor belt provides no acceleration, just constant velocity. [As an aside, also note that acceleration is independent of any fixed frame of reference].

We are allowed to decompose the velocity of the plane relative to ground (V_total) into the velocity component due to the conveyor belt (V_conveyor) and the velocity resulting from thrust acceleration (V_plane):

V_total = V_plane – V_conveyor

As velocity is the integral of acceleration, at the instant in time T=0 its (forward) velocity component due to the thrust force is still V_plane = 0: the plane is still moving backward relative to earth with velocity V_conveyor – V_plane = V_conveyor.

Time, T, continues to pass. Thrust F guarantees constant acceleration A=F/M, and the equation for the velocity of the plane due to thrust is V_plane=TA.

Thus V_total = TA – V_conveyor. The plane’s ground speed V_total will be zero only when

T_null = V_conveyor/A

Until this time the plane will continue to move backward. As more time passes, the plane’s forward velocity component due to thrust will continue to increase linearly with time, whereas the backward velocity component due to the conveyor belt motion will remain constant, and so the plane will progressively pick up forward velocity (V_total = TA – V_conveyor > 0 ), develop lift and take off.

Thus if the conveyor belt has constant velocity the plane will, of course, teventually take off. If the conveyor belt’s velocity increases linearly with time at a rate A, then the plane will not develop forward velocity and will not take off.

Tom, I’m afraid you just don’t seem to understand what role the wheels on a plane play. It has been described in perfect detail above (see the 13th comment from Ben why you are wrong), and the plane takes off.