A plane is on a conveyor belt which moves in the opposite direction. Can it take off?
This page is an appendix to the Physclips page Vectors and relative motion. It was written to answer an odd question put to me by several students. The question was asked with two very different conditions:
Question 1. A plane is on a conveyor belt which, when turned on, moves the plane backwards at a speed equal to the normal take-off speed of the plane. If the pilot uses the normal amount of throttle and other controls, can the plane take off?
Question 2. A plane is on a conveyor belt which, when turned on, moves the plane backwards. The belt speed is controlled and it can move sufficiently fast that, with respect to the ground, the plane never moves forwards. Can the plane take off?
As we shall see, these two sets of conditions are very different. Much of the confusion of the students who have asked this question, and the disagreements among them, can be traced to confusion over which (if either) of these conditions has been specified.
A few assumptions first: Let’s assume that the plane has wheels that roll on the ground, but that there is no engine to drive these wheels. The plane has either propellor or jet engines which push against the air, with negligible effect on the ground. The plane has wings (and other surfaces that we shan’t mention further) that generate lift as air passes over them. We also assume that there is no wind (except later when we consider the wind that may be generated by the conveyor belt itself).
Question 1: belt speed = normal take-off speed. (Technologically, this could be done using an aircraft carrier instead of the conveyor belt, and a plane with a very low take-off speed, such as the Gossamer Condor.) Let’s answer this first by making two approximations, and then deal with the approximations.
Question 2: plane assumed stationary with respect to ground. (This one may not be technologically possible.) This requires looking at some of the same effects, but under very different – and difficult – circumstances.
Question 3: plane assumed stationary with respect to centre of the Earth. (One could do this one experimentally with existing technologies.)