(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 256085, 7391]*) (*NotebookOutlinePosition[ 256720, 7413]*) (* CellTagsIndexPosition[ 256676, 7409]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Mathematica Lab Exercise #2", "Subtitle"], Cell[TextData[{ "In this lab we're going to solve equations of motion for some simple \ problems and plot the results.I begin the exercise by solving for motion of a \ projectile with constant acceleration (e.g., approximate motion on the \ surface of a planet).\n\nThe equations of motion are:\n\n \ ", StyleBox[" m ", Background->RGBColor[0, 1, 1]], Cell[BoxData[ FormBox[ StyleBox[\(\(\(d\^2\) x\)\/dt\^2\), FontSize->14], TraditionalForm]], Background->RGBColor[0, 1, 1]], StyleBox["= 0\n\n", Background->RGBColor[0, 1, 1]], " ", StyleBox["m ", Background->RGBColor[0, 1, 1]], Cell[BoxData[ FormBox[ StyleBox[\(\(\(d\^2\) y\)\/dt\^2\), FontSize->14], TraditionalForm]], Background->RGBColor[0, 1, 1]], StyleBox["= -mg,", Background->RGBColor[0, 1, 1]], "\n \nand we need to specify ", Cell[BoxData[ \(TraditionalForm\`x\_0, \ v\_x0\)]], ", ", Cell[BoxData[ \(TraditionalForm\`y\_0\)]], " and ", Cell[BoxData[ \(TraditionalForm\`v\_y0\)]], ". There are many ways to solve this problem in ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell["\<\ Let's start by using the analytic solution returned by \ DSolve.\ \>", "Text"], Cell[BoxData[ \(ClearAll["\"]\)], "Input"], Cell["\<\ Start with this invocation to clear any variables that might have \ been previously defined. Then use DSolve.\ \>", "Text"], Cell[BoxData[ \(sol[x0_, vx0_, y0_, vy0_] := \ DSolve[{\(x'\)[t] \[Equal] vx[t], \ \(vx'\)[t]\ \[Equal] \ 0, \ \(y'\)[t] == vy[t], \ \(vy'\)[t]\ \[Equal] \ \(-g\), x[0] == x0, vx[0] == vx0, y[0] == y0, vy[0] == vy0}, {x[t], vx[t], y[t], vy[t]}, t]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(sol[x0, vx0, y0, vy0]\)], "Input"], Cell[BoxData[ \({{x[t] \[Rule] t\ vx0 + x0, vx[t] \[Rule] vx0, y[t] \[Rule] 1\/2\ \((\(-g\)\ t\^2 + 2\ t\ vy0 + 2\ y0)\), vy[t] \[Rule] \(-g\)\ t + vy0}}\)], "Output"] }, Open ]], Cell["\<\ So DSolve returns a nested list of substitution rules. Let's make a \ parametric plot for the situation where a cannon is fired from a cliff, like \ the FCI problem. 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0ol00?ooooooo`_ooooo00<00000ooooooooool02_ooool002Gooooo00?o0000ool00?ooool0Jooo ool2o`00oooooooo3_ooool00`00003oooooooooo`0:ooooo`009Oooool00ol0003oo`00ooooo`1Y ooooo`;o003oooooool@ooooo`0300000?oooooooooo00[ooooo000Uooooo`03o`000?oo003ooooo 06Oooooo0_l00?oooooooa3ooooo0`00000"], ImageRangeCache->{{{0, 430.375}, {265.625, 0}} -> {-490.308, -10.9454, \ 13.1293, 0.424874}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ Problem 1. Set up the equations of motion when there is friction \ proportional to velocity-squared. Write the equations and some explanatory \ text in a text box. Use Format \[Rule] Style to select this as a Text box and \ then use Format\[Rule] Background Color to put a gray box around your text.\ \ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[TextData[{ "The equations of motion for a particle moving in a constant gravitational \ field and experiencing friction proportional to the square of the velocity \ are:\n\n\t", Cell[BoxData[ \(TraditionalForm\`\(\(d\^2\) x\)\/dt\^2\)]], "= -b ", Cell[BoxData[ \(TraditionalForm\`\@\(v\_x\^2 + v\_y\^2\)\)]], Cell[BoxData[ \(TraditionalForm\`v\_x\)]], ",\n\t\n\t", Cell[BoxData[ \(TraditionalForm\`\(\(d\^2\) y\)\/dt\^2\)]], "= -b ", Cell[BoxData[ \(TraditionalForm\`\@\(v\_x\^2 + v\_y\^2\)\)]], Cell[BoxData[ \(TraditionalForm\`v\_y\)]], "-g.\n\t\nWe have divided by the mass everywhere. We must specify \ gravitational acceleration, g, the initial position and the initial \ velocity." }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell["\<\ Problem 2. Following the example above use NDSolve to define a \ solution with friction. Note that the format for NDSolve is not identical to \ that for DSolve.\ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[BoxData[ \(ClearAll["\"]\)], "Input"], Cell[BoxData[ \(frictionsol[x0_, vx0_, y0_, vy0_, b_, g_, tmax_] := \ NDSolve[{\(x'\)[t] \[Equal] vx[t], \ \(vx'\)[t]\ \[Equal] \ \(-b\)*Sqrt[vx[t]^2 + vy[t]^2]* vx[t], \ \(y'\)[t] == vy[t], \ \(vy'\)[t]\ \[Equal] \ \(-g\) - b*Sqrt[vx[t]^2 + vy[t]^2]*vy[t], x[0] == x0, vx[0] == vx0, y[0] == y0, vy[0] == vy0}, {x, y, vx, vy}, {t, 0, tmax}]\)], "Input"], Cell["\<\ Problem 3. Use ParametricPlot to produce trajectories. 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