# The case of the plane and the conveyor belt

## by PhilipJ on 9 February 2006

Kottke is getting into the game of posting cute problems that make you think about what’s *really* going on: The case of the plane and conveyor belt (see Chad’s Tether-ball problem, and our very own Three cylinders problem for a couple of other good ones).

Reproduced for completeness:

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?

Jason flip-flops on the answer, so maybe we can help him out! Does the plane take off?

Only airspeed matters. If the plane needs 150 mph airspeed to lift off and it doesn’t get it, it stays on the ground. If there is a hurricane, it lifts against its moorings without engines. Peripheral issues are irrelevant. Velocity is relative locally.

One launches rockets near the Equator in the direction of the Earth’s rotation to get a free 1000 mph boost. An orbiting object doesn’t care if the Earth rotates (Gravity Probe-B slightly excepted).

There’s another long discussion of this problem at Advanced Physics Forums

My conclusion was that the problem is ill-posed and could favor either answer depending on how you interpret “tracks the plane speed”:

Look, the problem with this thread that undermines its interestingness is that nobody is agreeing on which problem we intend to solve. Both solutions are kicking around already. Is it:

1) The conveyor produces a FORCE equal to the force exerted by the plane’s engines. Solution: No take-off.

2) The conveyor moves at a VELOCITY with respect to the ground equal to minus the velocity of the plane with respect to the ground. Solution: The plane does take off.

There’s really no other controversy and no interesting physics in either solution.

The only thing that matters is speed with respect to the fluid that provides the force to displace the plane (i.e., the air). Thus, making the assumption that the average speed of the air wrt the ground is zero, the plane does not take off under this circumstance. The conveyor belt has the opposite velocity wrt to the ground as the airplane has. The plane doesn’t take off because, wrt to the air (which we have proxied by saying the flow speed of air wrt ground is zero), it’s not moving, so no significant aerodynamic forces exist to lift it off the ground. Basically, you get a whole lot of energy spent not moving anything large but getting a hell of a friction burn in the wheel bearings.

A more insightful question would be to ask what happens if one suddenly hit the brakes on the conveyor belt’s workings.

Xerxes is right that most people agree on what will happen once they decide if the plane is moving with respect to the air, but I disagree that the problem is ill-posed given a reasonable physical assumption: the frictional force due to the spinning wheels is small compared to the engine’s thrust. If you don’t agree that that is a VERY reasonable assumption you need to think about how the plane would take off on a regular runway.

Once you agree that that is a reasonable assumption (you have to! :) you will see that there is only one correct answer: the conveyor belt tracks the plane’s speed (not force, as clearly stated in the problem) producing a negligible force against the thrust and the plane takes off.

I think earlier correspondents have nailed this, but…

The plane engines force air backwards. In accordance with Newtons 3rd Law, the plane is subject to an equal ‘forwards’ force, causing it to accelerate and – ultimately – take off.

The only effect of the madly spinning conveyor belt runway is to make the planes wheels spin twice as fast as normal as it accelerates to take-off speed. (and a speedometer connected to the wheels would suggest that the plane is moving twice as fast as it actually is WRT to the airport.)

The trick here is that the plane is NOT “thrusting” against the runway (as would be the case of a car)in order to accelerate, hence the fact that the runway is moving is irrelevant.

In order for a plane to take off, it needs a vertical upward force. It is my understanding that this force is supplied by the wings forcing the air that goes across it downwards. The engines force air backward for the purpose of increasing the plane’s speed relative to the air. If the conveyor belt keeps the speed of the plane relative to air at zero, the plane can’t take off — right?

Ed,

The critical part of your statement is “If the conveyor belt keeps the speed of the plane relative to air at zero.” IF some device did that, the plane wouldn’t take off, but as I said in comment 4, a conveyor belt can’t do it.

OK – my company makes conveyor belt systems.

Anyone got a spare jumbo ?

Welcome to Blue Peter… and on TODAYS program… :D

The basic error in all those who argue that it won’t take off, is that are confusing the speeds. The speed of the airplane is a linear one, the speed in the wheels are angular. The speed of the belt is tangential.

And surprise: The tangential speed of the belt ALWAYS MATCH the tangential speed of the outer point of the wheel.

For an airplane accelerating over this conveyor belt, the angular speed of the wheels do not cause any effect, only a small

~~neglictible~~reduction of the speed by the friction.Conclusion: the airplane will take off

Here’s the thing: engine power alone can NOT lift a plane, otherwise planes could take off of 30ft ramps. It is the lift generated by the turbines PLUS the forward momentum relative to the air, which is why the plane needs to move forward relative to the air to take off. Even with the turbines moving air under the wings, it is not enough to keep it in the air. Think of a school bus going at 100mph, and you are on it and throw a ball straight up, the ball will not hit the back, because it will move relative to the bus REGARDLESS of how little friction. The same thing with the plane. There is not enough forward momentum pushing additional air around the wings to generate liftoff, and being that it will take off relative to the conveyor belt, even though the wheels generate little friction, it can only lift off vertically, and if it somehow did, once in the air, it will touch back down immediately due to lack of forward momentum generating the needed additional lift. Unless the conveyor belt is moving air with it, you’ve got no lift off.

Dear someone,

The plane has the needed momentum.

Who says it doesn’t? Not the question and the question doesn’t even allow the conveyor to stop the plane from moving forward.

Propellers are called propellers because they propell the plane forward, not the wheels.

What happens under the wheels makes no difference. The plane moves forward relative to the air.

someone is correct,

I this were true, aircraft carriers would have them. Its all about lift , no lift, no flight period, sorry Newtons 3rd Law

No, someone is wrong.

The plane in this is not stationary. It has nothing to do with stationary, the plane needs just as much room to take off as it would normally need. The key is the conveyer belt does NOT stop the plane from moving forward, the only effect the conveyer belt has is it makes the wheels spin faster (and due to friction, requires the plane to have a bit more thrust to reach it’s takeoff speed).

The people that are having trouble grasphing this are simply thinking of it wrong, it’s not like running on a treadmill. Here, the plane is actually moving forward, it’s not stationary. The only thing the Conveyer belt does is increase the speed the wheels spin at, it does not prohibit forward movement of the plane. If a plane is travelling at 200 MPH, it stays travelling at 200 MPH with a conveyer belt going 200 MPH in the opposite direction. Only difference is now the wheels are spinning at 400 MPH.

Moving forward and taking off are two different things. A plane moves forward from thrust but takes off with thrust and lift. Lift is created by air flow over the wings. Without enough airflow (the prop doesn’t supply enough or planes could take off instantly) it won’t take off. It is standing still relative to the surrounding air so will not take off.

It’s amazing how many people don’t understand it. The wheels have (virtually) no friction, and in fact are on the plane to reduce friction. They do not drive the plane.

In addition, the conveyor acts on the wheels, not on the air (except for minute amounts at the belt/air boundary).

The engines of the plane act on the AIR above the belt. Thus, since the air above the belt is unaffected, the airplane can begin moving forward, since it can move forward, it can generate lift and take off once it reaches a certain velocity.

Think of it this way, would it be possible for a plane to take off on a frictionless surface? Because that’s effectively what the wheels on the ground/conveyor belt create. The plane will certainly take off, in fact Mythbusters just busted this one.

The plane flew.

Could a plane take off on a frozen lake?

JOE. Again, the Plane is NOT standing still. It is impossible for a conveyor belt to stop a plane from moving forward, the only affect it will have is on the speed the wheels spin at.

It’s sort of a trick question in that it originally leads people to think of running on a treadmill where you are stationary in respect to the ground and air. In this case, the plane is NOT stationary, it gets its propulsion from it’s jet engines or propellers pushing against the air, not the wheels.

So again, This is one last time. The Conveyor belt DOES NOT stop the plane from moving forward. It cannot. The only affect it has is the speed the wheels spin at.