(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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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", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" has a single command, ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["ParametricPlot", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", which will do all of this for us.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "plot3 = ParametricPlot[{f[t], g[t]}, {t,0,16},\n \ PlotStyle->Green]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Use the command below to see that ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["ParametricPlot", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" does what it should.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Show[plot1,plot2,plot3]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Now use the same parametric equations in ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["ParametricPlot", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", but change the interval for the parameter to 0 \[LessEqual] ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ " \[LessEqual] 16. 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If a projectile is fired from ground level with initial \ velocity ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["v", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" at an angle of inclination ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["\[CapitalTheta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[", then its position after ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" seconds of flight is", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica 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Define these as our new ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[":", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "f[t_] := v0 Cos[theta] t;\ng[t_] := -16 t^2 + v0 Sin[theta] t\n"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["and choose particular values for ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["v", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["\[CapitalTheta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[":", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["v0 = 800;\ntheta = Pi/6;"], "Input", AspectRatioFixed->True], Cell[TextData[ "Now we can plot the trajectory of the projectile for 15 seconds, say, with \ the command"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["ParametricPlot[{f[t], g[t]}, {t,0,15}]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["For each of the angles ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["\[CapitalTheta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" = \[Pi]/6, ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["\[CapitalTheta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" = \[Pi]/4, and ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["\[CapitalTheta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " = \[Pi]/3, plot the trajectory of the projectile until it returns to the \ ground, and display all three trajectories on one plot. Print and label a \ copy and hand it in. For which of these values of ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["\[CapitalTheta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " does the projectile travel furthest horizontally? (Hint: The projectile \ hits ground when ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[") = 0 at some ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" > 0. Find the appropriate ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" value\[LongDash]it will be a function of ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["v", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["\[CapitalTheta]", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[LongDash]and make it the end of the ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" interval in your plot.) 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