(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "NeXT Mathematica Notebook Front End Version 2.2"; NeXTStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 12, "Times"; ; fontset = leftheader, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = leftfooter, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; noKeepOnOnePage; preserveAspect; startGroup] AVERAGE VELOCITY OF BICYCLIST :[font = section; inactive; preserveAspect; startGroup] BRIEF ABSTRACT :[font = subsection; inactive; preserveAspect; endGroup] This project has students gather (sample data is provided too) and analyze time-distance data from a bicycle ride to estimate the velocity by differencing the data over several time intervals. :[font = section; inactive; Cclosed; noKeepOnOnePage; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; endGroup] FileName: AVEVEL Full title: Average Velocity of a Bicyclist Last Update: 6/3/96 Developers: Lynn Kiaer, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA Aaron Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA Contact: Aaron Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8151. Email: Aaron.Klebanoff@rose-hulman.edu. 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 and the Arvin Foundation of Columbus IN. :[font = section; inactive; Cclosed; noKeepOnOnePage; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] 1. A bicyclist travels .5 mile. Obtain a data set consisting of (at least) 25 (time in sec, distance in ft) pairs of observations. (Or you can use the data set below.) :[font = input; noKeepOnOnePage; preserveAspect; endGroup] data = {{0, 0}, {6., 7.}, {12., 28.}, {18., 61.}, {24., 102.}, {30., 148.}, {36., 195.}, {42., 241.}, {48., 285.}, {54., 327.}, {60., 370.}, {66., 419.}, {72., 479.}, {78., 557.}, {84., 658.}, {90., 787.}, {96., 947.}, {102., 1137.}, {108., 1353.}, {114., 1590.}, {120., 1835.}, {126., 2075.}, {132., 2295.}, {138., 2476.}, {144., 2599.}, {150., 2644.}}; :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] 2. Plot every 8th data point. What was the average speed of the bicycle in each interval? Connect the data points with line segments. What is the slope of each segment? :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] 3. Plot every 4th, every other, and every data point, and answer the same questions. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] 4. Let the line segments that connect the data points be our `distance function'. What would our distance function look like if we took a data point every foot? every inch? if we had an electronic recording device that took observations continuously? :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] 5. What happens to the length of the line segments as we take measurements more frequently? What does the slope of the line segments tell us? If we have continuous observations, what slope (the slope of what line) would give us the same information? :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] 6. Estimate when the velocity ( in ft/sec and mi/hr) is greatest in this one-half mile cycling trip. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] 7. How would you decide how many observations to take? :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; endGroup] 8. In some research or industrial settings, observations may be very expensive or dangerous to obtain. Crash testing of automobiles and `weatherbeater' airplanes sent to obtain data about hurricanes are two examples. In these situations, additional factors play a role in determining the number of observations to obtain. Discuss some appropriate issues using these examples and/or some of your own. :[font = section; inactive; Cclosed; noKeepOnOnePage; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; endGroup] Average velocity, limits, derivative, difference quotient, data collection :[font = section; inactive; Cclosed; noKeepOnOnePage; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] ISSUES RELATED TO THE PROBLEM :[font = subsubsection; inactive; preserveAspect] 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. :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] In case gathering the data isn't worth the trouble, here's how we generated our data. :[font = text; inactive; preserveAspect] The data set given in the problem was created to ensure an interesting mathematical problem. We made every attempt to make the data appear real. :[font = text; inactive; preserveAspect] We try to generate an interesting curve for velocity so that our average velocity is less than 20 miles/hour for the bicyclist. In addition we require that the cyclist go exactly one-half of a mile (2640 ft) in 2.5 min or 150 seconds. :[font = input; preserveAspect] vtrial[t_] = b(t^3 Cos[ Pi/2 (t)/(60 2.5)] + 100000 Sin[ 3 Pi t/(2.5 60)]); :[font = input; preserveAspect] strial[t_] = Integrate[vtrial[T], {T,0, t}]; :[font = text; inactive; preserveAspect] Below, we prescribe that at time t = 2.5*60 = 150 sec the bicyclist must be at 2640 feet (one-half mile) to scale our trial velocity function. :[font = input; preserveAspect; startGroup] sol = Solve[strial[2.5 60]==5280/2,b] :[font = output; output; inactive; preserveAspect; endGroup] {{b -> 0.00006488669122712222}} ;[o] {{b -> 0.0000648867}} :[font = text; inactive; preserveAspect] Hence, by substituting the value of b (b = .000065) which puts us at 2640 ft at time 150 sec, we now have a reasonable position (s(t)) and velocity (v(t)) function with which to work. :[font = input; preserveAspect] v[t_] = .000065 (t^3 Cos[ Pi/2 (t)/(60 2.5)] + 100000 Sin[ 3 Pi t/(2.5 60)]); :[font = input; preserveAspect] s[t_] = Integrate[v[T], {T,0, t}]; :[font = text; inactive; preserveAspect] We note the average feet in ft/sec for covering the one-half mile course in an average speed of 20 mi/hr ( a reasonable average speed) is 29.33 ft/sec. :[font = input; preserveAspect; startGroup] avespeed = 20 mile/hour 5280 ft/mile /(3600 sec/hour)* (sec/ft) //N :[font = output; output; inactive; preserveAspect; endGroup] 29.33333333333333 ;[o] 29.3333 :[font = text; inactive; preserveAspect] The average speed for our velocity function we have constructed is 17.6 ft/sec which is very reasonable when compared to the 20 mi/hr average speed of 29.33 ft/sec. :[font = input; preserveAspect; startGroup] AveSpeed = Integrate[v[t],{t,0, 2.5 60}]/(2.5 60)//N :[font = output; output; inactive; preserveAspect; endGroup] 17.63073410532929 ;[o] 17.6307 :[font = text; inactive; preserveAspect] And here we finally construct the position function. :[font = input; preserveAspect] s[t_] = Integrate[v[T],{T,0, t}]; :[font = text; inactive; preserveAspect] We plot our velocity and our position functions. :[font = input; preserveAspect; startGroup] Plot[v[t], {t, 0, 2.5 60}, AxesLabel -> {"t [sec]", "velocity [ft/sec]"}] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Plot[s[t],{t,0,2.5 60}, AxesLabel -> {"t [sec]", "distance [ft]"}] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] And here we finally construct the position data - in a set called data - from which we shall take our data. We construct 25 data points equally spaced in time. :[font = input; preserveAspect] data = Table[{i,Floor[s[i]]},{i,0, 2.5 60,6}]//N; :[font = input; preserveAspect; startGroup] dataTable = TableForm[Table[{i,Floor[s[i]]},{i,0, 2.5 60,6}]//N, TableHeadings->{None,{"t - sec", "s - ft"}}] :[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup; endGroup] TableForm[{{0, 0}, {6., 7.}, {12., 28.}, {18., 61.}, {24., 102.}, {30., 148.}, {36., 195.}, {42.00000000000001, 241.}, {48., 285.}, {54.00000000000001, 327.}, {60., 370.}, {66., 419.0000000000001}, {72.00000000000001, 479.}, {78., 557.0000000000001}, {84., 658.}, {90., 787.}, {96., 947.}, {102., 1137.}, {108., 1353.}, {114., 1590.}, {120., 1835.}, {126., 2075.}, {132., 2295.}, {138., 2476.}, {144., 2599.}, {150., 2644.}}, TableHeadings -> {None, {"t - sec", "s - ft"}}] ;[o] t - sec s - ft 0 0 6. 7. 12. 28. 18. 61. 24. 102. 30. 148. 36. 195. 42. 241. 48. 285. 54. 327. 60. 370. 66. 419. 72. 479. 78. 557. 84. 658. 90. 787. 96. 947. 102. 1137. 108. 1353. 114. 1590. 120. 1835. 126. 2075. 132. 2295. 138. 2476. 144. 2599. 150. 2644. :[font = subsubsection; inactive; preserveAspect] 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. :[font = subsubsection; inactive; dontNoPageBreakBelow; noKeepOnOnePage; dontPreserveAspect] If the students will be doing all plots by hand, the distance should probably be kept to half a mile using 25 data points. :[font = subsubsection; inactive; preserveAspect; endGroup] This project is best done in groups because of the data collection component. :[font = subsection; inactive; dontNoPageBreakBelow; noKeepOnOnePage; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; dontNoPageBreakBelow; noKeepOnOnePage; preserveAspect; endGroup] Students need to be able to plot and take difference quotients and see limiting curves from data plots. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; dontNoPageBreakBelow; noKeepOnOnePage; preserveAspect; endGroup] 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. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect; endGroup] Students will enjoy gathering the data, and it will provide a greater sense of accomplishment once the problem is completed. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Future payoffs :[font = subsubsection; inactive; preserveAspect] Students will have a better understanding of the limiting process involved in differentiation. :[font = subsubsection; inactive; preserveAspect; endGroup] Students will gain a firm grasp on one of the most important interpretations of the derivative: the change in position with respect to time. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect] Consider asking about acceleration and when it turns negative as well as its significance there. :[font = subsubsection; inactive; preserveAspect] 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. :[font = subsubsection; inactive; preserveAspect] This project can be modified for cars as well leaving the opportunity for more variability in speeds and a simple solution for measuring distances. :[font = subsubsection; inactive; preserveAspect; endGroup] See CARDATA in this problem data base. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; endGroup] References and Sources :[font = section; inactive; Cclosed; noKeepOnOnePage; preserveAspect; startGroup] POSSIBLE SOLUTIONS :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] 1. A bicyclist travels .5 mile. Obtain a data set consisting of (at least) 25 (time in sec, distance in ft) pairs of observations. (Or you can use the data set below.) :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] There are several ways to obtain the data. We include here a dummy data set on {time (sec), position (ft)). :[font = input; noKeepOnOnePage; preserveAspect; endGroup] data = {{0, 0}, {6., 7.}, {12., 28.}, {18., 61.}, {24., 102.}, {30., 148.}, {36., 195.}, {42., 241.}, {48., 285.}, {54., 327.}, {60., 370.}, {66., 419.}, {72., 479.}, {78., 557.}, {84., 658.}, {90., 787.}, {96., 947.}, {102., 1137.}, {108., 1353.}, {114., 1590.}, {120., 1835.}, {126., 2075.}, {132., 2295.}, {138., 2476.}, {144., 2599.}, {150., 2644.}}; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; endGroup] Alternatively, the students may be asked to collect the data, and given specific instructions about how to do it: divide a quarter mile track into sections (not necessarily of equal length), and then write down the time at which the cyclist passes each mark, using a stopwatch. A third option is to simply ask the students to collect some minimum number of (time, distance) pairs, and accept the likelihood that the students may have to collect data more than once before obtaining a useful data set. The data will be more interesting if the cyclist varies his/her speed considerably. See Teacher Notes for more ideas. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] The intent of the next two questions is to have the students `discover' the relationship between slope and velocity (speed), and also to see the distance function become more smooth as more data points are included. This will help them to answer `smoothly curving' in response to the fourth question. It might be worthwhile to point out that the curve would pass through all of our data points. :[font = subsection; inactive; Cclosed; noKeepOnOnePage; preserveAspect; startGroup] 2. Plot every 8th data point. What was the average speed of the bicycle in each interval? Connect the data points with line segments. What is the slope of each segment? - The slope of each line segment is the average slope of the bicyclist in each interval. :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] We take every 8th data point and plot the position data - both unjoined and joined. :[font = input; noKeepOnOnePage; preserveAspect] sdata8 = Table[data[[i]],{i,1, Length[data],8}]; :[font = input; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[sdata8,PlotStyle->{PointSize[.02]}, PlotRange->{{-5,150},{-50,3000}}, AxesLabel->{"time [sec]","pos [ft]"}] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[sdata8,PlotStyle->{PointSize[.02]}, PlotJoined->True, PlotRange->{{-5,150},{-50,3000}}, AxesLabel->{"time [sec]","pos [ft]"}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] We compute the average velocity over these time intervals and plot the velocity at the right hand side of each sub-interval. :[font = input; noKeepOnOnePage; preserveAspect] vdata8 = Table[{sdata8[[i+1]][[1]], (sdata8[[i+1]][[2]] - sdata8[[i]][[2]])/ (sdata8[[i+1]][[1]] - sdata8[[i]][[1]])}, {i,1, Length[sdata8]-1}]; :[font = input; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[vdata8,PlotStyle->{PointSize[.02]}, AxesLabel->{"time [sec]","vel [ft/sec]"}] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[vdata8,PlotStyle->{PointSize[.02]}, PlotJoined->True, AxesLabel->{"time [sec]","vel [ft/sec]"}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; Cclosed; noKeepOnOnePage; preserveAspect; startGroup] 3. Plot every 4th, every other, and every data point, and answer the same questions. The slope of each line segment is the average slope of the bicyclist in each interval. :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] We take every 4th data point and plot the position data - both unjoined and joined. :[font = input; noKeepOnOnePage; preserveAspect] sdata4 = Table[data[[i]],{i,1, Length[data],4}]; :[font = input; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[sdata4,PlotStyle->{PointSize[.02]}, PlotRange->{{-5,150},{-50,3000}}, AxesLabel->{"time [sec]","pos [ft]"}] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[sdata4,PlotStyle->{PointSize[.02]}, PlotJoined->True, PlotRange->{{-5,150},{-50,3000}}, AxesLabel->{"time [sec]","pos [ft]"}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] We compute the average velocity over these time intervals and plot the velocity at the right hand side of each sub-interval. :[font = input; noKeepOnOnePage; preserveAspect] vdata4 = Table[{sdata4[[i+1]][[1]], (sdata4[[i+1]][[2]] - sdata4[[i]][[2]])/ (sdata4[[i+1]][[1]] - sdata4[[i]][[1]])}, {i,1, Length[sdata4]-1}]; :[font = input; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[vdata4,PlotStyle->{PointSize[.02]}, AxesLabel->{"time [sec]","vel [ft/sec]"}] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[vdata4,PlotStyle->{PointSize[.02]}, PlotJoined->True, AxesLabel->{"time [sec]","vel [ft/sec]"}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] We take every 2nd data point and plot the position data - both unjoined and joined. :[font = input; noKeepOnOnePage; preserveAspect] sdata2 = Table[data[[i]],{i,1, Length[data],2}]; :[font = input; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[sdata2,PlotStyle->{PointSize[.02]}, PlotRange->{{-5,150},{-50,3000}}, AxesLabel->{"time [sec]","pos [ft]"}] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[sdata2,PlotStyle->{PointSize[.02]}, PlotJoined->True, PlotRange->{{-5,150},{-50,3000}}, AxesLabel->{"time [sec]","pos [ft]"}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] We compute the average velocity over these time intervals and plot the velocity at the right hand side of each sub-interval. :[font = input; noKeepOnOnePage; preserveAspect] vdata2 = Table[{sdata2[[i+1]][[1]], (sdata2[[i+1]][[2]] - sdata2[[i]][[2]])/ (sdata2[[i+1]][[1]] - sdata2[[i]][[1]])}, {i,1, Length[sdata2]-1}]; :[font = input; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[vdata2,PlotStyle->{PointSize[.02]}, AxesLabel->{"time [sec]","vel [ft/sec]"}] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[vdata2,PlotStyle->{PointSize[.02]}, PlotJoined->True, AxesLabel->{"time [sec]","vel [ft/sec]"}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; startGroup] We take every data point and plot the position data - both unjoined and joined. :[font = input; noKeepOnOnePage; preserveAspect] sdataE = Table[data[[i]],{i,1, Length[data]}]; :[font = input; dontNoPageBreakInGroup; noKeepOnOnePage; preserveAspect; startGroup] sE = ListPlot[sdataE,PlotStyle->{PointSize[.02]}, PlotRange->{{-5,150},{-50,3000}}, AxesLabel->{"time [sec]","pos [ft]"}] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; dontNoPageBreakInGroup; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[sdataE,PlotStyle->{PointSize[.02]}, PlotJoined->True, PlotRange->{{-5,150},{-50,3000}}, AxesLabel->{"time [sec]","pos [ft]"}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; startGroup] We compute the average velocity over these time intervals and plot the velocity at the right hand side of each sub-interval. :[font = input; noKeepOnOnePage; preserveAspect] vdataE = Table[{sdataE[[i+1]][[1]], (sdataE[[i+1]][[2]] - sdataE[[i]][[2]])/ (sdataE[[i+1]][[1]] - sdataE[[i]][[1]])}, {i,1, Length[sdataE]-1}]; :[font = input; noKeepOnOnePage; preserveAspect; startGroup] vEPlot = ListPlot[vdataE,PlotStyle->{PointSize[.02]}, AxesLabel->{"time [sec]","vel [ft/sec]"}] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; dontNoPageBreakInGroup; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[vdataE,PlotStyle->{PointSize[.02]}, PlotJoined->True, AxesLabel->{"time [sec]","vel [ft/sec]"}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] 4. Let the line segments that connect the data points be our `distance function'. What would our distance function look like if we took a data point every foot? every inch? if we had an electronic recording device that took observations continuously? :[font = subsubsection; inactive; preserveAspect; endGroup] It would look something very much like our last plot for every data point, getting even smoother. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] 5. What happens to the length of the line segments as we take measurements more frequently? What does the slope of the line segments tell us? If we have continuous observations, what slope (the slope of what line) would give us the same information? :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; endGroup] The fifth question asks the students to apply their budding intuitive understanding of limits to the slope. Especially if the students are working in groups, they are likely to `discover' the tangent line slope as velocity without additional prodding. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] 6. Estimate when the velocity ( in ft/sec and mi/hr) is greatest in this one-half mile cycling trip. :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] We use the plot of velocity based on every data point. :[font = input; noKeepOnOnePage; preserveAspect; startGroup] ListPlot[vdataE, PlotStyle -> { PointSize[.02] }, PlotJoined -> True, AxesLabel -> {"time [sec]", "vel [ft/sec]"}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] From the plot of velocity based on every data point, it would appear that the point (120, 40.5) is the maximum on the velocity vs. time plot. Therefore it looks like the maximum velocity is 40.5 ft/sec which is approximately 27. 6 mi/hr. :[font = input; noKeepOnOnePage; preserveAspect; startGroup] 40.5 ft/sec 3600 sec/hr /(5280 ft/mi) :[font = output; output; inactive; noKeepOnOnePage; preserveAspect; endGroup; endGroup; endGroup] (27.61363636363636*mi)/hr ;[o] 27.6136 mi ---------- hr :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] The next two questions concern data collection. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] 7. How would you decide how many observations to take? :[font = subsubsection; inactive; preserveAspect; endGroup] The goal is to have the student say something along the lines of `Choose the interval so that there is not a large amount of change from one point to the next' or `so that a plot does not look very jagged.' By this time, it is not unreasonable to expect that some students will answer `so that the change in slope between adjacent segments is small.' :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] 8. In some research or industrial settings, observations may be very expensive or dangerous to obtain. Crash testing of automobiles and `weatherbeater' airplanes sent to obtain data about hurricanes are two examples. In these situations, additional factors play a role in determining the number of observations to obtain. Discuss some appropriate issues using these examples and/or some of your own. :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] This question raises some ethical questions in data collection, and is meant to stimulate thought and discussion. :[font = section; inactive; Cclosed; noKeepOnOnePage; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect; endGroup; endGroup] Due to the open-ended nature of the last two questions, before assigning them some consideration should be given as to how the answers will be scored so that both the instructor and the students know what the expectations are. ^*)