Project 3: A differential equations model of a Glider
Due Wednesday, 13 November

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. For most of this project, we will assume R=0.1.

One of our goals is to answer the following specific questions:

1.

When the glider is launched with an initial angle of 0, for what range of initial velocities (to the nearest 0.05) will the glider make exactly one loop?

2.

If the glider is launched from a height of 2 units with an inital velocity of 2, what angle (to the nearest hundredth of a radian) allows the glider to go the furthest before it hits the ground? What if the initial velocity is 0.5 instead?

3.

If the glider is launched from a height of 2 units, can you find an initial velocity and angle so that the glider lands exactly 20 horizontal units away? (There are many such initial conditions).

Is it possible to arrange it so that the glider ``lands gently'' instead of crashing? (That is, so that when the height is 0, the glider is approximately horizontal.) If it isn't possible, give a justification. (This question is a bit harder than the others, and is semi-optional. I would like you to at least try, however.)

For all the questions, you should give sufficient justification of your answers.

In addition to answering the above questions, a general description of what kinds of behaviours can be obtained for all initial conditions should be presented. Specifically, this means produce a good picture of the phase plane (the vs. v plane) including several representative trajectories, explain the different types of trajectories one can see and how they relate to different types of glider flight, and illustrate them with graphs of the corresponding glider's flight (in x vs h coordinates). I suspect that your write-up would make more sense if you do this part first, and then answer the specific questions. But that choice is up to you.

You might also want to compare what happens to the types of solutions when we change R. Are the types of glider flight dramatically different if R=0.5? How about if R=2? What happens for R=0? This last part is optional, but I encourage you to try it if you have time and interest.

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