First, the wording of the problem is confusing. What does "matching the speed of the wheels" mean? Let me pose the question a little differently. Is there ANY way that the belt can move in order to prevent the plane from taking off? In other words, can we instruct the belt to move backwards in such a way as to keep the plane stationary with respect to the ground, and therefore the air?
In order to figure this out, you can to ask “how can the belt apply backward FORCE to the plane (through the wheel axles) that is equal or opposite to the thrust of the propeller against the air”? But that turns out to be really, really hard, at least for me. Instead, I think it’s easier to think about whether there is a way that the belt can divert all of the POWER of the motor away from accelerating the plane.
First, the easy case. If the wheels have no friction and no mass, the belt is out of luck. There’s no way that it can apply any lateral force to the plane at all. So ALL of the motor power goes into accelerating the plane forward, and the plane takes off normally. In fact, better than normally, because there is no friction in the wheel bearings to worry about. This case is the mirror image of a man in slippery shoes on ice. His legs windmill, but he doesn’t move. Likewise, the belt moves under the plane, but the plane stays still by virtue of its inertia. With no force transfer between belt and plane, there can be no power transfer.
Now let’s add friction between the axle and the wheel (the wheel still has no mass). Remember that the motor normally has enough power to overcome the friction in the wheel bearing by a LONG way. That is, most of the power applied to the propeller goes into accelerating the plane, and a small amount goes into overcoming friction in the bearing (which heats the bearing). Let’s say that the amount of power stolen by friction near takeoff speed is about 1%, leaving 99% for acceleration. I actually think the number might be a little higher, but it doesn’t matter for our purposes.
To keep the plane still, the belt needs to increase the power dissipation in the bearings to be 100% of the motor’s power, so that no energy goes to acceleration. And in order to do that, the belt has to increase the wheel rotation speed by a factor of 100, because the power dissipated by friction is proportional to the relative speed of sliding parts. So with the throttle at max, the belt needs to go backwards at 100 times the normal takeoff speed of the plane. If it adjusts itself just right, the drag in the bearings will then exactly match the thrust. The wheels will be rolling at 100 times the takeoff speed, and the plane will be standing still because no energy remains to move it forward. Of course, with all the power going into the bearings, eventually the bearings will melt.
This same argument applies to rolling friction between tires and the belt. Now you are heating the tires and the belt as well as the bearing, and so the bearings will last a little longer.
Now let’s go back to assuming the wheels have no friction, but give them some mass. In this case, when the motor applies thrust, the power goes into two places: forward acceleration of the plane, and rotational acceleration of the wheels. That is, it takes power to increase the rotational speed of the wheels, in proportion to their moment of inertia.
So now what does the belt have to do in order to divert 100% of the motor power into rotating the wheels? If you apply constant power to a rotating mass, it will just spin faster and faster. So in order to keep constant power going to the wheels, the belt has to keep accelerating the wheel spin as long as the motor is on. The acceleration (I think) needs to be proportional to the square root of time, because rotational energy is proportional to the square of rotational velocity. In this case, however, the wheels will eventually fly apart at some large rotational velocity.
This last case is a little hard for me to grasp intuitively. With friction, you can see easily where the frictional force in the wheel will translate into a drag on the axles to counteract the thrust from the motor. Here it’s a little trickier. The plane is pulling forward on the axle. The belt is pulling backward on the bottom surface of the tire. The forces are hard to calculate, because they are applied to very different parts of the wheel, but in any case you can see that because the forces are applied obliquely to one another, the wheel will spin faster and faster and its kinetic energy will increase with time, just as we require.
Imagine that the belt can’t increase its speed forever. Let’s say it has a top speed of 1000 mph (and that the wheels can take these speeds). The belt keeps increasing its speed at first so that all the motor power goes into accelerating the wheel rotation. Then the belt maxes out. Can the plane now put power into accelerating itself instead of its wheels and begin to fly? I think so. There’s a huge amount of kinetic energy in the wheel rotation. The belt is whipping around. The wheels are spinning like crazy. But because there’s no friction, there’s no power required to maintain the wheel rotation. The airspeed of the plane is 0, and so much of its thrust will now go into moving the plane forward. But even in normal takeoff, SOME power goes into accelerating the wheel rotation. And the energy it takes to accelerate the wheel rotation from 1000 mph to 1010 mph is 20 times higher than the energy it takes to get them from 0 to 10 mph (because rotational energy is proportional to the square of velocity). So while the plane should eventually fly, the duration of its takeoff roll will be longer. And when it gets into the air, its wheels will keep rotating at 1010 mph. When you try to bank, there will be some wicked adverse yaw.
Finally, with both friction AND mass in the wheel, you’ll get some combination of the two effects, depending on bearing friction coefficients and wheel moments of inertia. Hard to tell whether the wheel will fly apart or melt first.
Addendum: There is a comment above from Aerospace Engineer (
http://pogue.blogs.nytimes.com/2006/...ill-conundrum/) about the belt inducing laminar flow over the wings. If that happens, then the plane WILL fly. The only question is how fast that airflow will be at the point the wheels melt/explode.