Project 2: A Differential Equations Model of a Glider
Due December 3 or 4

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.

Your write-up for this project should consist of two parts: an expository part describing the qualitative aspects of the behavior of the solutions, and a more computational part which answers some specific questions.

For the first part, you should discuss how the equations and their solutions correspond to glider flight. You should give any mathematical background you think necessary about vector fields, differential equations, and so on. Try to write a paper that you would have been able to understand had it been given to you last semester.

In the paper, you should relate the types of solution one sees in the -plane to general types of glider flight (level flight, looping, and so on). You should consider both the case R=0 and R>0. Any ``special'' solutions should be pointed out and discussed, as well as general categories of solutions and overall behaviors. 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- coordinates and in spatial (x-y) coordinates.

In particular, you should contrast what happens when R=0, when R is small (0.1 or 0.2), when R is moderate (around 1), and when R is large (R=3 or 4). Some questions you might want to discuss are: How does the flight of the glider change as you let R increase? 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'' you find for R=0 still present for positive R? 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, and/or what their linearizations are, although this is not strictly necessary. You may also want to discuss the what happens as , as we did in class via our change of variables.

Note that the exact selection of topics in your write-up is up to you. You needn't answer all of the above questions, or you may choose to answer slightly different ones. You should use some judgment in what to discuss. You will be graded on the clarity of your exposition, mathematical correctness, and overall discussion of the problem. Do not just include a bunch of maple commands and pictures. In fact, if you choose to include no maple at all for this part, that is fine. However, you almost certainly will need a number of well-chosen pictures.

For the second part, you are to answer the following specific questions. You will probably want to make use of (and possibly modify) the routines whereHit and bisect which were discussed in class (and appear on the class web page). Also, you may find it more helpful to do the problems in a slightly different order.

1.

Taking R=0.2, if the glider is launched from a height of 3 units with an initial velocity of 2, what angle should the glider be launched at (to the nearest 0.005 radians) so that it goes 18.25 units before hitting the ground? What should the initial velocity be if the initial angle is 0 (you need consider only )? In both cases, you should find all such solutions, and draw a graph of the flight of the corresponding glider flight in (x,y)-coordinates. (You can combine several on one set of axes).

2.

Again taking R=0.2, v(0)=2, x(0)=0, and y(0)=3, make a graph of the x coordinate when the glider hits as a function of the initial angle . Also make a graph of the landing coordinate as a function of the initial angle when R=0.1.

3.

Taking R=0.1, v(0)=2, x(0)=0, and y(0)=3, make a graph of the angle the glider lands with as a function of the initial angle. From this, can you give an approximate range of initial angles which cause the glider to make exactly one loop before hitting the ground? no loops? three loops?

For all the questions, you should give sufficient justification of your answers. Please try to make your presentation readable. Don't include your false starts, errors, and so on.

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