An interesting question was posed in my daily chatroom. It has caused much debate. At first I thought it wouldn’t work, but as I heard some arguments the other way, I changed my mind, and the more I think about the more sure I am that it will indeed work.

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?

The site I was originally pointed to is an airplane-related forum. The topic has a LOT of discussion.

After I wrote what you find below, I took a look at this page. It basically says what I did.

It’s my position that the plane will take off, a position not shared by many, even after having it explained thoroughly. Here’s my explanation about why it will take off:

Let’s presume that takeoff speed for this airplane is 100 miles per hour and the wheels have been over-engineered to safely roll at 250 miles per hour. The tire tread is excellent and the axles have very low friction. The wheels are free-spinning, the brakes are not applied, and the plane is not being held in place by any physical tether. If we can’t assume these things, the scenario is impossible even in fantasy and not worth considering.

When the engines are fired up, the plane begins to move forward, which causes the wheels to turn. Let’s say for ease of thinking that a plane on a normal runway will be going 1 mile per hour at 1 nanosecond after thrust is applied. This means that at T plus 1 nanosecond, the rolling speed of the wheel is 1 mile per hour.

The belt is going to match this speed as quickly as it can. Let’s assume that this only takes one nanosecond to happen. So, our little scenario is sitting at T plus 2 nanoseconds. The distance that the plane has moved forward is virtually immeasurable. It works out to 0.0000000176 inches.

In this instant, the speed of the wheels relative to solid ground is 1 mile per hour. The speed of the conveyor belt, rolling towards the rear of the plane, is also 1 mile per hour. The total rolling speed of the wheel relative to the conveyor belt surface is found by adding these two values, so it is 2 miles per hour.

What happens next will be determined by the difference between the force acting to push the plane forward and the force acting to push the plane backwards. If this were a car instead of a plane, the motion of the car relative to solid ground would be zero. The overall effect on the car is nullified. The car’s wheels exert a force on the conveyor belt. The conveyor belt simply moves out of the way.

In our airplane example, we have a large force pushing the plane forward – the engines. There is also a force acting in the other direction, but the error that many people make is in assuming that this force is equal to the engines.

The conveyor belt is moving towards the rear of the plane. This motion results in a force acting on the wheels … but what is the effect on the plane as a whole? Remember that we are using very low friction axles, and the wheels are free spinning.

Let’s put our current situation aside and consider another one – a completely unpowered and motionless airplane with the brakes disengaged, sitting on a belt that goes from zero speed to rolling forwards at 50 miles per hour in moments. The only contact the belt has with the airplane is the wheels, which will begin turning. Because the wheels spin freely and have very low-friction axles, and the plane is enormous with enormous inertia, very little of the power will be transmitted to the airplane itself. It will be almost motionless for several moments. It will take a significant and easily measurable amount of time for the airplane to reach 50 miles per hour. In this scenario, there is no other force acting, so over time the very small amount of friction causes the wheels to slow down and the motion of the plane to speed up. Eventually the wheels will not be turning and the plane will be going 50 miles per hour.

The only effect that the conveyor belt motion has on the plane’s body in our initial scenario is a very small amount because of friction in the axles. The net effect is thousands of pounds of thrust applied to the plane’s body – relative to the air, not the conveyor belt. As time passes, the plane will pick up speed, causing an increase in rolling speed, which speeds up the belt. The rolling speed of the tires in relation to the conveyor belt will always be twice the plane’s speed relative to the ground. When the theoretical takeoff speed of 100 MPH is reached, the wheels will be rolling along the belt at 200 MPH, which is less than our theoretical maximum.

There are two key pieces of information to understanding this – the force applied to the plane is not connected in any way to the conveyor belt, and the moving conveyor belt simply spins the wheels faster, barely affecting the plane.

In the real world, the airplane’s wheels might not be able to actually take being run at double takeoff speed, but assuming that’s not a problem, there’s no reason it wouldn’t be able to fly.

## 2 replies on “up up and away!”

I believe the plane will take off. The whole matter about the wheels is almost irrelevant because, unlike a car, the wheels do not propell an airplane, the force of moving air through the engines does and the movement of air over the wings provides the lift.

I can see why people get confused but if you render a proper force diagram (as we all learned in our physics classes) I think it becomes obvious.

Huh. You know, I tried this little idea problem out on a couple of people – I’m starting to think Elyo has chanced across a true intellectual litmus here. A *lot* of people just shrug and say “I dunno” – refuse to even consider the question. Some other people think about it, but figure the belt can stop the plane. So far, three people whose minds I respect have all reached the same (I believe correct) conclusion, using three different mental approaches.

Yeah. Forget college entrance exams – let’s just lay this on everyone graduating High School, and if they can’t figure it out, tell them to consider not spending the money on college.

-GB