(*^ ::[ 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. 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Where are we? Where are we heading? :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsubsection; inactive; preserveAspect; endGroup] FileName: INERTIAL Full title: Inertial Navigation Last Revision Date: 5 June 1996. Developer: Jerry Fine, Department of Mechanical Engineering, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8353. Email: fine@nextwork.rose-hulman.edu FAX: 812-877-3198 Contact: Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Aaron D. 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; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Description of How Inertial Navigation Works :[font = subsubsection; inactive; preserveAspect; startGroup] Inertial navigation measures accelerations. Using the acceleration data, it becomes possible to compute velocity and then position by integration. Inertial navigation systems have computers which perform these calculations. 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FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF grestore end grestore %End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect] The accelerometers are mounted on a gyro-stabilized platform, which remains level at all times and also maintains a constant orientation to true north. 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FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF grestore end grestore %End of Graphics MathPictureEnd :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Your Problem :[font = subsubsection; inactive; preserveAspect] You are at sea on a small inertial-navigation equipped sailing vessel. The inertial's computer (containing the four integrators shown above) breaks down, but the platform itself continues to function and produce acceleration data, which can be read out. You are equipped with a small personal computer. You will try to take the acceleration data and use it to obtain ship's velocity and position. :[font = subsubsection; inactive; preserveAspect] Acceleration Data -- Raw data from accelerometers. :[font = subsubsection; inactive; preserveAspect; startGroup] The cell below contains a set of acceleration data. :[font = input; preserveAspect; endGroup; endGroup] accelx = { {0, 0}, {0.1, -2.01}, {0.2, -3.94}, {0.3, -5.73}, {0.4, -7.31}, {0.5, -8.60}, {0.6, -9.56}, {0.7, -10.15}, {0.8, -10.34}, {0.9, -10.11}, {1., -9.45}, {1.1, -8.38}, {1.2, -6.92}, {1.3, -5.11}, {1.4, -2.99}, {1.5, -0.64}, {1.6, 1.88}, {1.7, 4.50}, {1.8, 7.13}, {1.9, 9.69}, {2., 12.09}, {2.1, 14.25}, {2.2, 16.08}, {2.3, 17.52}, {2.4, 18.51}, {2.5, 18.99}, {2.6, 18.93}, {2.7, 18.30}, {2.8, 17.11}, {2.9, 15.35}, {3.0, 13.06}, {3.1, 10.28}, {3.2, 7.07}, {3.3, 3.50}, {3.4, -0.34}, {3.5, -4.36}, {3.6, -8.46}, {3.7, -12.53}, {3.8, -16.45}, {3.9, -20.11}, {4.0, -23.40}, {4.1, -26.23}, {4.2, -28.49}, {4.3, -30.11}, {4.4, -31.02}, {4.5, -31.17}, {4.6, -30.53}, {4.7, -29.09}, {4.8, -26.85}, {4.9, -23.85}, {5., -20.14} }; accely = { {0, -0.21}, {0.1, -0.19}, {0.2, -0.18}, {0.3, -0.16}, {0.4, -0.15}, {0.5, -0.13}, {0.6, -0.12}, {0.7, -0.11}, {0.8, -0.09}, {0.9, -0.08}, {1., -0.07}, {1.1, -0.05}, {1.2, -0.0}, {1.3, -0.03}, {1.4, -0.02}, {1.5, -0.01}, {1.6, -0.00}, {1.7, 0.008}, {1.8, 0.01}, {1.9, 0.027}, {2., 0.036}, {2.1, 0.045}, {2.2, 0.056}, {2.3, 0.06}, {2.4, 0.06}, {2.5, 0.07}, {2.6, 0.08}, {2.7, 0.08}, {2.8, 0.09}, {2.9, 0.09}, {3.0, 0.10}, {3.1, 0.10}, {3.2, 0.10}, {3.3, 0.11}, {3.4, 0.11}, {3.5, 0.11}, {3.6, 0.11}, {3.7, 0.11}, {3.8, 0.12}, {3.9, 0.12}, {4.0, 0.12}, {4.1, 0.12}, {4.2, 0.12}, {4.3, 0.11}, {4.4, 0.11}, {4.5, 0.11}, {4.6, 0.11}, {4.7, 0.11}, {4.8, 0.11}, {4.9, 0.10}, {5., 0.10} }; :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Quantities to be Calculated :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] Find the x and y coordinates of position of the ship which was located at coordinates (0,1) at time 0. Assume that the ship's initial x-velocity is 3 and its initial y velocity is -0.2. :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Inertial navigation, accelerometer, Numerical integration, position, velocity, acceleration :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; preserveAspect] ISSUES RELATED TO THE PROBLEM :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect; endGroup] The student must know kinematics, in particular how to get from acceleration to velocity to position by integrating. The student must be cognizant of numerical integration techniques such as the rectangle rule, the trapezoidal rule, or Simpson's rule. The student must know enough about programming either a computer algebra system, or perhaps a sophisticated hand-held calculator, or perhaps a spreadsheet, to carry out the numerical integration with actual data values. The programming is modest beginner material, e.g,. looping, lists, and appending values to a list. It would be good if the student were familiar enough with the computing system to produce plots comparing the numerical solution to a true solution given below. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; preserveAspect; endGroup] This could be about a one-hour mini-project for groups. The most difficult part of the project would be programming the numerical integration schemes. It could take up quite a bit of time, particularly if Simpson's rule or even the trapezoidal rule were to be used. To get best advantage, students should try and compare at least two schemes. If interpolation of the acceleration data is used, then it it is possible to solve the problem with finer meshes and observe convergence of the schemes. If this is done, then the problem could wind up taking several hours. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect; endGroup] The student's knowledge of kinematics will be reinforced. The problem has enough of a practical flavor to interest some non-mathematical students in the use of numerical integration. The difficult part will be the actual programming of the methods. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Future payoffs :[font = subsubsection; inactive; preserveAspect; endGroup] The student should acquire more confidence in handling numerical data in various ways. The student will be more capable of implementing numerical algorithms. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect; endGroup] The guidance problem is an obvious extension. It is the inverse of this problem. Suppose that a planned trajectory is described. What accelerations must be programmed into the system to produce the motion? Another extension would be an in-depth comparison of several of the methods used to solve the problem. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] References and Sources :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] Numerical Methods for Engineers by Steven Chapra and Raymond Canale. McGraw Hill, New York, 1988. ISBN 0-07-079984-9 :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; preserveAspect] A number of solutions will now be given. 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.8869 .5764 L .89683 .56335 L .91667 .52969 L .93651 .48635 L .97619 .37389 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] Plot of y position vs time :[font = input; preserveAspect; startGroup] pty = Plot[ yy[t], {t,0,5}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.190476 0.148702 0.454617 [ [(1)] .21429 .1487 0 2 Msboxa [(2)] .40476 .1487 0 2 Msboxa [(3)] .59524 .1487 0 2 Msboxa [(4)] .78571 .1487 0 2 Msboxa [(5)] .97619 .1487 0 2 Msboxa [(-0.2)] .01131 .05778 1 0 Msboxa [(0.2)] .01131 .23963 1 0 Msboxa [(0.4)] .01131 .33055 1 0 Msboxa [(0.6)] .01131 .42147 1 0 Msboxa [(0.8)] .01131 .5124 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .1487 m .21429 .15495 L s P [(1)] .21429 .1487 0 2 Mshowa p .002 w .40476 .1487 m .40476 .15495 L s P [(2)] .40476 .1487 0 2 Mshowa p .002 w .59524 .1487 m .59524 .15495 L s P [(3)] .59524 .1487 0 2 Mshowa p .002 w .78571 .1487 m .78571 .15495 L s P [(4)] .78571 .1487 0 2 Mshowa p .002 w .97619 .1487 m .97619 .15495 L s P [(5)] .97619 .1487 0 2 Mshowa p .001 w .0619 .1487 m .0619 .15245 L s P p .001 w .1 .1487 m .1 .15245 L s P p .001 w .1381 .1487 m .1381 .15245 L s P p .001 w .17619 .1487 m .17619 .15245 L s P p .001 w .25238 .1487 m .25238 .15245 L s P p .001 w .29048 .1487 m .29048 .15245 L s P p .001 w .32857 .1487 m .32857 .15245 L s P p .001 w .36667 .1487 m .36667 .15245 L s P p .001 w .44286 .1487 m .44286 .15245 L s P p .001 w .48095 .1487 m .48095 .15245 L s P p .001 w .51905 .1487 m .51905 .15245 L s P p .001 w .55714 .1487 m .55714 .15245 L s P p .001 w .63333 .1487 m .63333 .15245 L s P p .001 w .67143 .1487 m .67143 .15245 L s P p .001 w .70952 .1487 m .70952 .15245 L s P p .001 w .74762 .1487 m .74762 .15245 L s P p .001 w .82381 .1487 m .82381 .15245 L s P p .001 w .8619 .1487 m .8619 .15245 L s P p .001 w .9 .1487 m .9 .15245 L s P p .001 w .9381 .1487 m .9381 .15245 L s P p .002 w 0 .1487 m 1 .1487 L s P p .002 w .02381 .05778 m .03006 .05778 L s P [(-0.2)] .01131 .05778 1 0 Mshowa p .002 w .02381 .23963 m .03006 .23963 L s P [(0.2)] .01131 .23963 1 0 Mshowa p .002 w .02381 .33055 m .03006 .33055 L s P [(0.4)] .01131 .33055 1 0 Mshowa p .002 w .02381 .42147 m .03006 .42147 L s P [(0.6)] .01131 .42147 1 0 Mshowa p .002 w .02381 .5124 m .03006 .5124 L s P [(0.8)] .01131 .5124 1 0 Mshowa p .002 w .02381 .60332 m .03006 .60332 L s P [(1)] .01131 .60332 1 0 Mshowa p .001 w .02381 .07596 m .02756 .07596 L s P p .001 w .02381 .09415 m .02756 .09415 L s P p .001 w .02381 .11233 m .02756 .11233 L s P p .001 w .02381 .13052 m .02756 .13052 L s P p .001 w .02381 .16689 m .02756 .16689 L s P p .001 w .02381 .18507 m .02756 .18507 L s P p .001 w .02381 .20326 m .02756 .20326 L s P p .001 w .02381 .22144 m .02756 .22144 L s P p .001 w .02381 .25781 m .02756 .25781 L s P p .001 w .02381 .27599 m .02756 .27599 L s P p .001 w .02381 .29418 m .02756 .29418 L s P p .001 w .02381 .31236 m .02756 .31236 L s P p .001 w .02381 .34873 m .02756 .34873 L s P p .001 w .02381 .36692 m .02756 .36692 L s P p .001 w .02381 .3851 m .02756 .3851 L s P p .001 w .02381 .40329 m .02756 .40329 L s P p .001 w .02381 .43966 m .02756 .43966 L s P p .001 w .02381 .45784 m .02756 .45784 L s P p .001 w .02381 .47603 m .02756 .47603 L s P p .001 w .02381 .49421 m .02756 .49421 L s P p .001 w .02381 .53058 m .02756 .53058 L s P p .001 w .02381 .54876 m .02756 .54876 L s P p .001 w .02381 .56695 m .02756 .56695 L s P p .001 w .02381 .58513 m .02756 .58513 L s P p .001 w .02381 .03959 m .02756 .03959 L s P p .001 w .02381 .02141 m .02756 .02141 L s P p .001 w .02381 .00322 m .02756 .00322 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .60332 m .06349 .5824 L .10317 .55793 L .14286 .53047 L .18254 .5006 L .22222 .46888 L .2619 .43583 L .30159 .40196 L .34127 .36774 L .38095 .33361 L .42063 .29999 L .46032 .26722 L .5 .23565 L .53968 .20555 L .57937 .17717 L .61905 .15072 L .65873 .12636 L .69841 .10422 L .7381 .08438 L .77778 .0669 L .81746 .05181 L .85714 .0391 L .89683 .02872 L .93651 .02062 L .97619 .01472 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; preserveAspect; startGroup] Preparation for Numerical Solutions :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Initial conditions :[font = input; preserveAspect; endGroup; endGroup] P0 = {0,1}; V0 = {3,-.2}; :[font = subsection; inactive; preserveAspect; startGroup] Using interpolation capability of CAS for acceleration. :[font = subsubsection; inactive; preserveAspect; startGroup] The following step facilitates numerical integration if you wish to use time steps other than 0.1. It also makes the programming a little easier. :[font = input; preserveAspect; startGroup] ax = Interpolation[accelx] :[font = output; output; inactive; preserveAspect; endGroup] InterpolatingFunction["<<>>"] ;[o] InterpolatingFunction[{0, 5.}, <>] :[font = input; preserveAspect; startGroup] ay = Interpolation[accely] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] InterpolatingFunction["<<>>"] ;[o] InterpolatingFunction[{0, 5.}, <>] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Possible Strategies :[font = subsubsection; inactive; preserveAspect] 1. Euler's Rectangle Method :[font = subsubsection; inactive; preserveAspect] 2. The trapezoidal rule :[font = subsubsection; inactive; preserveAspect; endGroup] 3. Simpson's Rule :[font = subsection; inactive; preserveAspect; startGroup] 1. Euler's Rectangle Method -- with Interpolation :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Time Step :[font = input; preserveAspect; endGroup] dt = .1; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Initial Velocity :[font = input; preserveAspect; endGroup] V1 = {V0}; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Get velocity in loop -- numerical integration -- the rectangle is based on the value of the function at the end of the step. We loop over times, starting with dt and ending with 5. :[font = input; preserveAspect; endGroup] Do[ AppendTo[ V1, Last[V1] + dt { ax[t], ay[t]} ], {t,dt,5,dt}] :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Interpolate Velocity to make things easier :[font = input; preserveAspect; endGroup] vx1 = Interpolation[ Table[{(i-1) dt, V1[[i,1]]}, {i,1,Length[V1]}] ]; vy1 = Interpolation[ Table[{(i-1) dt, V1[[i,2]]}, {i,1,Length[V1]}] ]; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Initialize position :[font = input; preserveAspect; startGroup] P1 = {P0} :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{0, 1}} ;[o] {{0, 1}} :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Get position in loop -- numerical integration :[font = input; preserveAspect; endGroup] Do[ AppendTo[ P1, Last[P1] + dt { vx1[t], vy1[t]} ], {t,dt,5,dt}] :[font = subsubsection; inactive; preserveAspect; startGroup] Comparisons of the solution we have computed with the true solution. :[font = input; preserveAspect; startGroup] Show[ ListPlot[P1], pt] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185; startGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.360073 0.0491577 0.111315 0.492004 [ [(-5)] .11428 .11132 0 2 Msboxa [(5)] .60586 .11132 0 2 Msboxa [(10)] .85165 .11132 0 2 Msboxa [(-0.2)] .34757 .01291 1 0 Msboxa [(0.2)] .34757 .20972 1 0 Msboxa [(0.4)] .34757 .30812 1 0 Msboxa [(0.6)] .34757 .40652 1 0 Msboxa [(0.8)] .34757 .50492 1 0 Msboxa [(1)] .34757 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .11428 .11132 m .11428 .11757 L s P [(-5)] .11428 .11132 0 2 Mshowa p .002 w .60586 .11132 m .60586 .11757 L s P [(5)] .60586 .11132 0 2 Mshowa p .002 w .85165 .11132 m .85165 .11757 L s P [(10)] .85165 .11132 0 2 Mshowa p .001 w .16344 .11132 m .16344 .11507 L s P p .001 w .2126 .11132 m .2126 .11507 L s P p .001 w .26176 .11132 m .26176 .11507 L s P p .001 w .31091 .11132 m .31091 .11507 L s P p .001 w .40923 .11132 m .40923 .11507 L s P p .001 w .45839 .11132 m .45839 .11507 L s P p .001 w .50755 .11132 m .50755 .11507 L s P p .001 w .5567 .11132 m .5567 .11507 L s P p .001 w .65502 .11132 m .65502 .11507 L s P p .001 w .70418 .11132 m .70418 .11507 L s P p .001 w .75333 .11132 m .75333 .11507 L s P p .001 w .80249 .11132 m .80249 .11507 L s P p .001 w .06513 .11132 m .06513 .11507 L s P p .001 w .01597 .11132 m .01597 .11507 L s P p .001 w .90081 .11132 m .90081 .11507 L s P p .001 w .94996 .11132 m .94996 .11507 L s P p .001 w .99912 .11132 m .99912 .11507 L s P p .002 w 0 .11132 m 1 .11132 L s P p .002 w .36007 .01291 m .36632 .01291 L s P [(-0.2)] .34757 .01291 1 0 Mshowa p .002 w .36007 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w .36007 .36716 m .36382 .36716 L s P p .001 w .36007 .38684 m .36382 .38684 L s P p .001 w .36007 .4262 m .36382 .4262 L s P p .001 w .36007 .44588 m .36382 .44588 L s P p .001 w .36007 .46556 m .36382 .46556 L s P p .001 w .36007 .48524 m .36382 .48524 L s P p .001 w .36007 .5246 m .36382 .5246 L s P p .001 w .36007 .54428 m .36382 .54428 L s P p .001 w .36007 .56396 m .36382 .56396 L s P p .001 w .36007 .58364 m .36382 .58364 L s P p .002 w .36007 0 m .36007 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p .008 w .36007 .60332 Mdot .37383 .59254 Mdot .38565 .58088 Mdot .39466 .56844 Mdot .40007 .55525 Mdot .40126 .54142 Mdot .39774 .52701 Mdot .38924 .51205 Mdot .37565 .49665 Mdot .35709 .48086 Mdot .33389 .46472 Mdot .30657 .44834 Mdot .27585 .43195 Mdot .24261 .41542 Mdot .20791 .39879 Mdot .17289 .38211 Mdot .13879 .36544 Mdot .10691 .3488 Mdot .07853 .33221 Mdot .05491 .31575 Mdot .03724 .29947 Mdot .02657 .28341 Mdot .02381 .26762 Mdot .02966 .25214 Mdot .04461 .23694 Mdot .06889 .22209 Mdot .10248 .20764 Mdot .14507 .19358 Mdot .19606 .17996 Mdot .2546 .16678 Mdot .31957 .1541 Mdot .38958 .14191 Mdot .46307 .13021 Mdot .53828 .11905 Mdot .61333 .10843 Mdot .68623 .09836 Mdot .75497 .08882 Mdot .81755 .07983 Mdot .87205 .07142 Mdot .91666 .06361 Mdot .94977 .05639 Mdot .96998 .04976 Mdot .97619 .04371 Mdot .9676 .03821 Mdot .94376 .03325 Mdot .90459 .02884 Mdot .85042 .02496 Mdot .78195 .02162 Mdot .70028 .01883 Mdot .60688 .01653 Mdot .50359 .01472 Mdot P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.356833 0.048684 0.148702 0.454617 [ [(-5)] .11341 .1487 0 2 Msboxa [(5)] .60025 .1487 0 2 Msboxa [(10)] .84367 .1487 0 2 Msboxa [(-0.2)] .34433 .05778 1 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.22053 m .1 .22428 L s P p .001 w .1381 .22053 m .1381 .22428 L s P p .001 w .17619 .22053 m .17619 .22428 L s P p .001 w .25238 .22053 m .25238 .22428 L s P p .001 w .29048 .22053 m .29048 .22428 L s P p .001 w .32857 .22053 m .32857 .22428 L s P p .001 w .36667 .22053 m .36667 .22428 L s P p .001 w .44286 .22053 m .44286 .22428 L s P p .001 w .48095 .22053 m .48095 .22428 L s P p .001 w .51905 .22053 m .51905 .22428 L s P p .001 w .55714 .22053 m .55714 .22428 L s P p .001 w .63333 .22053 m .63333 .22428 L s P p .001 w .67143 .22053 m .67143 .22428 L s P p .001 w .70952 .22053 m .70952 .22428 L s P p .001 w .74762 .22053 m .74762 .22428 L s P p .001 w .82381 .22053 m .82381 .22428 L s P p .001 w .8619 .22053 m .8619 .22428 L s P p .001 w .9 .22053 m .9 .22428 L s P p .001 w .9381 .22053 m .9381 .22428 L s P p .002 w 0 .22053 m 1 .22053 L s P p .002 w .02381 .07009 m .03006 .07009 L s P [(-5)] .01131 .07009 1 0 Mshowa p .002 w .02381 .37098 m .03006 .37098 L s P [(5)] .01131 .37098 1 0 Mshowa p .002 w .02381 .52142 m .03006 .52142 L s P [(10)] .01131 .52142 1 0 Mshowa p .001 w .02381 .10018 m .02756 .10018 L s P p .001 w .02381 .13027 m .02756 .13027 L s P p .001 w .02381 .16036 m .02756 .16036 L s P p .001 w .02381 .19045 m .02756 .19045 L s P p .001 w .02381 .25062 m .02756 .25062 L s P p .001 w .02381 .28071 m .02756 .28071 L s P p .001 w .02381 .3108 m .02756 .3108 L s P p .001 w .02381 .34089 m .02756 .34089 L s P p .001 w .02381 .40107 m .02756 .40107 L s P p .001 w .02381 .43115 m .02756 .43115 L s P p .001 w .02381 .46124 m .02756 .46124 L s P p .001 w .02381 .49133 m .02756 .49133 L s P p .001 w .02381 .04 m .02756 .04 L s P p .001 w .02381 .00992 m .02756 .00992 L s P p .001 w .02381 .55151 m .02756 .55151 L s P p .001 w .02381 .5816 m .02756 .5816 L s P p .001 w .02381 .61168 m .02756 .61168 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .008 w .02381 .22053 Mdot .04286 .22896 Mdot .0619 .23619 Mdot .08095 .2417 Mdot .1 .24502 Mdot .11905 .24574 Mdot .1381 .24359 Mdot .15714 .23839 Mdot .17619 .23007 Mdot .19524 .21871 Mdot .21429 .20451 Mdot .23333 .18779 Mdot .25238 .16898 Mdot .27143 .14864 Mdot .29048 .1274 Mdot .30952 .10596 Mdot .32857 .08509 Mdot .34762 .06558 Mdot .36667 .04821 Mdot .38571 .03375 Mdot .40476 .02294 Mdot .42381 .01641 Mdot .44286 .01472 Mdot .4619 .0183 Mdot .48095 .02745 Mdot .5 .04231 Mdot .51905 .06287 Mdot .5381 .08893 Mdot .55714 .12015 Mdot .57619 .15598 Mdot .59524 .19574 Mdot .61429 .2386 Mdot .63333 .28358 Mdot .65238 .32961 Mdot .67143 .37555 Mdot .69048 .42017 Mdot .70952 .46224 Mdot .72857 .50055 Mdot .74762 .53391 Mdot .76667 .56121 Mdot .78571 .58148 Mdot .80476 .59385 Mdot .82381 .59765 Mdot .84286 .59239 Mdot .8619 .5778 Mdot .88095 .55382 Mdot .9 .52067 Mdot .91905 .47876 Mdot .9381 .42877 Mdot .95714 .3716 Mdot .97619 .30838 Mdot P p p .004 w .02381 .22053 m .06349 .23843 L .08333 .2457 L .09325 .24864 L .10317 .25104 L .1131 .25281 L .11806 .25345 L .12054 .25371 L .12302 .25392 L .1255 .25408 L .12798 .2542 L .12922 .25424 L .13046 .25427 L .1317 .25429 L .13294 .2543 L .13418 .25429 L .13542 .25428 L .13666 .25425 L .1379 .25421 L .14038 .25409 L .14286 .25392 L .14534 .2537 L .14782 .25344 L .15278 .25275 L .15774 .25186 L .1627 .25076 L .17262 .24793 L .18254 .24425 L .20238 .2344 L .22222 .22131 L .2619 .18674 L .30159 .1444 L .34127 .10001 L .36111 .07915 L .38095 .06037 L .40079 .04455 L .42063 .03251 L .43056 .02815 L .43552 .02642 L .44048 .025 L .44296 .02442 L .44544 .02391 L .44792 .02349 L .4504 .02315 L .45164 .02302 L .45288 .0229 L .45412 .02281 L .45536 .02274 L .4566 .02269 L .45784 .02266 L Mistroke .45908 .02265 L .46032 .02267 L .46156 .02271 L .4628 .02277 L .46528 .02295 L .46652 .02308 L .46776 .02323 L .47024 .0236 L .4752 .02462 L .48016 .02601 L .49008 .02992 L .5 .03536 L .50992 .04236 L .51984 .05091 L .53968 .07261 L .57937 .13338 L .61905 .21351 L .65873 .30588 L .69841 .40107 L .7381 .48832 L .75794 .52552 L .77778 .5567 L .79762 .58068 L .80754 .58964 L .8125 .59331 L .81746 .59643 L .82242 .59898 L .8249 .60004 L .82738 .60096 L .82986 .60172 L .83234 .60234 L .83358 .60259 L .83482 .60281 L .83606 .60299 L .8373 .60313 L .83854 .60323 L .83978 .60329 L .84102 .60332 L .84226 .60331 L .8435 .60326 L .84474 .60317 L .84598 .60304 L .84722 .60287 L .8497 .60242 L .85218 .60182 L .85466 .60106 L .85714 .60014 L .8621 .59783 L .86706 .59489 L .87698 .5871 L Mistroke .8869 .57676 L .89683 .56389 L .91667 .53068 L .93651 .48793 L .97619 .37698 L Mfstroke P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 1. Euler's Rectangle Method -- without Interpolation :[font = input; preserveAspect] dt = .1; P1a = {P0}; V1a = {V0}; :[font = input; preserveAspect] Do[ AppendTo[ V1a, Last[V1a] + dt * { accelx[[i,2]], accely[[i,2]] } ], {i,1,Length[accelx]-1} ] :[font = input; preserveAspect] Do[ AppendTo[ P1a, Last[P1a] + dt * V1a[[i]] ], {i,1,Length[V1a]-1} ] :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Comparison with true solution :[font = input; preserveAspect; startGroup] p1a=ListPlot[P1a] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.330578 0.0491577 0.161052 0.442267 [ [(-5)] .08479 .16105 0 2 Msboxa [(5)] .57637 .16105 0 2 Msboxa [(10)] .82215 .16105 0 2 Msboxa [(-0.2)] .31808 .0726 1 0 Msboxa [(0.2)] .31808 .24951 1 0 Msboxa [(0.4)] .31808 .33796 1 0 Msboxa [(0.6)] .31808 .42641 1 0 Msboxa [(0.8)] .31808 .51487 1 0 Msboxa [(1)] .31808 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08479 .16105 m .08479 .1673 L s P [(-5)] .08479 .16105 0 2 Mshowa p .002 w .57637 .16105 m .57637 .1673 L s P [(5)] .57637 .16105 0 2 Mshowa p .002 w .82215 .16105 m .82215 .1673 L s P [(10)] .82215 .16105 0 2 Mshowa p .001 w .13395 .16105 m .13395 .1648 L s P p .001 w .1831 .16105 m .1831 .1648 L s P p .001 w .23226 .16105 m .23226 .1648 L s P p .001 w .28142 .16105 m .28142 .1648 L s P p .001 w .37974 .16105 m .37974 .1648 L s P p .001 w .42889 .16105 m .42889 .1648 L s P p .001 w .47805 .16105 m .47805 .1648 L s P p .001 w .52721 .16105 m .52721 .1648 L s P p .001 w .62552 .16105 m .62552 .1648 L s P p .001 w .67468 .16105 m .67468 .1648 L s P p .001 w .72384 .16105 m .72384 .1648 L s P p .001 w .773 .16105 m .773 .1648 L s P p .001 w .03563 .16105 m .03563 .1648 L s P p .001 w .87131 .16105 m .87131 .1648 L s P p .001 w .92047 .16105 m .92047 .1648 L s P p .001 w .96963 .16105 m .96963 .1648 L s P p .002 w 0 .16105 m 1 .16105 L s P p .002 w .33058 .0726 m .33683 .0726 L s P [(-0.2)] .31808 .0726 1 0 Mshowa p .002 w .33058 .24951 m .33683 .24951 L s P [(0.2)] .31808 .24951 1 0 Mshowa p .002 w .33058 .33796 m .33683 .33796 L s P [(0.4)] .31808 .33796 1 0 Mshowa p .002 w .33058 .42641 m .33683 .42641 L s P [(0.6)] .31808 .42641 1 0 Mshowa p .002 w .33058 .51487 m .33683 .51487 L s P [(0.8)] .31808 .51487 1 0 Mshowa p .002 w .33058 .60332 m .33683 .60332 L s P [(1)] .31808 .60332 1 0 Mshowa p .001 w .33058 .09029 m .33433 .09029 L s P p .001 w .33058 .10798 m .33433 .10798 L s P p .001 w .33058 .12567 m .33433 .12567 L s P p .001 w .33058 .14336 m .33433 .14336 L s P p .001 w .33058 .17874 m .33433 .17874 L s P p .001 w .33058 .19643 m .33433 .19643 L s P p .001 w .33058 .21412 m .33433 .21412 L s P p .001 w .33058 .23181 m .33433 .23181 L s P p .001 w .33058 .2672 m .33433 .2672 L s P p .001 w .33058 .28489 m .33433 .28489 L s P p .001 w .33058 .30258 m .33433 .30258 L s P p .001 w .33058 .32027 m .33433 .32027 L s P p .001 w .33058 .35565 m .33433 .35565 L s P p .001 w .33058 .37334 m .33433 .37334 L s P p .001 w .33058 .39103 m .33433 .39103 L s P p .001 w .33058 .40872 m .33433 .40872 L s P p .001 w .33058 .4441 m .33433 .4441 L s P p .001 w .33058 .46179 m .33433 .46179 L s P p .001 w .33058 .47948 m .33433 .47948 L s P p .001 w .33058 .49717 m .33433 .49717 L s P p .001 w .33058 .53256 m .33433 .53256 L s P p .001 w .33058 .55025 m .33433 .55025 L s P p .001 w .33058 .56794 m .33433 .56794 L s P p .001 w .33058 .58563 m .33433 .58563 L s P p .001 w .33058 .05491 m .33433 .05491 L s P p .001 w .33058 .03722 m .33433 .03722 L s P p .001 w .33058 .01953 m .33433 .01953 L s P p .001 w .33058 .00184 m .33433 .00184 L s P p .002 w .33058 0 m .33058 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p .008 w .33058 .60332 Mdot .34533 .59447 Mdot .36007 .5847 Mdot .37383 .57409 Mdot .38565 .56267 Mdot .39466 .55056 Mdot .40007 .53777 Mdot .40126 .52442 Mdot .39774 .51053 Mdot .38924 .49616 Mdot .37565 .48139 Mdot .35709 .46626 Mdot .33389 .45083 Mdot .30657 .43517 Mdot .27585 .41951 Mdot .24261 .40372 Mdot .20791 .38785 Mdot .17289 .37192 Mdot .13879 .356 Mdot .10691 .34012 Mdot .07853 .32428 Mdot .05491 .30855 Mdot .03724 .29299 Mdot .02657 .27762 Mdot .02381 .26251 Mdot .02966 .24766 Mdot .04461 .23307 Mdot .06889 .21879 Mdot .10248 .20487 Mdot .14507 .1913 Mdot .19606 .17813 Mdot .2546 .16536 Mdot .31957 .15303 Mdot .38958 .14114 Mdot .46307 .1297 Mdot .53828 .11874 Mdot .61333 .10826 Mdot .68623 .09828 Mdot .75497 .08878 Mdot .81755 .07976 Mdot .87205 .07128 Mdot .91666 .06333 Mdot .94977 .05591 Mdot .96998 .04902 Mdot .97619 .04266 Mdot .9676 .03678 Mdot .94376 .0314 Mdot .90459 .0265 Mdot .85042 .02208 Mdot .78195 .01816 Mdot .70028 .01472 Mdot P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Show[ p1a,pt] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.341591 0.0483191 0.161052 0.442267 [ [(-5)] .1 .16105 0 2 Msboxa [(5)] .58319 .16105 0 2 Msboxa [(10)] .82478 .16105 0 2 Msboxa [(-0.2)] .32909 .0726 1 0 Msboxa [(0.2)] .32909 .24951 1 0 Msboxa [(0.4)] .32909 .33796 1 0 Msboxa [(0.6)] .32909 .42641 1 0 Msboxa [(0.8)] .32909 .51487 1 0 Msboxa [(1)] .32909 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .1 .16105 m .1 .1673 L s P [(-5)] .1 .16105 0 2 Mshowa p .002 w .58319 .16105 m .58319 .1673 L s P [(5)] .58319 .16105 0 2 Mshowa p .002 w .82478 .16105 m .82478 .1673 L s P [(10)] .82478 .16105 0 2 Mshowa p .001 w .14831 .16105 m .14831 .1648 L s P p .001 w .19663 .16105 m .19663 .1648 L s P p .001 w .24495 .16105 m .24495 .1648 L s P p .001 w .29327 .16105 m .29327 .1648 L s P p .001 w .38991 .16105 m .38991 .1648 L s P p .001 w .43823 .16105 m .43823 .1648 L s P p .001 w .48655 .16105 m .48655 .1648 L s P p .001 w .53487 .16105 m .53487 .1648 L s P p .001 w .63151 .16105 m .63151 .1648 L s P p .001 w .67983 .16105 m .67983 .1648 L s P p .001 w .72814 .16105 m .72814 .1648 L s P p .001 w .77646 .16105 m .77646 .1648 L s P p .001 w .05168 .16105 m .05168 .1648 L s P p .001 w .00336 .16105 m .00336 .1648 L s P p .001 w .8731 .16105 m .8731 .1648 L s P p .001 w .92142 .16105 m .92142 .1648 L s P p .001 w .96974 .16105 m .96974 .1648 L s P p .002 w 0 .16105 m 1 .16105 L s P p .002 w .34159 .0726 m .34784 .0726 L s P [(-0.2)] .32909 .0726 1 0 Mshowa p .002 w .34159 .24951 m .34784 .24951 L s P [(0.2)] .32909 .24951 1 0 Mshowa p .002 w .34159 .33796 m .34784 .33796 L s P [(0.4)] .32909 .33796 1 0 Mshowa p .002 w .34159 .42641 m .34784 .42641 L s P [(0.6)] .32909 .42641 1 0 Mshowa p .002 w .34159 .51487 m .34784 .51487 L s P [(0.8)] .32909 .51487 1 0 Mshowa p .002 w .34159 .60332 m .34784 .60332 L s P [(1)] .32909 .60332 1 0 Mshowa p .001 w .34159 .09029 m .34534 .09029 L s P p .001 w .34159 .10798 m .34534 .10798 L s P p .001 w .34159 .12567 m .34534 .12567 L s P p .001 w .34159 .14336 m .34534 .14336 L s P p .001 w .34159 .17874 m .34534 .17874 L s P p .001 w .34159 .19643 m .34534 .19643 L s P p .001 w .34159 .21412 m .34534 .21412 L s P p .001 w .34159 .23181 m .34534 .23181 L s P p .001 w .34159 .2672 m .34534 .2672 L s P p .001 w .34159 .28489 m .34534 .28489 L s P p .001 w .34159 .30258 m .34534 .30258 L s P p .001 w .34159 .32027 m .34534 .32027 L s P p .001 w .34159 .35565 m .34534 .35565 L s P p .001 w .34159 .37334 m .34534 .37334 L s P p .001 w .34159 .39103 m .34534 .39103 L s P p .001 w .34159 .40872 m .34534 .40872 L s P p .001 w .34159 .4441 m .34534 .4441 L s P p .001 w .34159 .46179 m .34534 .46179 L s P p .001 w .34159 .47948 m .34534 .47948 L s P p .001 w .34159 .49717 m .34534 .49717 L s P p .001 w .34159 .53256 m .34534 .53256 L s P p .001 w .34159 .55025 m .34534 .55025 L s P p .001 w .34159 .56794 m .34534 .56794 L s P p .001 w .34159 .58563 m .34534 .58563 L s P p .001 w .34159 .05491 m .34534 .05491 L s P p .001 w .34159 .03722 m .34534 .03722 L s P p .001 w .34159 .01953 m .34534 .01953 L s P p .001 w .34159 .00184 m .34534 .00184 L s P p .002 w .34159 0 m .34159 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .008 w .34159 .60332 Mdot .35609 .59447 Mdot .37058 .5847 Mdot .38411 .57409 Mdot .39573 .56267 Mdot .40458 .55056 Mdot .4099 .53777 Mdot .41106 .52442 Mdot .40761 .51053 Mdot .39925 .49616 Mdot .38589 .48139 Mdot .36765 .46626 Mdot .34485 .45083 Mdot .31799 .43517 Mdot .28779 .41951 Mdot .25512 .40372 Mdot .22101 .38785 Mdot .18659 .37192 Mdot .15307 .356 Mdot .12173 .34012 Mdot .09384 .32428 Mdot .07063 .30855 Mdot .05326 .29299 Mdot .04277 .27762 Mdot .04006 .26251 Mdot .04581 .24766 Mdot .0605 .23307 Mdot .08437 .21879 Mdot .11739 .20487 Mdot .15924 .1913 Mdot .20937 .17813 Mdot .26691 .16536 Mdot .33077 .15303 Mdot .39959 .14114 Mdot .47183 .1297 Mdot .54575 .11874 Mdot .61952 .10826 Mdot .69118 .09828 Mdot .75874 .08878 Mdot .82026 .07976 Mdot .87383 .07128 Mdot .91768 .06333 Mdot .95022 .05591 Mdot .97009 .04902 Mdot .97619 .04266 Mdot .96774 .03678 Mdot .94431 .0314 Mdot .90581 .0265 Mdot .85257 .02208 Mdot .78526 .01816 Mdot .70498 .01472 Mdot P p p .004 w .34159 .60332 m .35651 .59361 L .37033 .58297 L .382 .57146 L .38673 .56541 L .39057 .55916 L .39342 .55273 L .39445 .54945 L .39486 .5478 L .3952 .54613 L .39546 .54445 L .39566 .54277 L .39572 .54192 L .39577 .54107 L .3958 .54022 L .39581 .53937 L 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L .95374 .06268 L .95474 .06189 L .95514 .06149 L .95549 .0611 L .95577 .06071 L .956 .06032 L .95616 .05994 L .95626 .05955 L .95631 .05917 L .95629 .05879 L .95621 .05842 L .95606 .05804 L .95586 .05767 L .95559 .0573 L .95487 .05657 L .9539 .05585 L .95268 .05513 L .9512 .05443 L .94749 .05304 L .94277 .05169 L .93026 .0491 L Mistroke .91366 .04664 L .89299 .04433 L .83966 .04012 L .80718 .03822 L .771 .03645 L .68819 .03332 L .59283 .03071 L Mfstroke P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] p1ay = ListPlot[ Table[{(i-1)*dt,P1a[[i,2]]}, {i,1,Length[P1a]}]] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.190476 0.161052 0.442267 [ [(1)] .21429 .16105 0 2 Msboxa [(2)] .40476 .16105 0 2 Msboxa [(3)] .59524 .16105 0 2 Msboxa [(4)] .78571 .16105 0 2 Msboxa [(5)] .97619 .16105 0 2 Msboxa [(-0.2)] .01131 .0726 1 0 Msboxa [(0.2)] .01131 .24951 1 0 Msboxa [(0.4)] .01131 .33796 1 0 Msboxa [(0.6)] .01131 .42641 1 0 Msboxa [(0.8)] .01131 .51487 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .16105 m .21429 .1673 L s P [(1)] .21429 .16105 0 2 Mshowa p .002 w .40476 .16105 m .40476 .1673 L s P [(2)] .40476 .16105 0 2 Mshowa p .002 w .59524 .16105 m .59524 .1673 L s P [(3)] .59524 .16105 0 2 Mshowa p .002 w .78571 .16105 m .78571 .1673 L s P [(4)] .78571 .16105 0 2 Mshowa p .002 w .97619 .16105 m .97619 .1673 L s P [(5)] .97619 .16105 0 2 Mshowa p .001 w .0619 .16105 m .0619 .1648 L s P p .001 w .1 .16105 m .1 .1648 L s P p .001 w .1381 .16105 m .1381 .1648 L s P p .001 w .17619 .16105 m .17619 .1648 L s P p .001 w .25238 .16105 m .25238 .1648 L s P p .001 w .29048 .16105 m .29048 .1648 L s P p .001 w .32857 .16105 m .32857 .1648 L s P p .001 w .36667 .16105 m .36667 .1648 L s P p .001 w .44286 .16105 m .44286 .1648 L s P p .001 w .48095 .16105 m .48095 .1648 L s P p .001 w .51905 .16105 m .51905 .1648 L s P p .001 w .55714 .16105 m .55714 .1648 L s P p .001 w .63333 .16105 m .63333 .1648 L s P p .001 w .67143 .16105 m .67143 .1648 L s P p .001 w .70952 .16105 m .70952 .1648 L s P p .001 w .74762 .16105 m .74762 .1648 L s P p .001 w .82381 .16105 m .82381 .1648 L s P p .001 w .8619 .16105 m .8619 .1648 L s P p .001 w .9 .16105 m .9 .1648 L s P p .001 w .9381 .16105 m .9381 .1648 L s P p .002 w 0 .16105 m 1 .16105 L s P p .002 w .02381 .0726 m .03006 .0726 L s P [(-0.2)] .01131 .0726 1 0 Mshowa p .002 w .02381 .24951 m .03006 .24951 L s P [(0.2)] .01131 .24951 1 0 Mshowa p .002 w .02381 .33796 m .03006 .33796 L s P [(0.4)] .01131 .33796 1 0 Mshowa p .002 w .02381 .42641 m .03006 .42641 L s P [(0.6)] .01131 .42641 1 0 Mshowa p .002 w .02381 .51487 m .03006 .51487 L s P [(0.8)] .01131 .51487 1 0 Mshowa p .002 w .02381 .60332 m .03006 .60332 L s P [(1)] .01131 .60332 1 0 Mshowa p .001 w .02381 .09029 m .02756 .09029 L s P p .001 w .02381 .10798 m .02756 .10798 L s P p .001 w .02381 .12567 m .02756 .12567 L s P p .001 w .02381 .14336 m .02756 .14336 L s P p .001 w .02381 .17874 m .02756 .17874 L s P p .001 w .02381 .19643 m .02756 .19643 L s P p .001 w .02381 .21412 m .02756 .21412 L s P p .001 w .02381 .23181 m .02756 .23181 L s P p .001 w .02381 .2672 m .02756 .2672 L s P p .001 w .02381 .28489 m .02756 .28489 L s P p .001 w .02381 .30258 m .02756 .30258 L s P p .001 w .02381 .32027 m .02756 .32027 L s P p .001 w .02381 .35565 m .02756 .35565 L s P p .001 w .02381 .37334 m .02756 .37334 L s P p .001 w .02381 .39103 m .02756 .39103 L s P p .001 w .02381 .40872 m .02756 .40872 L s P p .001 w .02381 .4441 m .02756 .4441 L s P p .001 w .02381 .46179 m .02756 .46179 L s P p .001 w .02381 .47948 m .02756 .47948 L s P p .001 w .02381 .49717 m .02756 .49717 L s P p .001 w .02381 .53256 m .02756 .53256 L s P p .001 w .02381 .55025 m .02756 .55025 L s P p .001 w .02381 .56794 m .02756 .56794 L s P p .001 w .02381 .58563 m .02756 .58563 L s P p .001 w .02381 .05491 m .02756 .05491 L s P p .001 w .02381 .03722 m .02756 .03722 L s P p .001 w .02381 .01953 m .02756 .01953 L s P p .001 w .02381 .00184 m .02756 .00184 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p .008 w .02381 .60332 Mdot .04286 .59447 Mdot .0619 .5847 Mdot .08095 .57409 Mdot .1 .56267 Mdot .11905 .55056 Mdot .1381 .53777 Mdot .15714 .52442 Mdot .17619 .51053 Mdot .19524 .49616 Mdot .21429 .48139 Mdot .23333 .46626 Mdot .25238 .45083 Mdot .27143 .43517 Mdot .29048 .41951 Mdot .30952 .40372 Mdot .32857 .38785 Mdot .34762 .37192 Mdot .36667 .356 Mdot .38571 .34012 Mdot .40476 .32428 Mdot .42381 .30855 Mdot .44286 .29299 Mdot .4619 .27762 Mdot .48095 .26251 Mdot .5 .24766 Mdot .51905 .23307 Mdot .5381 .21879 Mdot .55714 .20487 Mdot .57619 .1913 Mdot .59524 .17813 Mdot .61429 .16536 Mdot .63333 .15303 Mdot .65238 .14114 Mdot .67143 .1297 Mdot .69048 .11874 Mdot .70952 .10826 Mdot .72857 .09828 Mdot .74762 .08878 Mdot .76667 .07976 Mdot .78571 .07128 Mdot .80476 .06333 Mdot .82381 .05591 Mdot .84286 .04902 Mdot .8619 .04266 Mdot .88095 .03678 Mdot .9 .0314 Mdot .91905 .0265 Mdot .9381 .02208 Mdot .95714 .01816 Mdot .97619 .01472 Mdot P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Show[p1ay,pty] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.190476 0.161052 0.442267 [ [(1)] .21429 .16105 0 2 Msboxa [(2)] .40476 .16105 0 2 Msboxa [(3)] .59524 .16105 0 2 Msboxa [(4)] .78571 .16105 0 2 Msboxa [(5)] .97619 .16105 0 2 Msboxa [(-0.2)] .01131 .0726 1 0 Msboxa [(0.2)] .01131 .24951 1 0 Msboxa [(0.4)] .01131 .33796 1 0 Msboxa [(0.6)] .01131 .42641 1 0 Msboxa [(0.8)] .01131 .51487 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .16105 m .21429 .1673 L s P [(1)] .21429 .16105 0 2 Mshowa p .002 w .40476 .16105 m .40476 .1673 L s P [(2)] .40476 .16105 0 2 Mshowa p .002 w .59524 .16105 m .59524 .1673 L s P [(3)] .59524 .16105 0 2 Mshowa p .002 w .78571 .16105 m .78571 .1673 L s P [(4)] .78571 .16105 0 2 Mshowa p .002 w .97619 .16105 m .97619 .1673 L s P [(5)] .97619 .16105 0 2 Mshowa p .001 w .0619 .16105 m .0619 .1648 L s P p .001 w .1 .16105 m .1 .1648 L s P p .001 w .1381 .16105 m .1381 .1648 L s P p .001 w .17619 .16105 m .17619 .1648 L s P p .001 w .25238 .16105 m .25238 .1648 L s P p .001 w .29048 .16105 m .29048 .1648 L s P p .001 w .32857 .16105 m .32857 .1648 L s P p .001 w .36667 .16105 m .36667 .1648 L s P p .001 w .44286 .16105 m .44286 .1648 L s P p .001 w .48095 .16105 m .48095 .1648 L s P p .001 w .51905 .16105 m .51905 .1648 L s P p .001 w .55714 .16105 m .55714 .1648 L s P p .001 w .63333 .16105 m .63333 .1648 L s P p .001 w .67143 .16105 m .67143 .1648 L s P p .001 w .70952 .16105 m .70952 .1648 L s P p .001 w .74762 .16105 m .74762 .1648 L s P p .001 w .82381 .16105 m .82381 .1648 L s P p .001 w .8619 .16105 m .8619 .1648 L s P p .001 w .9 .16105 m .9 .1648 L s P p .001 w .9381 .16105 m .9381 .1648 L s P p .002 w 0 .16105 m 1 .16105 L s P p .002 w .02381 .0726 m .03006 .0726 L s P [(-0.2)] .01131 .0726 1 0 Mshowa p .002 w .02381 .24951 m .03006 .24951 L s P [(0.2)] .01131 .24951 1 0 Mshowa p .002 w .02381 .33796 m .03006 .33796 L s P [(0.4)] .01131 .33796 1 0 Mshowa p .002 w .02381 .42641 m .03006 .42641 L s P [(0.6)] .01131 .42641 1 0 Mshowa p .002 w .02381 .51487 m .03006 .51487 L s P [(0.8)] .01131 .51487 1 0 Mshowa p .002 w .02381 .60332 m .03006 .60332 L s P [(1)] .01131 .60332 1 0 Mshowa p .001 w .02381 .09029 m .02756 .09029 L s P p .001 w .02381 .10798 m .02756 .10798 L s P p .001 w .02381 .12567 m .02756 .12567 L s P p .001 w .02381 .14336 m .02756 .14336 L s P p .001 w .02381 .17874 m .02756 .17874 L s P p .001 w .02381 .19643 m .02756 .19643 L s P p .001 w .02381 .21412 m .02756 .21412 L s P p .001 w .02381 .23181 m .02756 .23181 L s P p .001 w .02381 .2672 m .02756 .2672 L s P p .001 w .02381 .28489 m .02756 .28489 L s P p .001 w .02381 .30258 m .02756 .30258 L s P p .001 w .02381 .32027 m .02756 .32027 L s P p .001 w .02381 .35565 m .02756 .35565 L s P p .001 w .02381 .37334 m .02756 .37334 L s P p .001 w .02381 .39103 m .02756 .39103 L s P p .001 w .02381 .40872 m .02756 .40872 L s P p .001 w .02381 .4441 m .02756 .4441 L s P p .001 w .02381 .46179 m .02756 .46179 L s P p .001 w .02381 .47948 m .02756 .47948 L s P p .001 w .02381 .49717 m .02756 .49717 L s P p .001 w .02381 .53256 m .02756 .53256 L s P p .001 w .02381 .55025 m .02756 .55025 L s P p .001 w .02381 .56794 m .02756 .56794 L s P p .001 w .02381 .58563 m .02756 .58563 L s P p .001 w .02381 .05491 m .02756 .05491 L s P p .001 w .02381 .03722 m .02756 .03722 L s P p .001 w .02381 .01953 m .02756 .01953 L s P p .001 w .02381 .00184 m .02756 .00184 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .008 w .02381 .60332 Mdot .04286 .59447 Mdot .0619 .5847 Mdot .08095 .57409 Mdot .1 .56267 Mdot .11905 .55056 Mdot .1381 .53777 Mdot .15714 .52442 Mdot .17619 .51053 Mdot .19524 .49616 Mdot .21429 .48139 Mdot .23333 .46626 Mdot .25238 .45083 Mdot .27143 .43517 Mdot .29048 .41951 Mdot .30952 .40372 Mdot .32857 .38785 Mdot .34762 .37192 Mdot .36667 .356 Mdot .38571 .34012 Mdot .40476 .32428 Mdot .42381 .30855 Mdot .44286 .29299 Mdot .4619 .27762 Mdot .48095 .26251 Mdot .5 .24766 Mdot .51905 .23307 Mdot .5381 .21879 Mdot .55714 .20487 Mdot .57619 .1913 Mdot .59524 .17813 Mdot .61429 .16536 Mdot .63333 .15303 Mdot .65238 .14114 Mdot .67143 .1297 Mdot .69048 .11874 Mdot .70952 .10826 Mdot .72857 .09828 Mdot .74762 .08878 Mdot .76667 .07976 Mdot .78571 .07128 Mdot .80476 .06333 Mdot .82381 .05591 Mdot .84286 .04902 Mdot .8619 .04266 Mdot .88095 .03678 Mdot .9 .0314 Mdot .91905 .0265 Mdot .9381 .02208 Mdot .95714 .01816 Mdot .97619 .01472 Mdot P p p .004 w .02381 .60332 m .06349 .58297 L .10317 .55916 L .14286 .53245 L .18254 .50339 L .22222 .47253 L .2619 .44038 L .30159 .40743 L .34127 .37414 L .38095 .34094 L .42063 .30823 L .46032 .27635 L .5 .24564 L .53968 .21636 L .57937 .18875 L .61905 .16302 L .65873 .13932 L .69841 .11777 L .7381 .09848 L .77778 .08148 L .81746 .0668 L .85714 .05443 L .89683 .04433 L .93651 .03645 L .97619 .03071 L s P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; preserveAspect; startGroup] 2. Trapezoidal Rule :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Time Step :[font = input; preserveAspect; endGroup] dt = .1; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Initial Velocity :[font = input; preserveAspect; endGroup] V2 = {V0}; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Get velocity in loop -- numerical integration :[font = input; preserveAspect; endGroup] Do[ AppendTo[ V2, Last[V2] + dt { ( ax[t-dt] + ax[t])/2., ( ay[t-dt] + ay[t])/2.} ], {t,dt,5,dt}] :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Interpolate Velocity to make things easier :[font = input; preserveAspect; endGroup] vx2 = Interpolation[ Table[{(i-1) dt, V2[[i,1]]}, {i,1,Length[V2]}] ]; vy2 = Interpolation[ Table[{(i-1) dt, V2[[i,2]]}, {i,1,Length[V2]}] ]; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Initialize position :[font = input; preserveAspect; startGroup] P2 = {P0} :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{0, 1}} ;[o] {{0, 1}} :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Get position in loop -- numerical integration :[font = input; preserveAspect; endGroup] Do[ AppendTo[ P2, Last[P2] + dt { ( vx2[t-dt] + vx2[t])/2., ( vy2[t-dt] + vy2[t])/2.} ], {t,dt,5,dt}] :[font = subsubsection; inactive; preserveAspect; startGroup] Comparison with True Solution :[font = input; preserveAspect; startGroup] p2=ListPlot[P2] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.345147 0.0494617 0.137793 0.465526 [ [(-5)] .09784 .13779 0 2 Msboxa [(5)] .59246 .13779 0 2 Msboxa [(10)] .83976 .13779 0 2 Msboxa [(-0.2)] .33265 .04469 1 0 Msboxa [(0.2)] .33265 .2309 1 0 Msboxa [(0.4)] .33265 .324 1 0 Msboxa [(0.6)] .33265 .41711 1 0 Msboxa [(0.8)] .33265 .51021 1 0 Msboxa [(1)] .33265 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .09784 .13779 m .09784 .14404 L s P [(-5)] .09784 .13779 0 2 Mshowa p .002 w .59246 .13779 m .59246 .14404 L s P [(5)] .59246 .13779 0 2 Mshowa p .002 w .83976 .13779 m .83976 .14404 L s P [(10)] .83976 .13779 0 2 Mshowa p .001 w .1473 .13779 m .1473 .14154 L s P p .001 w .19676 .13779 m .19676 .14154 L s P p .001 w .24622 .13779 m .24622 .14154 L s P p .001 w .29569 .13779 m .29569 .14154 L s P p .001 w .39461 .13779 m .39461 .14154 L s P p .001 w .44407 .13779 m .44407 .14154 L s P p .001 w .49353 .13779 m .49353 .14154 L s P p .001 w .54299 .13779 m .54299 .14154 L s P p .001 w .64192 .13779 m .64192 .14154 L s P p .001 w .69138 .13779 m .69138 .14154 L s P p .001 w .74084 .13779 m .74084 .14154 L s P p .001 w .7903 .13779 m .7903 .14154 L s P p .001 w .04838 .13779 m .04838 .14154 L s P p .001 w .88923 .13779 m .88923 .14154 L s P p .001 w .93869 .13779 m .93869 .14154 L s P p .001 w .98815 .13779 m .98815 .14154 L s P p .002 w 0 .13779 m 1 .13779 L s P p .002 w .34515 .04469 m .3514 .04469 L s P [(-0.2)] .33265 .04469 1 0 Mshowa p .002 w .34515 .2309 m .3514 .2309 L s P [(0.2)] .33265 .2309 1 0 Mshowa p .002 w .34515 .324 m .3514 .324 L s P [(0.4)] .33265 .324 1 0 Mshowa p .002 w .34515 .41711 m .3514 .41711 L s P [(0.6)] .33265 .41711 1 0 Mshowa p .002 w .34515 .51021 m .3514 .51021 L s P [(0.8)] .33265 .51021 1 0 Mshowa p .002 w .34515 .60332 m .3514 .60332 L s P [(1)] .33265 .60332 1 0 Mshowa p .001 w .34515 .06331 m .3489 .06331 L s P p .001 w .34515 .08193 m .3489 .08193 L s P p .001 w .34515 .10055 m .3489 .10055 L s P p .001 w .34515 .11917 m .3489 .11917 L s P p .001 w .34515 .15641 m .3489 .15641 L s P p .001 w .34515 .17504 m .3489 .17504 L s P p .001 w .34515 .19366 m .3489 .19366 L s P p .001 w .34515 .21228 m .3489 .21228 L s P p .001 w .34515 .24952 m .3489 .24952 L s P p .001 w .34515 .26814 m .3489 .26814 L s P p .001 w .34515 .28676 m .3489 .28676 L s P p .001 w .34515 .30538 m .3489 .30538 L s P p .001 w .34515 .34262 m .3489 .34262 L s P p .001 w .34515 .36125 m .3489 .36125 L s P p .001 w .34515 .37987 m .3489 .37987 L s P p .001 w .34515 .39849 m .3489 .39849 L s P p .001 w .34515 .43573 m .3489 .43573 L s P p .001 w .34515 .45435 m .3489 .45435 L s P p .001 w .34515 .47297 m .3489 .47297 L s P p .001 w .34515 .49159 m .3489 .49159 L s P p .001 w .34515 .52883 m .3489 .52883 L s P p .001 w .34515 .54746 m .3489 .54746 L s P p .001 w .34515 .56608 m .3489 .56608 L s P p .001 w .34515 .5847 m .3489 .5847 L s P p .001 w .34515 .02607 m .3489 .02607 L s P p .001 w .34515 .00745 m .3489 .00745 L s P p .002 w .34515 0 m .34515 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p .008 w .34515 .60332 Mdot .35974 .59354 Mdot .37334 .58287 Mdot .38502 .57137 Mdot .39388 .55912 Mdot .39917 .54618 Mdot .40024 .53262 Mdot .39663 .5185 Mdot .38805 .50388 Mdot .37441 .48883 Mdot .35582 .47341 Mdot .33261 .45768 Mdot .3053 .44175 Mdot .27461 .42572 Mdot .24143 .4096 Mdot .2068 .39339 Mdot .17187 .37713 Mdot .13789 .36087 Mdot .10613 .34464 Mdot .0779 .32847 Mdot .05443 .31242 Mdot .03692 .29654 Mdot .02641 .28087 Mdot .02381 .26545 Mdot .02982 .25031 Mdot .04492 .23546 Mdot .06935 .22093 Mdot .10306 .20676 Mdot .14577 .19298 Mdot .19686 .17961 Mdot .25548 .16666 Mdot .3205 .15417 Mdot .39055 .14215 Mdot .46406 .1306 Mdot .53926 .11955 Mdot .61427 .10902 Mdot .68711 .099 Mdot .75578 .08949 Mdot .81826 .0805 Mdot .87264 .07206 Mdot .91712 .06418 Mdot .95008 .05685 Mdot .97014 .05009 Mdot .97619 .04387 Mdot .96743 .03818 Mdot .94342 .033 Mdot .9041 .02833 Mdot .84977 .02417 Mdot .78115 .02053 Mdot .69935 .01738 Mdot .60583 .01472 Mdot P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Show[pt,p2] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.347759 0.049257 0.148702 0.454617 [ [(-5)] .10147 .1487 0 2 Msboxa [(5)] .59404 .1487 0 2 Msboxa [(10)] .84033 .1487 0 2 Msboxa [(-0.2)] .33526 .05778 1 0 Msboxa [(0.2)] .33526 .23963 1 0 Msboxa [(0.4)] .33526 .33055 1 0 Msboxa [(0.6)] .33526 .42147 1 0 Msboxa [(0.8)] .33526 .5124 1 0 Msboxa [(1)] .33526 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .10147 .1487 m .10147 .15495 L s P [(-5)] .10147 .1487 0 2 Mshowa p .002 w .59404 .1487 m .59404 .15495 L s P [(5)] .59404 .1487 0 2 Mshowa p .002 w .84033 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185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.190476 0.214927 0.0304425 [ [(1)] .21429 .21493 0 2 Msboxa [(2)] .40476 .21493 0 2 Msboxa [(3)] .59524 .21493 0 2 Msboxa [(4)] .78571 .21493 0 2 Msboxa [(5)] .97619 .21493 0 2 Msboxa [(-5)] .01131 .06271 1 0 Msboxa [(5)] .01131 .36714 1 0 Msboxa [(10)] .01131 .51935 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .21493 m .21429 .22118 L s P [(1)] .21429 .21493 0 2 Mshowa p .002 w .40476 .21493 m .40476 .22118 L s P [(2)] .40476 .21493 0 2 Mshowa p .002 w .59524 .21493 m .59524 .22118 L s P [(3)] .59524 .21493 0 2 Mshowa p .002 w .78571 .21493 m .78571 .22118 L s P [(4)] .78571 .21493 0 2 Mshowa p .002 w .97619 .21493 m .97619 .22118 L s P [(5)] .97619 .21493 0 2 Mshowa p .001 w .0619 .21493 m .0619 .21868 L s P p .001 w .1 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0 Mshowa p .002 w .02381 .51935 m .03006 .51935 L s P [(10)] .01131 .51935 1 0 Mshowa p .001 w .02381 .09316 m .02756 .09316 L s P p .001 w .02381 .1236 m .02756 .1236 L s P p .001 w .02381 .15404 m .02756 .15404 L s P p .001 w .02381 .18448 m .02756 .18448 L s P p .001 w .02381 .24537 m .02756 .24537 L s P p .001 w .02381 .27581 m .02756 .27581 L s P p .001 w .02381 .30625 m .02756 .30625 L s P p .001 w .02381 .3367 m .02756 .3367 L s P p .001 w .02381 .39758 m .02756 .39758 L s P p .001 w .02381 .42802 m .02756 .42802 L s P p .001 w .02381 .45847 m .02756 .45847 L s P p .001 w .02381 .48891 m .02756 .48891 L s P p .001 w .02381 .03227 m .02756 .03227 L s P p .001 w .02381 .00183 m .02756 .00183 L s P p .001 w .02381 .54979 m .02756 .54979 L s P p .001 w .02381 .58024 m .02756 .58024 L s P p .001 w .02381 .61068 m .02756 .61068 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .008 w .02381 .21493 Mdot .04286 .22391 Mdot .0619 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Simpson's Rule :[font = input; preserveAspect] dt = .1; :[font = input; preserveAspect] V3 = {V0}; :[font = input; preserveAspect] Do[ AppendTo[ V3, Last[V3] + dt { ( ax[t-dt] + 4. ax[t-dt/2]+ ax[t])/6., ( ay[t-dt] + 4. ay[t-dt/2]+ ay[t])/6.} ], {t,dt,5,dt}] :[font = input; preserveAspect] vx3 = Interpolation[ Table[{(i-1) dt, V3[[i,1]]}, {i,1,Length[V3]}] ]; vy3 = Interpolation[ Table[{(i-1) dt, V3[[i,2]]}, {i,1,Length[V3]}] ]; :[font = input; preserveAspect] P3 = {P0}; :[font = input; preserveAspect] Do[ AppendTo[ P3, Last[P3] + dt { ( vx3[t-dt] + 4. vx3[t-dt/2]+ vx3[t])/6., ( vy3[t-dt] + 4. vy3[t-dt/2]+ vy3[t])/6.} ], {t,dt,5,dt}] :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Comparisons with true solution :[font = input; preserveAspect; startGroup] p3 = ListPlot[P3] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 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.38248 Mdot .13972 .36661 Mdot .10776 .35077 Mdot .07931 .33499 Mdot .05564 .31932 Mdot .03792 .30381 Mdot .02721 .28852 Mdot .02442 .27347 Mdot .03024 .25869 Mdot .04518 .24419 Mdot .06945 .23002 Mdot .10304 .21619 Mdot .14563 .20274 Mdot .19663 .18969 Mdot .25518 .17706 Mdot .32015 .16487 Mdot .39017 .15314 Mdot .46367 .14187 Mdot .53888 .1311 Mdot .61391 .12082 Mdot .68679 .11105 Mdot .7555 .10177 Mdot .81803 .093 Mdot .87247 .08477 Mdot .917 .07709 Mdot .95001 .06995 Mdot .97011 .06336 Mdot .97619 .05731 Mdot .96745 .05176 Mdot .94344 .04672 Mdot .9041 .04217 Mdot .84973 .03812 Mdot .78105 .03458 Mdot .69916 .03153 Mdot .60554 .02893 Mdot P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] p3x = ListPlot[ Table[{(i-1)*dt, P3[[i,1]]}, {i,1,Length[P3]}]] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.190476 0.214912 0.0304972 [ [(1)] .21429 .21491 0 2 Msboxa [(2)] .40476 .21491 0 2 Msboxa [(3)] .59524 .21491 0 2 Msboxa [(4)] .78571 .21491 0 2 Msboxa [(5)] .97619 .21491 0 2 Msboxa [(-5)] .01131 .06243 1 0 Msboxa [(5)] .01131 .3674 1 0 Msboxa [(10)] .01131 .51988 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .21491 m .21429 .22116 L s P [(1)] .21429 .21491 0 2 Mshowa p .002 w .40476 .21491 m .40476 .22116 L s P [(2)] .40476 .21491 0 2 Mshowa p .002 w .59524 .21491 m .59524 .22116 L s P [(3)] .59524 .21491 0 2 Mshowa p .002 w .78571 .21491 m .78571 .22116 L s P [(4)] .78571 .21491 0 2 Mshowa p .002 w .97619 .21491 m .97619 .22116 L s P [(5)] .97619 .21491 0 2 Mshowa p .001 w .0619 .21491 m .0619 .21866 L s P p .001 w .1 .21491 m .1 .21866 L s P p .001 w .1381 .21491 m .1381 .21866 L s P p .001 w .17619 .21491 m .17619 .21866 L s P p .001 w .25238 .21491 m .25238 .21866 L s P p .001 w .29048 .21491 m .29048 .21866 L s P p .001 w .32857 .21491 m .32857 .21866 L s P p .001 w .36667 .21491 m .36667 .21866 L s P p .001 w .44286 .21491 m .44286 .21866 L s P p .001 w .48095 .21491 m .48095 .21866 L s P p .001 w .51905 .21491 m .51905 .21866 L s P p .001 w .55714 .21491 m .55714 .21866 L s P p .001 w .63333 .21491 m .63333 .21866 L s P p .001 w .67143 .21491 m .67143 .21866 L s P p .001 w .70952 .21491 m .70952 .21866 L s P p .001 w .74762 .21491 m .74762 .21866 L s P p .001 w .82381 .21491 m .82381 .21866 L s P p .001 w .8619 .21491 m .8619 .21866 L s P p .001 w .9 .21491 m .9 .21866 L s P p .001 w .9381 .21491 m .9381 .21866 L s P p .002 w 0 .21491 m 1 .21491 L s P p .002 w .02381 .06243 m .03006 .06243 L s P [(-5)] .01131 .06243 1 0 Mshowa p .002 w .02381 .3674 m .03006 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Mdot .04286 .22396 Mdot .0619 .23239 Mdot .08095 .23963 Mdot .1 .24513 Mdot .11905 .2484 Mdot .1381 .24906 Mdot .15714 .24681 Mdot .17619 .24148 Mdot .19524 .233 Mdot .21429 .22146 Mdot .23333 .20704 Mdot .25238 .19007 Mdot .27143 .17101 Mdot .29048 .15039 Mdot .30952 .12887 Mdot .32857 .10716 Mdot .34762 .08602 Mdot .36667 .06626 Mdot .38571 .04866 Mdot .40476 .03402 Mdot .42381 .02307 Mdot .44286 .01644 Mdot .4619 .01472 Mdot .48095 .01832 Mdot .5 .02755 Mdot .51905 .04257 Mdot .5381 .06334 Mdot .55714 .08968 Mdot .57619 .12122 Mdot .59524 .15743 Mdot .61429 .19761 Mdot .63333 .24091 Mdot .65238 .28636 Mdot .67143 .33287 Mdot .69048 .37927 Mdot .70952 .42434 Mdot .72857 .46684 Mdot .74762 .50551 Mdot .76667 .53917 Mdot .78571 .56671 Mdot .80476 .58713 Mdot .82381 .59956 Mdot .84286 .60332 Mdot .8619 .59791 Mdot .88095 .58307 Mdot .9 .55873 Mdot .91905 .52511 Mdot .9381 .48264 Mdot .95714 .432 Mdot .97619 .3741 Mdot P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Show[p3x,ptx] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.190476 0.215159 0.0304778 [ [(1)] .21429 .21516 0 2 Msboxa [(2)] .40476 .21516 0 2 Msboxa [(3)] .59524 .21516 0 2 Msboxa [(4)] .78571 .21516 0 2 Msboxa [(5)] .97619 .21516 0 2 Msboxa [(-5)] .01131 .06277 1 0 Msboxa [(5)] .01131 .36755 1 0 Msboxa [(10)] .01131 .51994 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .21516 m .21429 .22141 L s P [(1)] .21429 .21516 0 2 Mshowa p .002 w .40476 .21516 m .40476 .22141 L s P [(2)] .40476 .21516 0 2 Mshowa p .002 w .59524 .21516 m 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-Graphics- :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect; endGroup; endGroup] The programming of the solutions requires what one might call "advanced beginner" skills. The main question in the solution is whether the computer programming detracts from learning the mathematics, or whether it enhances. The opinion of the problem's author is that the latter case is true. ^*)