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## A Plane on a Conveyor Belt

Because MythBusters was supposed to show the "plane on a conveyor belt" show last night, I stumbled onto the original question at Jason Kottke's blog. The question is as follows:

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?

It was interesting reading the comments to see people going back and forth, but the answer is really quite easy and simple: the plane takes off.

The solution and a very quick explanation below, if you want to read it…

The problem some people have is that they seem to believe the conveyor belt going backwards can counter the effect of the airplane pulling itself forward through the air. In other words, they believe the plane remains stationary.

The best way I can think to describe the result is simply to imagine the plane resting on a frictionless conveyor belt. The belt can move a billion miles an hour in any direction and the plane won't move (frictionless). Wheels aren't frictionless, but they're close. That's the whole purpose of putting wheels on a plane. The small amount of friction that wheels add (particularly when they've got fresh ball bearings and grease) are nowhere near enough to slow the plane's forward movement.

Some people are misled by earlier comments that imply that the engines or propellers would have to push air over the entire wing structure. They miss out on the fact that in this case, the propellers and engines act on the air, and pull the entire plane forward. That's what generates the airspeed - the plane moving against a motionless block of air, not air moving over a stationary wing.

Or try this: imagine sitting in a wagon on a conveyor belt moving backwards. You're holding onto a rope that's anchored to a wall in front of you. The conveyor belt starts spinning. You begin pulling on the rope. You move closer to the wall. ((Engines or props on a plane push against the air behind the plane rather than pulling forward, but since they're equal and opposite, the resulting force is in the same direction. Pulling forward is the same as pushing backwards.))

### 13 Responses to "A Plane on a Conveyor Belt"

1. The answer is above, and this comment has little to do with that answer:

This comment gets something else wrong, I believe:

In fact, since the conveyor belt would drag along a bit of air due to simple friction the plane would actually get into the air _quicker_ than if it was taking off from a normal runway. The conveyor moving in the opposite direction would drag a bit of air backwards with it, increasing the relative airspeed of the wing and increasing its lift .

Unfortunately, the conveyor belt would likely only speed up the air along the bottom of the wing, reducing lift (by a small amount). But whatever…

Oh, and the comment about aircraft carriers requiring the use of a catapult? Priceless! Planes have to be catapulted because they don't have a long enough runway to build up airspeed.

Anyway, wow. I can't believe how many people in the Kottke thread simply think the plane is stationary. To a stationary observer, the takeoff appears almost normal, except the tires will be spinning twice as fast as normal.

2. I don't understand why people say the plane takes off.

The problem states that the "conveyor tracks the plane speed", not the wheel speed. So as the propeller(s) are acting to pull the plane forward, the conveyor is matching that movement.

Doesn't that mean it appears to stay stationary. Even if the wheels are spinning twice as fast as normal?

3. I think most people think the plane stands still because of the way the question is worded. A casual reading makes it sound like the conveyor belt is rigged to make sure the plane doesn't move forward.

4. [quote comment="44884"]I think most people think the plane stands still because of the way the question is worded. A casual reading makes it sound like the conveyor belt is rigged to make sure the plane doesn't move forward.[/quote]

I don't think it's confusing at all. It says "the plane moves forward." It doesn't say "the conveyor moves fast enough to keep the plane stationary in relation to the ground." It simply says the plane moves forward.

If the plane's engines (or propellers) pull the plane through the air (and past the stationary ground) at 250 MPH, the conveyor belt moves backwards at 250 MPH. The wheels spin as fast as if the plane was rolling 500 MPH on a traditional runway.

Alistair: Put on a pair of skates and stand on a treadmill. Turn the treadmill on. Heck, crank the treadmill to 1000 miles per hour. Now turn on the jet pack you've strapped to your back. You'll go forward.

The force of friction on wheels is not nearly strong enough to counter-act the force of thrust from the engines/propellers.

5. Erik, isn't the idea that the conveyor adjusts its speed? The problem states that the conveyor tracks the speed, so the speed changes, right?

When I turn my jet pack on, doesn't the treadmill just speed up to match my change in speed?

6. [quote comment="44889"]Erik, isn't the idea that the conveyor adjusts its speed? The problem states that the conveyor tracks the speed, so the speed changes, right?[/quote]

The conveyor belt could be moving a million miles an hour. So long as the wheels and tires could withstand the heat and spin that fast, the plane could probably still take off.

[quote comment="44889"]When I turn my jet pack on, doesn't the treadmill just speed up to match my change in speed?[/quote]

I'll try it one more time. Wheels are designed to be pretty frictionless. Spinning a wheel doesn't take much effort ("require much force"). The "speed" at which the ground moves is largely irrelevant. The speed at which the plane is pulled through the air (by the engine/propeller) is important. Once a plane reaches an air speed of X, it can take off.

So you've got this massive force pushing the plane forward through the air (near the ground). You have two much smaller forces pushing back: air resistance and the friction in rotating of the wheels.

Force 1 > (Force 2 + Force 3)

When forces are not equal, acceleration occurs. The plane accelerates down the runway. It's slowed down probably less than 1% due to the small increase in friction because the wheels are spinning twice as fast.

7. [quote comment="44889"]When I turn my jet pack on, doesn't the treadmill just speed up to match my change in speed?[/quote]

But the treadmill's effect on moving you backwards is negligible as compared to the affect of the jetpack moving you forward.

Let's draw a free body diagram (plane is pointing left, treadmill is going right):

```                 ----------  >> Fd
<<<<<<<<<<< Fe  (  Plane  (
----------  >>> Fw```

Fe is the force of the engines pushing the plane forward. Fd is the force of drag from air pushing the plane back and Fw is the force of the friction from the wheels pushing the plane back. The engines produce a much larger force which exceeds to the other two forces therefore the resulting net force pushes the plane forward.

If you'd like to see actual empirical testing check out this video (skip forward to 2:48), it might help you visualize what is going on better.

8. Sorry I get it now. Thanks.

9. That has to be one of the most over-complicated explanations I've heard in a while!

The plane propels itself forward through air. The conveyor's motion acts against the plane's tires, which spin (relatively) freely. Regardless how fast the conveyor moves, it's wasted energy because the spinning tires are reacting to absorb its active force. There is no countering force in the air (not stated in the scenario, anyway) to prevent it from moving forward, so the "runway that can move" is just a diversion from where the actual center of action is.

Now, if the brakes were engaged, or the landing gear were retracted, it probably wouldn't be able to lift off.

10. [...] (in)famous "Plane on a Conveyor Belt" experiment was run last night on Mythbusters and, as physics dictates, the plane takes off. That [...]

11. My Question is,
Can the Jet Plane stand still in air Like helicopters?

12. [quote comment="46150"]Can the Jet Plane stand still in air Like helicopters?[/quote]

Irrelevant question. The plane doesn't "stand still."

1. Why would he ask that question? It makes literally no sense.