Project 1: A differential equations model of a Glider
Due Thursday, 29 February

As we have discussed in class, the flight of a balsa wood glider can be approximately described by the system

where v>0 is the speed of the glider, and is the angle the nose makes with the horizontal. The and terms represent the effects of gravity, and the and account for drag and lift, respectively. The parameter R adjusts the strength of the drag on the plane due to air resistance.

The goal of this project is to understand and describe how this model relates to the flight of a glider, and how the parameter R for air resistance affects what types of flight are possible. Your writeup should discuss the model, and relate the types of solution one sees in the phase plane to the types of behaviors of the glider, first for R=0 and then for R>0. Below are listed some questions which you should discuss and answer in your paper. However, your project should be presented as a paper, not as a list of answers to questions. You should include in your writeup a number of relevant pictures and graphs, properly labeled and referred to. These graphs should include both pictures in the v- coordiates and in spatial (x-y) coordinates.

For the case with no drag (R=0). Notice that in the phase plane (v- coordinates) there are two major types of solution trajectories; discuss what type of motion of the glider each of these corresponds to. There are also an additional two special solutions, one at , and another which ``divides'' the two types of solutions. Discuss what these correspond to, and give (approximately or exactly) initial conditions which correspond to the ``dividing case''. Note that to get a good picture of this special solution, you will probably have to adjust the stepsize somewhat, as the default stepsize 0.1 is too large for small values of v.

For the case with drag included (R>0). How does the flight of the glider change if you let R be positive? Is level flight still possible? Is it still possible to make the glider ``loop the loop''? How about making it go around more than once? Are the ``special solutions'' discussed for R=0 still present? (Your answer may depend on what values of R you choose. You should consider at least 3 cases, namely small R (say, around .1 or so), moderate R (for example, R=1), and large values (like R=4). If you know something about fixed points from your differential equations class, you might want to discuss the location and type of the fixed points, although this is not strictly necessary.

You might also want to discuss which sorts of trajectories allow for the furthest flight in each of the cases. For example, suppose you throw the glider from a height of 10 (x=0, y=10). If R=.1 and you throw the glider with an initial velocity of v=2, what is the angle that allows the glider to go the farthest before it hits the ground? What sort of path does it take? You might also want to explore how the answer changes for different initial velocities, or for different values of R. Another interesting question to explore might be which initial conditions allow the glider to go furthest in a fixed amount of time -- is this the same answer as above? You are encouraged to explore these or any other aspect which seems interesting to you.