BATTERUP BRIEF ABSTRACT This is an application for definite integrals. The problem uses data which leaves a good reason to use a numerical method such as the trapezoid rule. GENERAL INFORMATION FileName: BatterUp Full title: Batter Up: The Physics of Power in Baseball Last Update: 5/23/96 Developers: Aaron D. Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Sandra K. Dawson, Mathematics, Glenbrook South High School, Glenview, IL 60025 USA Rosaline Secrest, Physics, Terre Haute South Vigo High School, Terre Haute, IN 47802 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. STATEMENT OF PROBLEM When a batter comes to the plate in baseball and swings at a pitch, the swing usually takes 0.2 seconds to reach the plate. During this time, the batter is transferring energy to the bat. The rate at which the energy is transferred to the bat increases from 0 to 9 horsepower during the first 0.15 seconds and then decreases to zero as the bat comes across the plate. (Note: one horsepower equals 746 joules/second.) The following data was collected for a particular swing: Time Power 0 0.0 .025 0.8 .05 2.0 .075 3.0 .1 3.0 .125 7.5 .15 9.0 .175 3.0 .2 0.0 1. Connect the data points in a piecewise linear graph on a Cartesian coordinate system with power (in horsepower) versus time (in seconds). 2. Using your graph, predict the time interval in which the power is increasing the fastest. 3. Use the trapezoid rule to estimate the area under the graph which represents the total energy transferred to the bat in a given time interval. Be certain to label the units of measure for total energy transferred. 4. Find a polynomial function which fits this data. 5. Based on your polynomial, at what instant is the rate of energy transfer increasing the fastest? 6. Based on your polynomial, find a new function which represents the cumulative energy transfer as a function of time and plot the graph. Note: if you plot ten times the cumulative energy transfer, you can plot both graphs on the same set of axes (understanding that the vertical scale represents different quantities.) KEYWORDS Curve fitting, differentiation, trapezoid rule, power, energy. TEACHER NOTES ISSUES RELATED TO THE PROBLEM While this problem entails curve fitting, we leave it up to the instructor to decide whether the students can use their computer software to perform curve fits or if they must derive the formulas for themselves. Prerequisites A knowledge of distance = rate * time, accumulation concept which leads to the integral. Interpretations of the definite integral. Some experience with numerical integration. Some experience with curve fitting (at least with computer software). Time allotment - time management Expectations Future payoffs Extensions There are numerous ways to fit curves through the data: splines and polynomials of various orders are only starters. Ask students what shape they expect the power function to be, and have them fit a function of that shape. This, by the way will force them away from typical packaged curve fitting software. It will also give them the opportunity to familiarize themselves with how functions change shape as a function of their parameters. References and Sources Adair, Robert K.,"The Physics of Baseball", Physics Today, p. 26-31, May, 1995 POSSIBLE SOLUTION(S) 1. Piecewise linear graph of the data. In the proposed solution, we have created a list, a graphic for the enlarged points, and a second graphic for the connecting line segments. Finally, we show the graphs on the same set of axes. thePoints={{0, 0}, {.025, .8},{.05, 2}, {.075, 3}, {.1, 3},{.125, 7.5}, {.15, 9}, {.175, 3}, {.2, 0}}; s=ListPlot[thePoints, PlotStyle->PointSize[1/40], DisplayFunction->Identity, AxesLabel -> {TIME, POWER}, PlotLabel -> BATTING POWER]; t=ListPlot[thePoints, PlotJoined->True, DisplayFunction->Identity]; In[6]:= Show[{s,t}, DisplayFunction->$DisplayFunction] Out[6]= -Graphics- 2. We are looking for the time at which the rate of transfer of energy to the bat is increasing at the fastest. From the graph, it appears that the largest rate of increase is between t = .1 seconds and t = .125 seconds. 3. Use the trapezoid rule to estimate the area under the graph which represents the total energy transferred to the bat in a given time interval. Be certain to label the units of measure for total energy transferred. Below is the usual formula for the Trapezoid Rule where h = (b - a)/n is the step size and the function f(x) is approximated over n equally spaced subintervals inside the interval [a, b]. We first input the data set. In[125]:= n = Length[thePoints] - 1 a = thePoints[[1, 1]] b = thePoints[[n+1, 1]] y0 = thePoints[[1, 2]] y = Table[thePoints[[i, 2]], {i, 2, n}] yn = thePoints[[n+1, 2]] h = (b - a)/n Out[125]= 8 Out[126]= 0 Out[127]= 0.2 Out[128]= 0 Out[129]= {0.8, 2, 3, 3, 7.5, 9, 3} Out[130]= 0 Out[131]= 0.025 In[133]:= TrapezoidRule = (h/2) (y0 + 2 Sum[y[[i]], {i, 1, n-1}] + yn) Out[133]= 0.7075 Accounting for units, the Total Energy is approximately 0.7075 hp seconds. 4. Find a polynomial function which fits this data. After experimenting with various polynomials, a seventh degree polynomial gives a reasonable shape for the curve including extreme values and concavity. In[134]:= f[x_]= Fit[thePoints,{1,x,x^2,x^3,x^4,x^5,x^6,x^7}, x] Out[134]= 2 3 0.00780109 + 2.18079 x - 393.385 x + 127454. x - 6 4 7 5 8 6 3.64084 10 x + 4.07979 10 x - 1.99528 10 x + 8 7 3.54337 10 x In[138]:= fPlot = Plot[f[x],{x,0,.19}, AxesLabel -> {TIME, POWER}, PlotLabel -> BattingPower, DisplayFunction -> Identity]; In[139]:= Show[fPlot, s, DisplayFunction -> $DisplayFunction] Out[139]= -Graphics- 5. The energy transfer is increasing at the fastest rate when its change (derivative) is maximized. That happens when the second derivative is zero while the first derivative is positive. In[149]:= sol = Solve[f''[x] == 0, x] Out[149]= {{x -> 0.00109607}, {x -> 0.0315948}, {x -> 0.0707862}, {x -> 0.122078}, {x -> 0.176662}} In[152]:= xs = {x /. sol[[1, 1]], x /. sol[[2, 1]], x /. sol[[3, 1]], x /. sol[[4, 1]], x /. sol[[5, 1]]} Out[152]= {0.00109607, 0.0315948, 0.0707862, 0.122078, 0.176662} In[154]:= {f'[xs[[1]]], f'[xs[[2]]], f'[xs[[3]]], f'[xs[[4]]], f'[xs[[5]]]} Out[154]= {1.75891, 67.7416, 2.89308, 165.359, -408.922} Of the candidates, we see that when x = 0.122078, that f(x) is increasing the fastest at a rate of 165.359 hp/second. 6. The cumulative energy transfer is found by summing the power*time so the desired function which we call g[x] is the integral of x from time t=0 to t=x. g[x_] = Integrate[f[t], {t, 0, x}]; Plot[{f[x], 10 g[x]}, {x, 0, .19}, AxesLabel->{TIME,POWER/ENERGY}, PlotLabel->BATTING POWER with ENERGY [x10]]; ISSUES IN SOLUTION We used Mathematica's Fit[] function to perform a least squares fit. The problem is much more substantial if you require the students to find the best fitting curves by working a least squares problem.