This was my primary objective when I began the process of modernizing Math 242 in Fall 1995. Two fundamental changes introduced in the first pilot sections of the revised course, supported by USC's Lilly Teaching Fellow program and the Colleges of Engineering and of Science and Mathematics, were the addition of several group projects and the use of the computer algebra system Maple. The projects served several purposes: to expose students to problems that could not be discussed in the typical 50-minute class period, to give students experience working in groups, and to require students to prepare a technical report summarizing their model, analysis, and results. The students were expected to demonstrate the appropriate use of technology to support their work and prepare their final report. Maple was selected as the official course software on the basis of its wide availability at USC and its ability to combine analytic, numeric, and graphic methods.
The parachute problem that is emphasized in this proposal is an example of a problem around which a number of different projects can be created. The basic problem appears in many ODE textbooks (e.g., Edwards and Penney, Nagle and Saff, Kostelich and Armbruster, and Abell and Braselton). The feature that makes this problem of interest is the piecewise-defined coefficient of air resistance. A simple project is to compare the different problems found in these books and to evaluate the merits of each problem. At the next level of difficulty, students can be asked to research the physics of the situation (e.g., the time it takes for a parachute to deploy and the forces that can be comfortably withstood by the human body) and to develop a modified model that more accurately models the real world.
Pedagogical Objective:
One of the objectives I emphasize in my introduction to this
course is that we live in a world containing many interesting,
yet non-trivial, applications of ODEs. The parachute problem
is ideal for this discussion because almost all students have
a general understanding of the different stages of a parachute
jump (freefall, deployment, and final descent) and the modelling
involves nothing more complicated than Newton's Second Law of
Motion. Specific data for the model can be found in easily
accessible public documents (e.g., Air Force Academy Student
Handbook for Airmanship 490, 1990.)
While it is possible to obtain an explicit solution to this model, the resulting formulae for position and velocity are not generally useful. Instead, it is recommended that Maple be used to graphically (and numerically) simulate the solution. The characteristics observed in the numerical solutions can then be verified using basic analytical tools for ODEs. (Mature students might be tempted to skip this stage, but I think all students should take the time to understand how the displacement, velocity, and acceleration differ for different models for the air resistance.)
One specific mathematical issue that needs to be addressed is the connection between the choice of model parameters, the smoothness of solutions, and the survivability of the jump. This is one area in which almost all of the textbook problems are deficient. The key to this analysis is to realize that it is not necessary to have explicit formulae for the solution.
Pedagogical Evaluation: (200 wds)
Traditional sections of Math 242 tend to emphasize methods
for finding explicit solutions to a differential equation.
On the other hand, the new version of Math 242 emphasizes
the understanding of what a differential equation is and how
ODEs can be used to answer questions about real-world problems
even if an explicit solution is not available.
The ability to decide when to use the computer is almost as important as learning individual Maple commands. To emphasize this point, the new course is taught in a computer classroom. While the students can use the computer at any time during the course, they quickly learn that many problems are simple enough to work by hand.
The parachute module has not yet been implemented in my ODE course. Similar projects that have been used in the classroom have been well received, once the initial objections to working in groups, preparing a report that explains their approach and their results, and generally being unfamiliar with this type of assignment are addressed. Student comments, both confidential and direct, confirm that these exercises are worth the extra effort required of both faculty and student.
Two journal papers relating to the parachute problem have already been accepted for publication. The SIAM Review paper (to appear) focuses more on the mathematical analysis and specific questions that can be addressed at various stages of the analysis. The MapleTech paper (1997) touches on similar issues, but is clearly directed towards the development and use of specific Maple routines for the analysis of this problem. The control problem introduced at the end of the MapleTech paper is particularly interesting. Both papers can be viewed on the WWW as either PostScript or Interactive DVI (IDVI).
Computational Problem:
The basic model for a parachute jump is obtained from Newton's
Second Law of Motion with air resistance proportional to the
projectile's velocity. For simplicity, only the vertical component
of the motion is analyzed. The focus of the modeling and the
analysis is on the connection between the smoothness of the
coefficient of air resistance, the smoothness of the solution,
and the survivability of the jump.
Computational Models Used:
The model is considered as both a second-order equation for
the height above ground and a first-order system for the
height and velocity. Appropriate initial conditions are
specified in either case. The coefficient of air resistance
is defined differently during each of the different stages
(freefall, chute deployment, and final descent) of the jump.
Typical questions of interest include: what is the terminal velocity during freefall and at landing? at what altitude is the parachute deployed? how long does the jump last? Questions relating to the strength of the forces acting on the skydiver when the parachute is deployed and at landing are less common, but quite relevant to this project.
Computational Methods Used: (200 wds)
The graphical and numerical approximations produced by Maple use
Maple's default ODE solver (a Fehlberg fourth-fifth order Runge-Kutta
method); the simpler Euler's method could be used with only nominal
degradation in the quality of the approximate solution.
The smoothness analysis is more interesting. One of the basic facts found in the skydiving literature is that the "opening shock" is strong but smooth. The mathematical translation of this is that, at the time of deployment, the acceleration is continuously differentiable. This is the source of my favorite subtitle for this work: jerks don't jump. The real beauty of this analysis is that a formula for the jerk can be obtained directly from the equation of motion and conditions under which the jerk will be smooth can be (easily) determined without knowing explicit formulae for the position or velocity. The fact that real information is obtained directly from the ODE - not the solution - is very attractive.
Computational Implementation:
Maple V, Release 4 (and Release 3) worksheets can be downloaded
from the WWW. The journal papers contain sufficient explanation
to facilitate implementation in Matlab, Mathematica, or almost
any other scientific programming environment. Utilities implemented
in one of the traditional programming languages (Fortran, Pascal,
C, C++) could be designed, but only for the numeric and graphical
components of the analysis. The modules in a software package such
as Interactive Differential Equations (IDE) generally do not provide
sufficient flexibility to allow the user to test models that are
encountered in problems such as the parachute problem.
Computational Assessment:
The parachute problem is ideally suited for analysis with Maple.
The assembly of the complete model and the general analysis of
the smoothness of solutions benefits greatly from Maple's symbolic
features. The numeric and graphical tools provided by Maple are
well suited for solving the system for specific sets of parameter
values. I firmly believe that all three types of analysis are
critical when real-world intuition is being combined with mathematical
theory and rigor.