Parachute Problem

pulling of the ripcord this precisely and there is no time to consider using the reserve chute in an emergency, this does explain why some skydivers survive jumps when their chutes deploy only seconds before impact. In other words, it is almost never too late!

Conclusion

In this paper we have demonstrated the use of Maple in the analysis of a common textbook version of the parachute problem. Through a combination of graphic, numeric, and symbolic methods, the analysis identified several non-physical features of the original model and, subsequently, suggested the formulation of an improved model. While explicit solutions can be found for the new model, all analysis of the motions resulting from the improved model was either numerical or graphical. Many of the questions addressed in this paper — including parameter sensitivity and solution of a control problem — are not reasonable to attempt to solve using manual techniques. Maple, with its combination of symbolic, numeric, and graphic features, is the ideal software tool to use on these problems. The complete worksheet (Release 3 and Release 4) from which this article is produced is available from the author’s WWW home page.

Acknowledgment

Financial support for this project has been received from NSF grants DMS-9404488 and OSR-9108722.

References

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

[2] R. M. Corless, G. H. Gonnet, D. E. G. Hare, and D. J. Jeffrey, Lambert’s W function in Maple, MapleTech, Issue 9, 1993, pp. 12–22.

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

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

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

[6] D. B. Meade, ODE models for the parachute problem, SIAM Review, (submitted).

[7] D. B. Meade, B. S. Haran, and R. E. White, The shooting technique for the solution of two-point boundary value problems, MapleTech, vol. 3, no. 1, 1996, pp. 85–93.

[8] 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.

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

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

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

Biography

Douglas B. Meade

Douglas B. Meade received a Ph.D. in Mathematics from Carnegie Mellon University in 1989, then spent two years as a Research Assistant Professor in the Center of Applied Mathematics at Purdue University. Since 1991 he has been an Assistant Professor in the Department of Mathematics at the University of South Carolina. Research interests are presently directed towards numerical methods for PDEs, in particular, wave propagation on unbounded domains and nonlinear reaction–diffusion equations. Dr. Meade has been active in the development of Maple resources for use in a variety of undergraduate and graduate courses. A Maple supplement to the Nagle/Saff ODE text was completed in 1996; a Maple module for the Engineer’s Toolkit series, co-authored with Etan Bourkoff, will be published by Addison–Wesley in 1997.

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ditional tests should be conducted to ascertain the reliability and accuracy of these results.