(* Content-type: application/vnd.wolfram.mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 9.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       157,          7]
NotebookDataLength[    110449,       2106]
NotebookOptionsPosition[    108218,       2028]
NotebookOutlinePosition[    108562,       2043]
CellTagsIndexPosition[    108519,       2040]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["Catapult Problem", "Title",
 CellChangeTimes->{{3.5770432430825787`*^9, 3.57704324680157*^9}}],

Cell["\<\
A medieval city has the shape of a square and is protected by walls with \
length 500 meters and height 15 meters.You are the commander of an attacking \
army and the closest you can get to the wall is 100 meters.Your plan is to \
set fire to the city by catapulting heated rocks over the wall with an \
initial speed of 80 meters per second.At what range of angles should you tell \
your men to set the catapult?Assume that you will be firing from exactly 100 \
meters from the wall and that you will be firing perpendicular to the wall.We \
also assume that the walls of the city are impervious to heated rocks.\
\>", "Text",
 CellChangeTimes->{3.577043269224084*^9}],

Cell[BoxData[
 GraphicsBox[
  TagBox[RasterBox[CompressedData["
1:eJzs3QdcU9cCBnD31qp1W+ustVqx1lZrbavY2ldHS7XD2tLaSbXaau0Sq8he
[encoded data, 85766 characters]
AAAAAAAAAAAAAAAS5v/Xd11H
    "], {{0, 580}, {875, 0}}, {0, 255},
    ColorFunction->RGBColor],
   BoxForm`ImageTag["Byte", ColorSpace -> ColorProfileData[CompressedData["
1:eJyVlwdQU+kWx797b3qhBSIgJfTepUsJPRRBOohKSAKEEkJCaHZFXMEVRUQE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      "], "RGB", 
      "XYZ"], Interleaving -> True],
   Selectable->False],
  BaseStyle->"ImageGraphics",
  ImageSizeRaw->{875, 580},
  PlotRange->{{0, 875}, {0, 580}}]], "Input"],

Cell[BoxData[""], "Input",
 CellChangeTimes->{{3.577043260079913*^9, 3.577043264425056*^9}}],

Cell["\<\
The projectile sometimes hits the front wall - and therefore doesn\
\[CloseCurlyQuote]t get into the city. The projectile sometiimes goes past \
the back wall and doesn\[CloseCurlyQuote]t get into the city. \
\>", "Text",
 CellChangeTimes->{{3.5770440648868837`*^9, 3.577044120293538*^9}}],

Cell[TextData[{
 "There is a range of angles that we are looking for\nFrom (0,",
 Cell[BoxData[
  FormBox[
   SubscriptBox["\[Theta]", "1"], TraditionalForm]],
  FormatType->"TraditionalForm"],
 ") the projectile hits the front wall.\nFrom (",
 Cell[BoxData[
  FormBox[
   SubscriptBox["\[Theta]", "1"], TraditionalForm]],
  FormatType->"TraditionalForm"],
 ",",
 Cell[BoxData[
  FormBox[
   SubscriptBox["\[Theta]", "2"], TraditionalForm]],
  FormatType->"TraditionalForm"],
 ") the projectile makes it into the city.\nFrom (",
 Cell[BoxData[
  FormBox[
   SubscriptBox["\[Theta]", "2"], TraditionalForm]],
  FormatType->"TraditionalForm"],
 ",",
 Cell[BoxData[
  FormBox[
   SubscriptBox["\[Theta]", "3"], TraditionalForm]],
  FormatType->"TraditionalForm"],
 ") the projectile misses the city and flies past the back wall.\nFrom (",
 Cell[BoxData[
  FormBox[
   SubscriptBox["\[Theta]", "3"], TraditionalForm]],
  FormatType->"TraditionalForm"],
 ",",
 Cell[BoxData[
  FormBox[
   SubscriptBox["\[Theta]", "4"], TraditionalForm]],
  FormatType->"TraditionalForm"],
 ") the projectile again hits the interior of the city.\nFrom (",
 Cell[BoxData[
  FormBox[
   SubscriptBox["\[Theta]", "4"], TraditionalForm]],
  FormatType->"TraditionalForm"],
 ",90) the projectile does not make to to the front wall."
}], "Text",
 CellChangeTimes->{{3.577044468202783*^9, 3.57704472222932*^9}}],

Cell[BoxData[""], "Input",
 CellChangeTimes->{{3.577044459512603*^9, 3.577044460363768*^9}}],

Cell[CellGroupData[{

Cell["The Model", "Section",
 CellChangeTimes->{{3.577044453915156*^9, 3.57704445550483*^9}}],

Cell[TextData[{
 "Parametric Equations for the Path of the Projectile\n\n",
 Cell[BoxData[
  FormBox[
   RowBox[{"x", "=", 
    RowBox[{"80", "*", "t", "*", "cos", 
     RowBox[{"\[LeftAngleBracket]", "\[Theta]", "\[RightAngleBracket]"}]}]}], 
   TraditionalForm]]],
 "\n",
 Cell[BoxData[
  FormBox[
   RowBox[{"y", "=", 
    RowBox[{"80", "*", "t", "*", "sin", 
     RowBox[{"{", "\[Theta]", "}"}]}]}], TraditionalForm]]],
 " - 4.9",
 Cell[BoxData[
  FormBox[
   RowBox[{"*", 
    SuperscriptBox["t", "2"]}], TraditionalForm]]]
}], "Text",
 CellChangeTimes->{{3.577043498528852*^9, 3.577043685506493*^9}, {
  3.5771765692642*^9, 3.5771765720566*^9}}],

Cell["", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Manipulate", "[", 
  RowBox[{
   RowBox[{"ParametricPlot", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{
         RowBox[{"80", "t", "*", 
          RowBox[{"Cos", "[", "theta", "]"}]}], ",", 
         RowBox[{
          RowBox[{"80", "t", "*", 
           RowBox[{"Sin", "[", "theta", "]"}]}], "-", 
          RowBox[{"4.9", "\[IndentingNewLine]", 
           RowBox[{"t", "^", "2"}]}]}]}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{
         RowBox[{"100", "+", 
          RowBox[{"10", "t"}]}], ",", "15"}], "}"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"t", ",", "0", ",", "50"}], "}"}], ",", 
     RowBox[{"PlotRange", "\[Rule]", 
      RowBox[{"{", 
       RowBox[{
        RowBox[{"{", 
         RowBox[{"0", ",", "700"}], "}"}], ",", 
        RowBox[{"{", 
         RowBox[{"0", ",", "200"}], "}"}]}], "}"}]}]}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"theta", ",", "0", ",", 
     RowBox[{"Pi", "/", "2"}]}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.57703590678804*^9, 3.57703604896598*^9}, {
   3.577042096342194*^9, 3.577042096700368*^9}, {3.577042245908386*^9, 
   3.577042271841467*^9}, {3.577042302971974*^9, 3.577042337315282*^9}, {
   3.5770423687058573`*^9, 3.577042391384057*^9}, {3.577042529488461*^9, 
   3.577042535534046*^9}, {3.577043426283783*^9, 3.577043426393469*^9}, 
   3.577176470561*^9}],

Cell[BoxData[
 TagBox[
  StyleBox[
   DynamicModuleBox[{$CellContext`theta$$ = 0.9707521299592461, 
    Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, 
    Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 
    1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{
      Hold[$CellContext`theta$$], 0, Rational[1, 2] Pi}}, Typeset`size$$ = {
    360., {57., 60.}}, Typeset`update$$ = 0, Typeset`initDone$$, 
    Typeset`skipInitDone$$ = True, $CellContext`theta$369$$ = 0}, 
    DynamicBox[Manipulate`ManipulateBoxes[
     1, StandardForm, "Variables" :> {$CellContext`theta$$ = 0}, 
      "ControllerVariables" :> {
        Hold[$CellContext`theta$$, $CellContext`theta$369$$, 0]}, 
      "OtherVariables" :> {
       Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 
        Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 
        Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$,
         Typeset`skipInitDone$$}, "Body" :> 
      ParametricPlot[{{
         80 $CellContext`t Cos[$CellContext`theta$$], 
          80 $CellContext`t Sin[$CellContext`theta$$] - 
          4.9 $CellContext`t^2}, {
         100 + 10 $CellContext`t, 15}}, {$CellContext`t, 0, 50}, 
        PlotRange -> {{0, 700}, {0, 200}}], 
      "Specifications" :> {{$CellContext`theta$$, 0, Rational[1, 2] Pi}}, 
      "Options" :> {}, "DefaultOptions" :> {}],
     ImageSizeCache->{407., {114., 119.}},
     SingleEvaluation->True],
    Deinitialization:>None,
    DynamicModuleValues:>{},
    SynchronousInitialization->True,
    UnsavedVariables:>{Typeset`initDone$$},
    UntrackedVariables:>{Typeset`size$$}], "Manipulate",
   Deployed->True,
   StripOnInput->False],
  Manipulate`InterpretManipulate[1]]], "Output",
 CellChangeTimes->{{3.577035946361485*^9, 3.577035972204008*^9}, {
   3.577036035710704*^9, 3.577036049438677*^9}, 3.577042098208205*^9, {
   3.577042249810299*^9, 3.577042273109788*^9}, {3.577042315458764*^9, 
   3.577042337873473*^9}, 3.577042536438631*^9, 3.57704342699065*^9, 
   3.5771764785326*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Graphics", "[", 
  RowBox[{"Polygon", "[", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"100", ",", "0"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"100", ",", "15"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"600", ",", "15"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"600", ",", "0"}], "}"}]}], "}"}], "]"}], "]"}]], "Input",
 CellChangeTimes->{{3.577036378554247*^9, 3.5770364336005077`*^9}, {
  3.577036636309875*^9, 3.5770366411593246`*^9}, {3.577043375348157*^9, 
  3.5770433957456713`*^9}}],

Cell[BoxData[
 GraphicsBox[
  PolygonBox[{{100, 0}, {100, 15}, {600, 15}, {600, 0}}]]], "Output",
 CellChangeTimes->{{3.577036410474072*^9, 3.577036434107877*^9}, 
   3.577036641676306*^9, {3.577043383267153*^9, 3.577043396076097*^9}}]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Front Wall", "Section",
 CellChangeTimes->{{3.57704399910394*^9, 3.5770440011671267`*^9}}],

Cell[TextData[{
 "When will the projectile clear the front wall?\n\nThe projectile will hit \
the top of the front wall when ",
 StyleBox["x",
  FontSlant->"Italic"],
 " = 100 and when ",
 StyleBox["y",
  FontSlant->"Italic"],
 " = 15.\n\n"
}], "Text",
 CellChangeTimes->{{3.577043473656802*^9, 3.577043488966304*^9}, {
   3.577043699510313*^9, 3.577043737202834*^9}, {3.577043795412178*^9, 
   3.5770438228102407`*^9}, 3.57704416218396*^9, {3.577044205412635*^9, 
   3.5770442094365597`*^9}, {3.577044260468281*^9, 3.577044261691495*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Solve", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"80", "*", "t", "*", 
     RowBox[{"Cos", "[", "theta", "]"}]}], "==", "100"}], ",", "t"}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.577043745565235*^9, 3.577043777133658*^9}}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"{", 
   RowBox[{"t", "\[Rule]", 
    FractionBox[
     RowBox[{"5", " ", 
      RowBox[{"Sec", "[", "theta", "]"}]}], "4"]}], "}"}], "}"}]], "Output",
 CellChangeTimes->{{3.57704376674619*^9, 3.577043777911663*^9}, 
   3.5771765284058*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"NSolve", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{"80", "*", "t", "*", 
       RowBox[{"Cos", "[", "theta", "]"}]}], "==", "100"}], ",", 
     RowBox[{
      RowBox[{
       RowBox[{"80", "t", "*", 
        RowBox[{"Sin", "[", "theta", "]"}]}], "-", 
       RowBox[{"4.9", "\[IndentingNewLine]", 
        RowBox[{"t", "^", "2"}]}]}], "\[Equal]", "15"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"t", ",", "theta"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.577043842236166*^9, 3.57704390104712*^9}, {
  3.577044280838449*^9, 3.577044281366375*^9}}],

Cell[BoxData[
 RowBox[{
  StyleBox[
   RowBox[{"NSolve", "::", "ifun"}], "MessageName"], 
  RowBox[{
  ":", " "}], "\<\"Inverse functions are being used by \[NoBreak]\\!\\(NSolve\
\\)\[NoBreak], so some solutions may not be found; use Reduce for complete \
solution information. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", \
ButtonStyle->\\\"Link\\\", ButtonFrame->None, \
ButtonData:>\\\"paclet:ref/NSolve\\\", ButtonNote -> \\\"NSolve::ifun\\\"]\\)\
\"\>"}]], "Message", "MSG",
 CellChangeTimes->{{3.577043879859068*^9, 3.577043901490953*^9}, 
   3.577044282419467*^9, 3.5771765793417997`*^9}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"t", "\[Rule]", 
      RowBox[{"-", "16.08687475952258`"}]}], ",", 
     RowBox[{"theta", "\[Rule]", 
      RowBox[{"-", "1.648577829896174`"}]}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"t", "\[Rule]", 
      RowBox[{"-", "1.282814610305484`"}]}], ",", 
     RowBox[{"theta", "\[Rule]", 
      RowBox[{"-", "2.9149212028790186`"}]}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"t", "\[Rule]", "1.282814610305484`"}], ",", 
     RowBox[{"theta", "\[Rule]", "0.22667145071077463`"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"t", "\[Rule]", "16.08687475952258`"}], ",", 
     RowBox[{"theta", "\[Rule]", "1.4930148236936192`"}]}], "}"}]}], 
  "}"}]], "Output",
 CellChangeTimes->{{3.577043879859997*^9, 3.577043901492482*^9}, 
   3.5770442824219933`*^9, 3.5771765793574*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"t1deg", "=", 
  RowBox[{
   RowBox[{"0.226671", "/", "Pi"}], "*", "180"}]}]], "Input",
 CellChangeTimes->{{3.577043939708416*^9, 3.577043964139162*^9}, 
   3.577044290383047*^9}],

Cell[BoxData["12.987291638009886`"], "Output",
 CellChangeTimes->{3.577043965090088*^9, 3.577044291788028*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"t2deg", "=", 
  RowBox[{
   RowBox[{"1.49301", "/", "Pi"}], "*", "180"}]}]], "Input",
 CellChangeTimes->{{3.577043967044531*^9, 3.577043977824946*^9}, 
   3.5770443053439903`*^9}],

Cell[BoxData["85.54317177082704`"], "Output",
 CellChangeTimes->{3.5770439788172407`*^9, 3.577044306502891*^9}]
}, Open  ]],

Cell[TextData[{
 "So we need our angles to be larger than 12.99 degrees and less than 85.54 \
degrees.\n\nThese two angles are ",
 Cell[BoxData[
  FormBox[
   SubscriptBox["\[Theta]", "1"], TraditionalForm]],
  FormatType->"TraditionalForm"],
 "and ",
 Cell[BoxData[
  FormBox[
   SubscriptBox["\[Theta]", "4"], TraditionalForm]]],
 "."
}], "Text",
 CellChangeTimes->{{3.577044219623129*^9, 3.577044249412202*^9}, {
  3.5770442974857883`*^9, 3.577044313227323*^9}, {3.577044736325201*^9, 
  3.577044781476297*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Back Wall", "Section",
 CellChangeTimes->{{3.57704399910394*^9, 3.5770440126471148`*^9}}],

Cell[TextData[{
 "When will the projectile go past the back wall?\n\nThe projectile will hit \
the top of the back wall when ",
 StyleBox["x",
  FontSlant->"Italic"],
 " = 600 and when ",
 StyleBox["y",
  FontSlant->"Italic"],
 " = 15\n\n"
}], "Text",
 CellChangeTimes->{{3.577043473656802*^9, 3.577043488966304*^9}, {
  3.577043699510313*^9, 3.577043737202834*^9}, {3.577043795412178*^9, 
  3.5770438228102407`*^9}, {3.5770440305034113`*^9, 3.577044032782607*^9}, {
  3.577044177423098*^9, 3.5770442011885443`*^9}, {3.577044254843997*^9, 
  3.57704425881177*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"NSolve", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{"80", "*", "t", "*", 
       RowBox[{"Cos", "[", "theta", "]"}]}], "\[Equal]", "600"}], ",", 
     RowBox[{
      RowBox[{
       RowBox[{"80", "t", "*", 
        RowBox[{"Sin", "[", "theta", "]"}]}], "-", 
       RowBox[{"4.9", "\[IndentingNewLine]", 
        RowBox[{"t", "^", "2"}]}]}], "\[Equal]", "15"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"t", ",", "theta"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.577043842236166*^9, 3.57704390104712*^9}, {
  3.577044274495667*^9, 3.577044274789934*^9}, {3.57704431785366*^9, 
  3.5770443180449867`*^9}}],

Cell[BoxData[
 RowBox[{
  StyleBox[
   RowBox[{"NSolve", "::", "ifun"}], "MessageName"], 
  RowBox[{
  ":", " "}], "\<\"Inverse functions are being used by \[NoBreak]\\!\\(NSolve\
\\)\[NoBreak], so some solutions may not be found; use Reduce for complete \
solution information. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", \
ButtonStyle->\\\"Link\\\", ButtonFrame->None, \
ButtonData:>\\\"paclet:ref/NSolve\\\", ButtonNote -> \\\"NSolve::ifun\\\"]\\)\
\"\>"}]], "Message", "MSG",
 CellChangeTimes->{{3.577043879859068*^9, 3.577043901490953*^9}, 
   3.577044318839102*^9, 3.577176712617*^9}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"t", "\[Rule]", 
      RowBox[{"-", "13.206512522017062`"}]}], ",", 
     RowBox[{"theta", "\[Rule]", 
      RowBox[{"-", "2.174750573233465`"}]}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"t", "\[Rule]", 
      RowBox[{"-", "9.274760366650153`"}]}], ",", 
     RowBox[{"theta", "\[Rule]", 
      RowBox[{"-", "2.5126436135323047`"}]}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"t", "\[Rule]", "9.274760366650153`"}], ",", 
     RowBox[{"theta", "\[Rule]", "0.6289490400574884`"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"t", "\[Rule]", "13.206512522017062`"}], ",", 
     RowBox[{"theta", "\[Rule]", "0.9668420803563283`"}]}], "}"}]}], 
  "}"}]], "Output",
 CellChangeTimes->{{3.577043879859997*^9, 3.577043901492482*^9}, 
   3.577044318841535*^9, 3.577176712617*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"t1deg", "=", 
  RowBox[{
   RowBox[{"0.628949", "/", "Pi"}], "*", "180"}]}]], "Input",
 CellChangeTimes->{{3.577043939708416*^9, 3.577043964139162*^9}, {
  3.577044328384617*^9, 3.577044349797514*^9}}],

Cell[BoxData["36.036123228973615`"], "Output",
 CellChangeTimes->{
  3.577043965090088*^9, {3.577044336457951*^9, 3.577044350634899*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"t2deg", "=", 
  RowBox[{
   RowBox[{"0.966842", "/", "Pi"}], "*", "180"}]}]], "Input",
 CellChangeTimes->{{3.577043967044531*^9, 3.577043977824946*^9}, {
  3.5770443555249443`*^9, 3.57704435725312*^9}}],

Cell[BoxData["55.39596605598754`"], "Output",
 CellChangeTimes->{3.5770439788172407`*^9, 3.577044358151308*^9}]
}, Open  ]],

Cell[TextData[{
 "We have just computed ",
 Cell[BoxData[
  FormBox[
   SubscriptBox["\[Theta]", "2"], TraditionalForm]],
  FormatType->"TraditionalForm"],
 "and ",
 Cell[BoxData[
  FormBox[
   SubscriptBox["\[Theta]", "3"], TraditionalForm]]],
 "."
}], "Text",
 CellChangeTimes->{
  3.577044379621958*^9, {3.5770448031162567`*^9, 3.5770448256777687`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Conclusion", "Section",
 CellChangeTimes->{{3.57704399910394*^9, 3.5770440126471148`*^9}, {
   3.577044837277952*^9, 3.577044839083811*^9}, 3.577045088301484*^9}],

Cell["\<\
Therefore

For angles between (0, 12.99\[Degree]) the projectile strikes the front wall \
and misses the city.

For angles between (12.99\[Degree], 36.04\[Degree]) the projectile makes it \
into the city.

For angles between (36.04\[Degree], 55.40\[Degree]) the projectiles clears \
the city and strikes the ground past the back wall.\[LineSeparator]\
\[LineSeparator]For angles between (55.40\[Degree], 85.54\[Degree]) the \
projectile again strikes the city.

For angles between (85.54\[Degree], 90\[Degree]) the projectile hits the \
ground in front of the front wall.
\
\>", "Text",
 CellChangeTimes->{{3.577043473656802*^9, 3.577043488966304*^9}, {
  3.577043699510313*^9, 3.577043737202834*^9}, {3.577043795412178*^9, 
  3.5770438228102407`*^9}, {3.5770440305034113`*^9, 3.577044032782607*^9}, {
  3.577044177423098*^9, 3.5770442011885443`*^9}, {3.577044254843997*^9, 
  3.57704425881177*^9}, {3.577044865605749*^9, 3.57704508830179*^9}}]
}, Open  ]]
}, Open  ]]
},
WindowSize->{1008, 629},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
FrontEndVersion->"9.0 for Microsoft Windows (64-bit) (November 20, 2012)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[579, 22, 100, 1, 90, "Title"],
Cell[682, 25, 679, 10, 106, "Text"],
Cell[1364, 37, 89964, 1480, 548, "Input"],
Cell[91331, 1519, 92, 1, 31, "Input"],
Cell[91426, 1522, 299, 5, 49, "Text"],
Cell[91728, 1529, 1382, 43, 125, "Text"],
Cell[93113, 1574, 92, 1, 31, "Input"],
Cell[CellGroupData[{
Cell[93230, 1579, 93, 1, 79, "Section"],
Cell[93326, 1582, 651, 21, 86, "Text"],
Cell[93980, 1605, 16, 0, 30, "Text"],
Cell[CellGroupData[{
Cell[94021, 1609, 1413, 37, 52, "Input"],
Cell[95437, 1648, 2073, 41, 250, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[97547, 1694, 556, 15, 31, "Input"],
Cell[98106, 1711, 235, 4, 40, "Output"]
}, Open  ]]
}, Open  ]],
Cell[CellGroupData[{
Cell[98390, 1721, 96, 1, 79, "Section"],
Cell[98489, 1724, 539, 13, 106, "Text"],
Cell[CellGroupData[{
Cell[99053, 1741, 250, 7, 31, "Input"],
Cell[99306, 1750, 279, 8, 46, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[99622, 1763, 616, 17, 52, "Input"],
Cell[100241, 1782, 596, 12, 21, "Message"],
Cell[100840, 1796, 906, 25, 52, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[101783, 1826, 202, 5, 31, "Input"],
Cell[101988, 1833, 110, 1, 31, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[102135, 1839, 203, 5, 31, "Input"],
Cell[102341, 1846, 111, 1, 31, "Output"]
}, Open  ]],
Cell[102467, 1850, 514, 15, 68, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[103018, 1870, 95, 1, 79, "Section"],
Cell[103116, 1873, 564, 14, 106, "Text"],
Cell[CellGroupData[{
Cell[103705, 1891, 672, 18, 52, "Input"],
Cell[104380, 1911, 591, 12, 21, "Message"],
Cell[104974, 1925, 904, 25, 52, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[105915, 1955, 225, 5, 31, "Input"],
Cell[106143, 1962, 137, 2, 31, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[106317, 1969, 226, 5, 31, "Input"],
Cell[106546, 1976, 111, 1, 31, "Output"]
}, Open  ]],
Cell[106672, 1980, 356, 13, 30, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[107065, 1998, 168, 2, 79, "Section"],
Cell[107236, 2002, 954, 22, 239, "Text"]
}, Open  ]]
}, Open  ]]
}
]
*)

(* End of internal cache information *)
