CARDATA Distance and Velocity Graphs from Car Ride Data BRIEF ABSTRACT Here's a fun group project for differential calculus students. Students go on a drive, record speed and distance, and then analyze their data. GENERAL INFORMATION FileName: CarData Full title: Distance and Velocity Graphs from Car Ride Data. Last Update: 6/3/96 Developer: Aaron D. Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA, E-mail: Aaron.Klebanoff@Rose-Hulman.Edu. Contact: Aaron D. Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA, E-mail: Aaron.Klebanoff@Rose-Hulman.Edu. Phone: 812-877-8151. FAX: 812-877-3198. Support: The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849: Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering. STATEMENT OF PROBLEM Form groups of two to three people. Go on a 20 minute drive in which you cover various speeds. Each person should have at least one task. For example, one person is the driver, and should pay attention to only one thing -- driving. The other person or persons should concentrate on reading the odometer and speedometer, and record the data. For three person teams, make one person the observer, and the other the recorder. Make sure that the driver takes a route which allows for variability in speed. A poor route would be one that entailed a 20 minute cruise down a highway at a constant speed. Keep track of your route, the stops you made, and any events that may have affected your speed. Every 15 seconds, record your speed and distance traveled from the start. You'll have to round and/or interpolate to get the most accurate data. Graph the distance and velocity data against the same time scale on separate graphs. Then, connect the data points with a SMOOTH curve. (1) Write a brief story about your trip describing what happened. Incorporate into your story any and all blips in your graphs carefully explaining what happened. (2) Compare the graphs. Do you need both graphs, or could you recreate the trip from either graph separately? Explain. Extra Credit: Produce a working method/device to measure the data as accurately as possible. Then, repeat steps one and two. KEYWORDS Antiderivatives, Data Collection, Derivatives, Distance, Velocity TEACHER NOTES Issues Related to the Problem Make sure that you are free from liability before you set out to do this assignment. This problem gets all of the students involved, is easy for everyone to do. It is a highly recommended problem for early in a course on differential calculus. Basically, the problem is for the students to go for a drive and record speed and distance data. Afterwards, of course, they are to study the data that they have collected. Two other reasons I like this exercise: This is an excellent opportunity for the students to submit early writing samples. The students don't mind, since they have a good understanding (or at least they think they do) of what they are writing about, and each group's paper is personalized. This problem requires the use of groups. So, if you haven't found a nice group problem to get the class going, this is a good problem to try. I purposely do not make this problem too hard. The goal is to give the students a (good) memorable example to carry with them through calculus. The harder problems build on this problem later in the course. Prerequisites: Students must be able to drive and have access to a car. This project is designed to be done soon before the derivative is introduced. Time Allotment - Time Management Give part of a class period for the students to map out their game plan. Provide about a week (including a weekend) for the students to gather their data. Provide another week or so for students submit rough and final drafts. Expectations This is a simple problem that the students enjoy. Future Payoffs This is an excellent opportunity for the students to submit early writing samples. The students don't seem to mind since they have a good understanding (or at least they think they do) of what they are writing about, and each group's paper is personalized. The nature of the problem requires the use of groups. So, if you haven't found a nice group problem to get the class going, this is a good problem to try. This problem provides a memorable experience with several important concepts from calculus: velocity is the derivative of position, the derivative can be graphically interpreted as slope. And lesser known, that antidifferentiation tends to smooth data or conversely, that differentiation of real data tends to result in very complicated curves. This is fun for the instructor to grade. Extensions The students can do differencing with their data to see how closely the numerical derivative of position fits the recorded speed data. Further differencing allows students estimate the acceleration. Graphing the curve smoothly offers a chance to talk about interpolation. If you are using Mathematica for example, you could discuss the mathematics involved in creating the interpolation function. Here's some more suggestions offered by Brian Winkel. How about integrating the velocity function to see how good it fits the position function? Or differentiating the position function? Or differentiating the velocity function to see the "G" forces they were subjected to? References POSSIBLE SOLUTIONS A typical student's solution (name withheld). The story These two graphs represent my quest for dinner in Sacramento. I left my home in West Sacramento to go to the Spaghetti Factory in downtown Sacramento. On my way to Route 160, I encountered two stoplights (see zero velocities at time 1.25 and time 1.75). My assistant proceeded onto the ramp for Route 160 (see increasing speed around three minutes) and drove a steady 55 mph for about 45 seconds. Then, at a little over 4 minutes, there is a decrease in our speed as we crossed the tower bridge into Sacramento. From 4.75 minutes until 5.25 minutes, we sat at the stoplight by the bridge. After this we drove down J Street at a moderate speed (20-30 mph) toward the Spaghetti Factory. At around 10 minutes, as we neared the Spaghetti Factory, we decided that the wait looked far too long, and that we should go to the Good Earth instead. At this time, we drove back toward Route 160 and increased our speed until my accomplice got a little out of hand at 70 miles per hour (luckily no CHPs). Then, we exited at Arden, our speed decreased to zero two more times as waited for traffic lights. Finally, at time 19.00 we parked in the Good Earth parking lot and headed off to our dinner. It is difficult to note from these graphs that velocity is the derivative of position. I believe this is because the velocities were taken at 15 second intervals and could only be approximations of the average speed for that interval. At points, however, this relationship is very clear. For example, when the position does not change over time (at stop lights) the velocity is zero. This makes a lot of sense. Also, when the position increases rapidly, such as from time 2.00 to time 4.00, the increase in velocity is apparent. The data time = Table[N[i/4], {i, 0, 75}] {0, 0.25, 0.5, 0.75, 1., 1.25, 1.5, 1.75, 2., 2.25, 2.5, 2.75, 3., 3.25, 3.5, 3.75, 4., 4.25, 4.5, 4.75, 5., 5.25, 5.5, 5.75, 6., 6.25, 6.5, 6.75, 7., 7.25, 7.5, 7.75, 8., 8.25, 8.5, 8.75, 9., 9.25, 9.5, 9.75, 10., 10.25, 10.5, 10.75, 11., 11.25, 11.5, 11.75, 12., 12.25, 12.5, 12.75, 13., 13.25, 13.5, 13.75, 14., 14.25, 14.5, 14.75, 15., 15.25, 15.5, 15.75, 16., 16.25, 16.5, 16.75, 17., 17.25, 17.5, 17.75, 18., 18.25, 18.5, 18.75} velocity = {0, 10, 25, 10, 30, 0, 3, 0, 12, 25, 20, 30, 15, 45, 55, 55, 55, 40, 39, 0, 0, 0, 30, 0, 30, 30, 25, 20, 20, 32, 19, 20, 26, 26, 28, 26, 20, 18, 19, 0, 0, 21, 26, 19, 30, 28, 29, 29, 37, 38, 42, 39, 52, 56, 62, 59, 65, 63, 62, 70, 69, 55, 45, 40, 2, 0, 10, 30, 18, 0, 25, 35, 15, 20, 3, 0}; distance = {0, 0, 0.1, 0.2, 0.3, 0.3, 0.3, 0.4, 0.4, 0.5, 0.6, 0.7, 0.8, 1.0, 1.2, 1.4, 1.6, 1.9, 2.0, 2.0, 2.0, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.6, 3.7, 3.7, 3.8, 3.9, 4.0, 4.1, 4.2, 4.3, 4.4, 4.5, 4.7, 4.9, 5.0, 5.2, 5.4, 5.7, 6.0, 6.2, 6.5, 6.6, 7.0, 7.3, 7.6, 7.8, 8.0, 8.1, 8.1, 8.1, 8.2, 8.3, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.8}; The graphs Remark: The assignment requests that the curve fit be SMOOTH. This was emphasized to be more realistic. As a result, students were told to do point plots and connect the dots "by hand" in a smooth fashion. We show both smooth and not smooth below. DistancePoints = Table[{time[[i]], distance[[i]]}, {i, 1, 76}]; VelocityPoints = Table[{time[[i]], velocity[[i]]}, {i, 1, 76}]; ListPlot[DistancePoints, PlotJoined -> True, AxesLabel -> {"t [min]", "distance [miles]"}, PlotLabel -> "Jagged Distance Graph"] -Graphics- ListPlot[VelocityPoints, PlotJoined -> True, AxesLabel -> {"t [min]", "vel [mph]"}, PlotLabel -> "Jagged Velocity Graph"] -Graphics- DistFunc = Interpolation[DistancePoints]; VelFunc = Interpolation[VelocityPoints]; Plot[DistFunc[t], {t, 0, 18.75}, AxesLabel -> {"t [min]", "distance [miles]"}, PlotLabel -> "Smooth Distance Graph"] -Graphics- Plot[VelFunc[t], {t, 0, 18.75}, AxesLabel -> {"t [min]", "vel [mph]"}, PlotLabel -> "Smooth Velocity Graph", PlotRange -> {0, 80}] -Graphics- Further Remarks The solution above was not the best I have seen nor the worst, but it solved the problem and earned a high score. Many students get quite creative in their story line, and some are very careful to choose a route conducive to an interesting story. In more recent trials of this problem, I have told the students up front how much credit will be allotted to creativity which has noticeably improved the stories and made grading more enjoyable. I have never had a group of students answer the Extra Credit problem, although being at an Engineering College, I feel inclined to keep offering. ISSUES IN THE SOLUTION While the problem calls for groups of two or three, three is ideal and four works fine for this project too. Try to keep the group size no greater than four, however, since the problem is relatively easy to do and you want to keep everyone involved. Since this problem is fairly easy for the students, I warn them that they must use proper grammar in their story and explanations. Neatness should also be stressed. Some students may wish to graph the data on computers. This is fine, but warn them again against connecting the dots with straight lines. In every case that I have seen, the graphs that the students create show that the speed graph is (quite amazingly, actually) very squiggly, while the distance graph is smooth and increasing. This offers an opportunity to discuss smoothing issues associated with integration and some of the properties of integration versus differentiation. Especially in the days of computer plotting utilities, most students will graph their data on a machine. If it is important to you that the graphs be smooth, students should be offered methods for creating a smooth plot with ease since the mathematics behind the curve fitting created in our solution is beyond the scope of the problem.