6 D. B. MEADE

4. Conclusion. This note identifies a number of concerns about the parachute problem as it appears in several ODE texts. Real-life data is presented and used to create an improved model that can be analyzed using similar methods. The predictions obtained from the new model are consistent with the physics of skydiving. The modified model is still relatively simple. For example, while the true motion is threedimensional, the current models consider only the vertical component of the motion. Further extensions of the model make good project assignments.

For jumps that begin at altitudes higher than 25,000′ above sea level it begins to be reasonable to consider including the altitude dependence of the air density, air pressure, and gravitational constant in the model. A simple investigation of the sensitivity of the solution to the different parameters is useful when deciding which (if any) of the parameters should be allowed to be altitude dependent (see [5]).

Another extension of the problem is to consider models with nonlinear (quadratic) air resistance during one or more of the three stages of the jump. While the numerical and graphical analysis can proceed virtually unchanged, explicit solutions can be more difficult to obtain (manually) since the individual IVPs become (nonlinear) Bernoulli equations.

The list of physically interesting situations that can be analyzed is almost endless. One obvious example is to formulate a model consistent with the information about the reserve chute and to check that the stated constraints can be satisfied. (What parameters are appropriate for descent under the reserve chute? How smooth is the jerk? Can the cubic model for k_{d} be used?) Other sets of problems can be constructed around different criteria for chute deployment: e.g., at a specified velocity or altitude. Two control problems that can be particularly instructive are: Given a desired duration of the jump, when should the ripcord be pulled so that the landing occurs at the desired time? and What is the latest time the ripcord can be pulled so that the landing velocity is below a given threshold?

The computations and graphs in this paper were prepared using Maple. Full details, including solutions to some of the problems posed above, can be found in [5].

Acknowledgments. The author would like to thank Professors Eric Kostelich, Dieter Armbruster, and David Rocheleau for the discussions which motivated and improved this paper.

REFERENCES

[1] M. L. Abell and J. P. Braselton, Modern Differential Equations: Theory, Applications, Technology, Harcourt Brace College Publishers, 1995.

[2] J. Drucker, Minimal time of descent, The College Mathematics Journal, 26 (1995), pp. 232– 235.

[3] C. H. Edwards, Jr. and D. E. Penney, Differential Equations and Boundary Value Problems: Computing and Modeling, Prentice Hall, 1996.

[4] E. J. Kostelich and D. Armbruster, Introductory Differential Equations: From Linearity to Chaos, Addison–Wesley, 1996.

[5] D. B. Meade, Maple and the parachute problem: modelling with an impact, MapleTech, (in press).

[6] R. Melka and D. Farrior, Exploration of the parachute problem with STELLA, Newsletter for the Consortium for Ordinary Differential Equations Experiments, Summer–Fall 1995, pp. 5–6.

[7] R. K. Nagle and E. B. Saff, Fundamentals of Differential Equations, Fourth Edition, Addison–Wesley, 1996.

[8] Student Handbook for Airmanship 490: Basic Free Fall Parachuting, USAF Academy, May 1990.

[9] H. S. Zim, Parachutes, Harcourt, Brace and Company, 1942.