TEACHER NOTES

ISSUES RELATED TO THE PROBLEM

This exercise is meant to be somewhat open ended. The goal is to get the students thinking about how the equations can represent a physical situation.

If the students haven't already been introduced to a simple free fall model (with linear drag), then the instructor should lead the students through the first problem. However, students should be able to figure out (2) on their own, and (3) then makes for a good group project in class.

One could also ask for a comparison between the models in (2) and (3) and ask if either is more realistic than the other in terms of graphical output say.

Prerequisites

Linear 1st order differential equations OR the derivative and a graphical DE solver tool such as DSolve in Mathematica.

Time allotment - time management

This project should take 1 day in class to finish most of (1) and (2); and then a week out of class for (3).

Expectations

The first question should be straight forward if the students have been introduced to similar problems before. Some students may not be sure exactly what the format of their answer is -- you may wish to clarify what you would deem an acceptable answer.

The second question is difficult for students, but a group of 2 or 3 students would likely succeed.

Students will likely need tips to account for non-instantaneous transition times unless you work related examples in class.

If this is the student's first attempt at modeling, it may take a while for students to have any idea of where to start. You may wish to help them through this and ask them to further improve the model. (See extensions too.)

Future payoffs

Reinforces Newton's 2nd Law as applied to dynamics models.

If carried far enough (esp, see Extensions), the student should begin to get comfortable with how different components of a model affect the solution.

Reinforces the notion that the equations are models -- not reality, and opens the door to experimentation with better models.

Extensions

This IS an extension of the classical free-fall problem.

Here are some further extensions that make for good projects beyond the original assignment:

(4) Experimentation has shown that the turbulence in the air in fact causes a drag force approximated by a constant of proportionality times the square of the speed. Repeat (1), (2) and (3) with this modification.

(5) Assume that in fact, the wind drag force is proportional to the speed raised to SOME power between 1 and 3. Determine how sensitive the results are to changes in the exponent.

See ISSUES IN SOLUTION before assigning these extensions.

References and Sources

This is an extension of an exercise suggested by Kostelich and Armbruster of Arizona State University. Tabulated data from Benson, H., 1991, University Physics. New York: John Wiley & Sons, p. 109.