ODE MODELS FOR THE PARACHUTE PROBLEM^{∗}

DOUGLAS B. MEADE^{†}

Abstract. Classroom discussions relating to the modelling of physical phenomena has experienced a resurgence in recent years. One problem that is accessible to students and mathematically tractable is the motion of skydiver. The basic “parachute problem” is formulated and solved. A simple analysis of the problem raises questions about the applicability of this model. Real-life data is used to propose an extended model. The modified model is shown to be more physically realistic and is no more complicated than the original problem. This discussion gives equal emphasis to both the modelling and the analysis of the problem.

Key words. modelling, IVPs for ODEs, parachute problem, piecewise-defined data

AMS subject classifications. 34-01, 34A12

1. Introduction. The “parachute problem” is discussed in numerous differential equations textbooks (e.g., [1, (p. 141, #19 and 20)], [3, (p. 95, #10, 11, 20, and 21)], [4, (p. 109, #20)], [7, (p. 112–114, Example 3 and #8)]) and journal articles (e.g., [2], [6]). The appeal of the parachute problem is a combination of the facts that the basic model (Newtonian mechanics with resistance) is relatively simple for students to understand and that working with piecewise-defined functions, with which many students have some difficulty, is good preparation for the future discussion of Laplace transform methods. A common formulation of the parachute problem is:

A skydiver drops from a helicopter hovering at a specified height, x_{0}, above the ground and falls toward the Earth under the influence of gravity. Assume the force due to air resistance is proportional to the velocity of the parachutist, with different constant of proportionality when the chute is closed (free-fall) and open (final descent). Given the condition that determines when the chute is deployed, how long does the jump last?

Typical questions to be addressed in the analysis of the problem include:

• what are the terminal velocities of the different stages of the jump? what is the velocity when the chute is opened? at impact?

• what is the latest time that the parachute can be opened while keeping the impact velocity below a specified threshold?

• compare the motions for jumps when the parachute is opened after a fixed amount of time, at a specified altitude, and when a given velocity is attained.

• find the corresponding model with quadratic air resistance, with coefficients selected so that the pre- and post-deployment terminal velocities are the same as for the linear model; how do the two motions compare?

The purpose of this note is to present an analysis of the traditional parachute problem that coordinates graphical solutions with the theory for initial value problems for a system of first-order ODEs. While this initial discussion is rather elementary, it does emphasize a number of important points: parameters identification, dimensional analysis, verification of solutions. The problem becomes more interesting at the end of Section 2, when the accuracy of the model is considered. Information from an Air Force Academy Training Guide [8] is used, in Section 3, to derive and analyze an

∗This work was supported by NSF awards DMS–9404488 and OSR–9108722.

†Dept. of Mathematics, University of South Carolina, Columbia, SC 29208 (meade@math.sc.edu).