Aircraft and Treadmills
In this post, I want to address an interesting thought experiment that I “found on the internet”™. The problem is usually phrased in a few different ways but here is the version I have chosen to attack.
It was a calm, sunny day at Horsehead Airport. An aircraft, ready to fly to a foreign country lines up on the airport runway, carrying out final preparations for takeoff. On this calmest of days, the evil Dr. Horrible has decided to discombobulate the pilot of our aircraft by swapping out the runway for a long conveyor belt. This belt is powered by a very powerful motor that can drive the conveyor belt and anything on it at enormous speeds. As the pilot throttles up for takeoff, he powers up his conveyor belt to move backwards. His aim is to move the conveyor belt backwards just fast enough to cancel the aircraft’s forward motion, hence preventing the aircraft from achieving takeoff velocity. Does he succeed in doing so? Can the pilot still takeoff his aircraft or is he at the mercy of the evil Dr. Horrible?
What do you think would happen? Take an intuitive guess. Part of being a good scientist is getting good at making defensible guesses. Also part of being a good scientist is realizing when your common sense and gut feeling are going to fail and training to guess more systematically. When I first ran into this problem many years ago, my gut feeling was “Hmm.. if the conveyor belt moves as fast as the aircraft but in the other direction, then the two would cancel each other out and the aircraft would be at rest, unable to take off!”. My second thought was “Wait a minute.. maybe the engines can produce enough airflow over the wings that the plane can lift off even if stationary!”. The Mythbusters investigated a similar question a while back (here is a sneak peek).
In a reasonably realistic scenario, the pilot would takeoff with no difficulty. The only force that Dr. Horrible can affect is the force of friction/rolling resistance that acts on the wheels of the aircraft through the conveyor belt. We can show that this force, under a remotely realistic model, could not counter the thrust generated by the aircraft engines. The speed of the conveyor belt is only consequential insofar as it may affect the resistive force on the wheel, which we show is not likely.
Elucidating the Problem
Before we charge head-first into gritty math and physics equations, we need to do a little work in choosing our model and rephrasing the informal question clearly. What do I mean by “choosing a model” for this problem? When we are a given a problem like this, we need to clarify the assumptions under which we are working. The better our assumptions, the closer we are to a realistic answer. There is a common joke in academia: “To a biologist, animals come in a variety of shapes, sizes and colors; to a physicist, all animals are spheres with uniformly distributed mass”. As ridiculous as the previous statement sounds, one can derive a surprising number of accurate results having made sensible simplifying assumptions. After all, these simplifying assumptions are made to neglect higher-order effects that only contribute marginally to the answer. If you want to know how far a cannon ball will fly, it doesn’t really matter if you do not take the imperfections on the surface of the ball, the spinning of the ball, or air resistance into account. The result is reasonably accurate. If on the other hand, you are firing the ball into a tornado, it might make sense to not ignore wind and air resistance
Let us analyze this problem to take into account all the forces involved. Once we have this understanding, we can then make some simplifying assumptions.
The figure shown above describes a simplified free body diagram of the forces acting on an aircraft as it is throttling up for take off. The six forces are:
- Thrust: The horizontal force applied on the aircraft from its engines.
- Drag: The horizontal retarding force applied on the aircraft due to air resistance. This force depends on the airspeed of the aircraft.
- Lift: The vertical force on the aircraft due to airflow over the wing. The magnitude of this force changes with the airspeed of the aircraft.
- Gravity: The vertical downward pull on the aircraft by mother earth.
- Normal Force: The vertical upward push of the ground on the aircraft.
- Friction: The retarding force applied on the wheels of the aircraft from the ground due to friction.
Let us see how each of these forces are developed and applied. Thrust is developed by the engine pushing gasses at a high velocity out towards the rear of the aircraft. This causes a forward thrust. Note that this thrust is independent of the velocity of the conveyor belt. The thrust is controlled solely by the throttles on the engine. The drag on the aircraft is a function of aircraft geometry and its velocity with respect to air. It is pretty small compared to the maximum thrust at take off speeds. We know this, of course, because planes routinely manage to take off, even when the pilot does not command maximum thrust. The lift on the aircraft is solely a matter of airspeed across the airfoil. We are explicitly assuming that there are no weird air pockets or other artifacts of the wind stream that affect the lift. This assumption reasonably holds during takeoff during calm weather. Since we also assume that there is no wind (calm weather), we don’t have to worry about head or tail winds. This essentially means that we have conditions where the airspeed of the aircraft is equal to its ground speed. The lift of the aircraft, hence, depends entirely on the speed on the aircraft relative to the ground. Gravity is a fixed force that is equal to the take-off mass of the aircraft times the local acceleration due to gravity. It acts downwards through the center of mass of the aircraft. The normal force is a reactive force from the ground that balances gravity. Its magnitude changes to match the net downward force exerted by the aircraft on the ground. It exists to keep the aircraft from sinking into the earth. Last but not least, there are frictional forces acting on the wheels. Technically, I should not call this force friction. I should call it rolling resistance. This is because this force is unlike any frictional force that one learns about in a high school or introductory college physics class. It is not static or dynamic friction. It is a retarding force that is caused due to hysteresis of the structure of the wheel. A bunch of NASA engineers did a number of tests on how this rolling resistance changes with respect to a number of parameters (tire inflation, speed, yaw etc). They concluded that, within operation parameters, an aircraft wheel’s rolling resistance for straight line motion is approximately 2% of the load on the wheel (I am simplifying a lot but this is essentially correct) and is independent of the ground speed. There are similar formulas to show that the resistance in the bearings that couple the wheels to the airframe is quite small when within the operational range of the wheels and the bearings.
After our long analysis, let’s take a stab at trying to answer the question. The only force that Dr. Horrible can affect is the force of fiction. All the other forces involved do not depend on the conveyor belt at all. The crucial element in answering this is deciding what assumptions we would like to keep. As we can noted earlier, throughout the operating range of the aircraft tires, its rolling resistance does not depend on aircraft ground speed. This is a very non-idealized statement. It says nothing about what could happen if the wheel could survive higher velocities. There is no idealized wheel here. All this data is empirical. If we assume that Dr. Horrible does not wish to kill the occupants of the aircraft by driving his treadmill so fast that the tires explode and the wheels collapse, we can safely say that the thrust of the engine will always overcome the resistive forces involved. Hence the aircraft will always take off. As an example, the maximum takeoff weight of a Boeing 747 is about 350 tonnes. This translates to a normal force of 3.5 MN (mega newtons). The maximum rolling resistance would thus be approximately 70 kN (kilo newtons). The 747’s four engines develop around 900 kN of peak thrust. This is enough to overcome any resistive force and drag during take-off. “What happens to the wheels?”, you ask. The backwards movement of the treadmill simply spins the wheel faster. At some point, the speed is so high that the wheel is operating out of its intended range and it fails catastrophically. The resistive force of the wheel doesn’t increase as the aircraft speeds up. In fact, it decreases as the load on the wheels decrease due to increasing lift as the aircraft approaches takeoff speed. This is one possible explanation for what would happen, and is the most practical of explanations. There are alternate explanations that posit infinitely strong wheels with various rolling resistance curves. Some of them allow the plane to take off because they make assumptions similar to ours. Some of them overwhelm the engines eventually because they assume that the force of friction can increase with the increase in the difference in speed between the wheel and the conveyor belt. I do not believe that the latter situation is possible with any reasonable engineered material we have but would be happy to be proven wrong. In other words, unless your wheels are made out of wonderflonium, you’ll be airborne or in a ditch and on fire. There is no realistic “hold the airplane in one place” situation.
Notice I have managed to avoid discussing the mechanics of how the wheel will rotate. Calculating the moments and the torque on the wheel is a whole different can of worms. The wheel does not spin without slip in a situation such as this. Not being a mechanical engineer and knowing very little about tire engineering, I cannot comment on what would realistically happen. All of the toy models that involve solid cylinders and artificial coefficients of friction provide results that diverge from reality quite quickly. Another quick point – a common argument that I commonly read (usually used to justify that the aircraft would take off) is to say that the engines would still push air over the wings. Shown here is a picture of an F-16 flying. Its engine inlets are under its body and the outlets are behind the wings (and the elevators). The F-16 happens to fly quite well. The airflow generated by an engine is simply not enough to provide lift if it were pushed over the wing. Wings generates lift because the aircraft moves forward through the air, not because of engine exhaust (or intake) flowing over the wing.