(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 20921, 612] NotebookOptionsPosition[ 19123, 548] NotebookOutlinePosition[ 19463, 563] CellTagsIndexPosition[ 19420, 560] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell["\<\ Per Morin 3.53 we fire a projectile with intial speed v0 at angle \[Theta], \ including the effects both of gravity and of air resistance:\ \>", "Text", CellChangeTimes->{{3.433851572436*^9, 3.433851574736*^9}, {3.433851680567*^9, 3.4338517209230003`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sx", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "''"}], "[", "t", "]"}], "\[Equal]", " ", RowBox[{ RowBox[{"-", " ", 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Defining a \"rule\" ...\ \>", "Text", CellChangeTimes->{{3.4338518157530003`*^9, 3.433851908749*^9}, { 3.4338519471470003`*^9, 3.4338519753929996`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Alpha]rule", " ", "=", " ", RowBox[{"\[Alpha]", " ", "\[Rule]", " ", RowBox[{"g", "/", "v0"}]}]}]], "Input", CellChangeTimes->{{3.433851937551*^9, 3.4338519629189997`*^9}, { 3.433852513731*^9, 3.433852515157*^9}}], Cell[BoxData[ RowBox[{"\[Alpha]", "\[Rule]", FractionBox["g", "v0"]}]], "Output", CellChangeTimes->{3.4338527971289997`*^9}] }, Open ]], Cell["\<\ ... we then ask for the time at which we peak, i.e. when y[t] is a maximum:\ \>", "Text", CellChangeTimes->{{3.433851970699*^9, 3.433852001882*^9}, { 3.4338521994639997`*^9, 3.433852208013*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"st", " ", "=", " ", RowBox[{ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{"y", "[", "t", "]"}], "/.", " ", "sy"}], ",", "t"}], "]"}], "\[Equal]", "0"}], ",", "t"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.433852213264*^9, 3.433852251334*^9}, { 3.4338522945810003`*^9, 3.43385229798*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Solve", "::", "\<\"ifun\"\>"}], RowBox[{ ":", " "}], "\<\"Inverse functions are being used by \\!\\(Solve\\), so \ some solutions may not be found; use Reduce for complete solution \ information. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", \ ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ ButtonData:>\\\"paclet:ref/message/Solve/ifun\\\", ButtonNote -> \ \\\"Solve::ifun\\\"]\\)\"\>"}]], "Message", "MSG", CellChangeTimes->{3.433852797281*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"t", "\[Rule]", FractionBox[ RowBox[{"Log", "[", FractionBox[ RowBox[{"g", "+", RowBox[{"v0", " ", "\[Alpha]", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}], "g"], "]"}], "\[Alpha]"]}], "}"}]], "Output", CellChangeTimes->{3.433852797303*^9}] }, Open ]], Cell["What is the x-range as we peak?", "Text", CellChangeTimes->{{3.433852281308*^9, 3.4338522876879997`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"d", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "[", "t", "]"}], "/.", "sx"}], "/.", "st"}], " ", "//", "Simplify"}]}]], "Input", CellChangeTimes->{{3.433852289329*^9, 3.4338523187869997`*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ SuperscriptBox["v0", "2"], " ", RowBox[{"Cos", "[", "\[Theta]", "]"}], " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}], RowBox[{"g", "+", RowBox[{"v0", " ", "\[Alpha]", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}]]], "Output", CellChangeTimes->{3.433852797333*^9}] }, Open ]], Cell["\<\ For the special value of \[Alpha] in this problem:\ \>", "Text", CellChangeTimes->{{3.4338524457060003`*^9, 3.4338524619309998`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"d", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "[", "t", "]"}], "/.", "sx"}], "/.", "st"}], " ", "/.", "\[Alpha]rule"}], "//", "Simplify"}]}]], "Input", CellChangeTimes->{{3.4338524689*^9, 3.433852482582*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ SuperscriptBox["v0", "2"], " ", RowBox[{"Cos", "[", "\[Theta]", "]"}], " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}], RowBox[{"g", "+", RowBox[{"g", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}]]], "Output", CellChangeTimes->{3.433852797363*^9}] }, Open ]], Cell["And finally, the optimal angle is given by", "Text", CellChangeTimes->{{3.433852347771*^9, 3.433852357742*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Theta]rule", " ", "=", " ", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"D", "[", RowBox[{"d", ",", "\[Theta]"}], "]"}], "\[Equal]", "0"}], ",", "\[Theta]"}], "]"}]}]], "Input", CellChangeTimes->{{3.433852414236*^9, 3.433852423892*^9}, { 3.4338526361359997`*^9, 3.433852639701*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Solve", "::", "\<\"verif\"\>"}], RowBox[{ ":", " "}], "\<\"Potential solution \\!\\({\[Theta] \[Rule] \\(\\(-\\(\\(\ \[Pi]\\/2\\)\\)\\)\\)}\\) (possibly discarded by verifier) should be checked \ by hand. May require use of limits. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\ \", ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ ButtonData:>\\\"paclet:ref/message/Solve/verif\\\", ButtonNote -> \ \\\"Solve::verif\\\"]\\)\"\>"}]], "Message", "MSG", CellChangeTimes->{3.433852797441*^9}], Cell[BoxData[ RowBox[{ RowBox[{"Solve", "::", "\<\"ifun\"\>"}], RowBox[{ ":", " "}], "\<\"Inverse functions are being used by \\!\\(Solve\\), so \ some solutions may not be found; use Reduce for complete solution \ information. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", \ ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ ButtonData:>\\\"paclet:ref/message/Solve/ifun\\\", ButtonNote -> \ \\\"Solve::ifun\\\"]\\)\"\>"}]], "Message", "MSG", CellChangeTimes->{3.4338527974639997`*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"\[Theta]", "\[Rule]", RowBox[{"ArcCos", "[", RowBox[{"-", SqrtBox[ RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SqrtBox["5"]}], ")"}]}]]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"\[Theta]", "\[Rule]", RowBox[{"ArcCos", "[", SqrtBox[ RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SqrtBox["5"]}], ")"}]}]], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"\[Theta]", "\[Rule]", RowBox[{"-", RowBox[{"ArcCos", "[", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", SqrtBox[ RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"1", "+", SqrtBox["5"]}], ")"}]}]]}], "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"\[Theta]", "\[Rule]", RowBox[{"-", RowBox[{"ArcCos", "[", RowBox[{"\[ImaginaryI]", " ", SqrtBox[ RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"1", "+", SqrtBox["5"]}], ")"}]}]]}], "]"}]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.433852797488*^9}] }, Open ]], Cell["And as advertized, the solutions have the following cosines:", "Text", CellChangeTimes->{{3.433852647028*^9, 3.433852703602*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Cos", "[", "\[Theta]", "]"}], "/.", "\[Theta]rule"}]], "Input", CellChangeTimes->{{3.433852672212*^9, 3.433852685909*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", SqrtBox[ RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SqrtBox["5"]}], ")"}]}]]}], ",", SqrtBox[ RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SqrtBox["5"]}], ")"}]}]], ",", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", SqrtBox[ RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"1", "+", SqrtBox["5"]}], ")"}]}]]}], ",", RowBox[{"\[ImaginaryI]", " ", SqrtBox[ RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"1", "+", SqrtBox["5"]}], ")"}]}]]}]}], "}"}]], "Output", CellChangeTimes->{3.4338527975290003`*^9}] }, Open ]], Cell["of which the first two are plus/minus the Golden Ratio.", "Text", CellChangeTimes->{{3.433852716991*^9, 3.433852729311*^9}}] }, WindowSize->{769, 517}, WindowMargins->{{44, Automatic}, {Automatic, 56}}, FrontEndVersion->"6.0 for Microsoft Windows (32-bit) (May 21, 2008)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[568, 21, 270, 5, 29, "Text"], Cell[CellGroupData[{ Cell[863, 30, 799, 23, 31, "Input"], Cell[1665, 55, 517, 16, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2219, 76, 845, 25, 31, "Input"], Cell[3067, 103, 971, 26, 51, "Output"] }, Open ]], Cell[4053, 132, 195, 4, 29, "Text"], Cell[CellGroupData[{ Cell[4273, 140, 374, 9, 31, "Input"], Cell[4650, 151, 301, 8, 43, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4988, 164, 423, 13, 31, "Input"], Cell[5414, 179, 5767, 101, 218, "Output"] }, Open ]], Cell[11196, 283, 274, 6, 47, "Text"], Cell[CellGroupData[{ Cell[11495, 293, 249, 5, 31, "Input"], Cell[11747, 300, 129, 3, 43, "Output"] }, Open ]], Cell[11891, 306, 207, 4, 29, "Text"], Cell[CellGroupData[{ Cell[12123, 314, 448, 13, 31, "Input"], Cell[12574, 329, 487, 10, 39, "Message"], Cell[13064, 341, 322, 10, 57, "Output"] }, Open ]], Cell[13401, 354, 112, 1, 29, "Text"], Cell[CellGroupData[{ Cell[13538, 359, 248, 7, 31, "Input"], Cell[13789, 368, 326, 9, 50, "Output"] }, Open ]], Cell[14130, 380, 144, 3, 29, "Text"], Cell[CellGroupData[{ Cell[14299, 387, 281, 8, 31, "Input"], Cell[14583, 397, 308, 9, 50, "Output"] }, Open ]], Cell[14906, 409, 118, 1, 29, "Text"], Cell[CellGroupData[{ Cell[15049, 414, 343, 9, 31, "Input"], Cell[15395, 425, 525, 10, 54, "Message"], Cell[15923, 437, 492, 10, 39, "Message"], Cell[16418, 449, 1377, 47, 99, "Output"] }, Open ]], Cell[17810, 499, 136, 1, 29, "Text"], Cell[CellGroupData[{ Cell[17971, 504, 158, 3, 31, "Input"], Cell[18132, 509, 841, 33, 54, "Output"] }, Open ]], Cell[18988, 545, 131, 1, 29, "Text"] } ] *) (* End of internal cache information *)