TEACHER NOTES
ISSUES RELATED TO THE PROBLEM
It is best to gather the data if possible. A bicycle equipped with a cycle-computer which can measure to the 100th of a mile is helpful. The distance and time period can be extended if it is too difficult to obtain fine measurements.
This is really a laboratory rather than a problem. It can be used in place of a lecture or two on average velocity and an intuitive introduction to limits.
If the students will be doing all plots by hand, the distance should probably be kept to half a mile using 25 data points.
This project is best done in groups because of the data collection component.
Prerequisites
Students need to be able to plot and take difference quotients and see limiting curves from data plots.
Time allotment - time management
After collection of data, one could accomplish the differencing and discussion in one class period with groups of students working on the problem. Then ask for write ups in a few days.
Expectations
Students will enjoy gathering the data, and it will provide a greater sense of accomplishment once the problem is completed.
Future payoffs
Students will have a better understanding of the limiting process involved in differentiation.
Students will gain a firm grasp on one of the most important interpretations of the derivative: the change in position with respect to time.
Extensions
Consider asking about acceleration and when it turns negative as well as its significance there.
The data leaves us with the opportunity to perform a curve fit. With modern computer software, there is no reason to wait for the curve fitting techniques of multi-variate calculus.
This project can be modified for cars as well leaving the opportunity for more variability in speeds and a simple solution for measuring distances.
See CARDATA in this problem data base.
References and Sources