(* 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[ 12493, 378] NotebookOptionsPosition[ 11851, 354] NotebookOutlinePosition[ 12272, 372] CellTagsIndexPosition[ 12229, 369] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[StyleBox["Elastic Collision in Two Dimensions", FontColor->RGBColor[0, 0, 1]]], "Text", CellChangeTimes->{{3.4123736616820927`*^9, 3.412373670243538*^9}}, TextAlignment->Center, TextJustification->0., FontSize->18, FontWeight->"Bold"], Cell[TextData[{ "Elastic collisions of two known masses in two dimensions conserve momentum \ in the two dimensions, and also kinetic energy. Therefore there are three \ equations which can be written. If the initial system is completely \ specified, then there are 4 known initial kinematic quanties, in addition to \ the two masses: magnitude and direction of the initial velocity of the first \ mass, and the magnitude and direction of initial velocity of the second mass.\ \n\nThere will be 4 final kinematic quantities: magnitude and direction of \ the final velocity of the first mass, and the magnitude and direction of the \ final velocity of the second mass. If just one of these final kinematic \ quantities is specified, then the other three final kinematic quantities can \ be derived from the conservation equations.\n\nAs an example of a two \ dimensional elastic collision, consider the collision of a first mass ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["m", RowBox[{"1", " "}]], FontSize->18], TraditionalForm]]], "traveling along the horizontal direction (x) with speed ", Cell[BoxData[ FormBox[ SubscriptBox["v", RowBox[{"1", "i", " "}]], TraditionalForm]]], "and then striking a second mass ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["m", "2"], FontSize->18], TraditionalForm]]], "which is a rest. After the collision the first mass is seen to be traveling \ at an angle \[Theta] above the horizontal direction. Calculate the \ magnitudes of the final velocities ", Cell[BoxData[ FormBox[ SubscriptBox["v", RowBox[{"1", "f"}]], TraditionalForm]]], "and ", Cell[BoxData[ FormBox[ SubscriptBox["v", RowBox[{"2", "f"}]], TraditionalForm]]], " as well as the direction angle \[Phi] of the second mass.\n\nThe three \ conservation equations are:" }], "Text", CellChangeTimes->{{3.41237373432697*^9, 3.41237413348452*^9}, { 3.4123742945088873`*^9, 3.412374325361549*^9}, {3.41237436050741*^9, 3.412374596058243*^9}, {3.412374629231833*^9, 3.41237480978942*^9}, { 3.412374852441931*^9, 3.412374867710471*^9}}, TextAlignment->Left, TextJustification->1., FontSize->18, FontColor->RGBColor[0, 0, 1]], Cell[BoxData[{ FormBox[ StyleBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["m", "1"], " ", SubscriptBox["v", RowBox[{"1", "i", " "}]]}], "=", " ", RowBox[{ RowBox[{ StyleBox[ SubscriptBox["m", RowBox[{"1", " "}]], FontColor->RGBColor[0, 0, 1]], SubscriptBox["v", RowBox[{"1", "f", " "}]], "cos", " ", StyleBox["\[Theta]", FontSlant->"Plain"]}], StyleBox[" ", FontSlant->"Plain"], StyleBox["+", FontSlant->"Plain"], " ", RowBox[{ SubscriptBox["m", RowBox[{"2", " "}]], SubscriptBox["v", RowBox[{"2", "f", " "}]], "cos", " ", "\[Phi]"}]}]}], StyleBox[" ", FontSlant->"Plain"]}], FontSize->18, FontColor->RGBColor[0, 0, 1]], TraditionalForm], "\[IndentingNewLine]", FormBox[ StyleBox[ RowBox[{ RowBox[{ StyleBox["0", FontSlant->"Plain"], "=", " ", RowBox[{ RowBox[{ SubscriptBox["m", RowBox[{"1", " "}]], SubscriptBox["v", RowBox[{"1", "f", " "}]], "sin", " ", StyleBox["\[Theta]", FontSlant->"Plain"]}], StyleBox[" ", FontSlant->"Plain"], StyleBox["-", FontSlant->"Plain"], " ", RowBox[{ SubscriptBox["m", RowBox[{"2", " "}]], SubscriptBox["v", RowBox[{"2", "f", " "}]], "sin", " ", "\[Phi]"}]}]}], StyleBox[" ", FontSlant->"Plain"]}], FontSize->18, FontColor->RGBColor[0, 0, 1]], TraditionalForm], "\[IndentingNewLine]", FormBox[ StyleBox[ RowBox[{ RowBox[{ FractionBox["1", "2"], SubscriptBox["m", "1"], " ", SuperscriptBox[ SubscriptBox["v", RowBox[{"1", "i", " "}]], "2"]}], "=", " ", RowBox[{ RowBox[{ FractionBox["1", "2"], StyleBox[ SubscriptBox["m", RowBox[{"1", " "}]], FontColor->RGBColor[0, 0, 1]], SuperscriptBox[ SubscriptBox["v", RowBox[{"1", "f", " "}]], "2"]}], StyleBox[" ", FontSlant->"Plain"], StyleBox["+", FontSlant->"Plain"], " ", RowBox[{ FractionBox["1", "2"], SubscriptBox["m", RowBox[{"2", " "}]], SuperscriptBox[ SubscriptBox["v", RowBox[{"2", "f", " "}]], "2"]}]}]}], FontSize->18, FontColor->RGBColor[0, 0, 1]], TraditionalForm]}], "Text", CellChangeTimes->{{3.4123748214883633`*^9, 3.4123748426296053`*^9}, { 3.412374889776425*^9, 3.4123749874858837`*^9}, {3.412375067257622*^9, 3.412375180288906*^9}}, TextAlignment->Center, TextJustification->0.], Cell[TextData[{ "As an example, let's take the two masses to be proton masses, 1.67 x ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "27"}]], TraditionalForm]]], "kg, with an initial speed of the first proton mass to be 3.5 x ", Cell[BoxData[ FormBox[ SuperscriptBox["10", "5"], TraditionalForm]]], " m/s, and the second proton is at rest. After the collision the first \ proton is seen to have a velocity with an angle \[Theta] = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["37", "o"], " ", "above", " ", "the", " ", "horizontal"}], TraditionalForm]]], ". What are the magnitudes of the two final velocities, and what is the \ direction angle \[Phi] (below the horizontal) of the second proton's velocity \ after the collision.\n\nWe can solve this with ", StyleBox["Mathematica", FontSlant->"Italic"], " after first assigning the known kinematic variables, including the \ conversion of the ", "\[Theta]", " angle to radians:" }], "Text", CellChangeTimes->{{3.412376429409556*^9, 3.4123767076751947`*^9}, { 3.412376895611278*^9, 3.412376908145584*^9}, {3.412376999523193*^9, 3.412376999527369*^9}}, FontSize->18, FontColor->RGBColor[0, 0, 1]], Cell[BoxData[{ StyleBox[ RowBox[{ RowBox[{"mass1", "=", RowBox[{"1.67", "*", RowBox[{"10", "^", RowBox[{"-", "27"}]}]}]}], ";"}], FontSize->18, FontColor->RGBColor[0, 0, 1]], "\n", StyleBox[ RowBox[{ RowBox[{"mass2", "=", " ", "mass1"}], ";"}], FontSize->18, FontColor->RGBColor[0, 0, 1]], "\n", StyleBox[ RowBox[{ RowBox[{"V1Initial", "=", RowBox[{"3.5", "*", RowBox[{"10", "^", "5"}]}]}], ";"}], FontSize->18, FontColor->RGBColor[0, 0, 1]], "\n", StyleBox[ RowBox[{ RowBox[{"\[Theta]", "=", RowBox[{"37.0", "*", RowBox[{"Pi", "/", "180"}]}]}], ";"}], FontSize->18, FontColor->RGBColor[0, 0, 1]]}], "Input", CellChangeTimes->{{3.412377097849381*^9, 3.412377109727332*^9}, 3.412377227662321*^9, {3.412377344262877*^9, 3.4123773737131577`*^9}, { 3.412377408183988*^9, 3.412377410164557*^9}, {3.4123777198397713`*^9, 3.412377722523993*^9}, 3.412378017684993*^9}], Cell[TextData[{ StyleBox["Now we use the ", FontSize->18, FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontSize->18, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" simultaneous equation solver, and show the one correct solution:", FontSize->18, FontColor->RGBColor[0, 0, 1]] }], "Text", CellChangeTimes->{{3.412378268620329*^9, 3.412378276303863*^9}, { 3.4123783396866703`*^9, 3.412378367631044*^9}}], Cell[BoxData[ RowBox[{ StyleBox["Solve", FontColor->RGBColor[1, 0, 0]], StyleBox["[", FontColor->RGBColor[1, 0, 0]], RowBox[{ RowBox[{ StyleBox["{", FontColor->GrayLevel[0]], RowBox[{ StyleBox[ RowBox[{ RowBox[{"mass1", "*", "V1Initial"}], "\[Equal]", " ", RowBox[{ RowBox[{"mass1", "*", "V1Final", "*", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], " ", "+", " ", RowBox[{"mass2", "*", "V2Final", "*", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}]}], FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontColor->RGBColor[0, 0, 1]], StyleBox["\[IndentingNewLine]", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[ RowBox[{ RowBox[{ RowBox[{"mass1", "*", "V1Final", "*", RowBox[{"Sin", "[", "\[Theta]", "]"}]}], " ", "-", " ", RowBox[{"mass2", "*", "V2Final", "*", RowBox[{"Sin", "[", "\[Phi]", "]"}]}]}], "\[Equal]", "0"}], FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontColor->RGBColor[0, 0, 1]], StyleBox["\[IndentingNewLine]", FontColor->RGBColor[0, 0, 1]], StyleBox[ RowBox[{ RowBox[{"mass1", "*", "V1Initial", "*", "V1Initial"}], "==", RowBox[{ RowBox[{"mass1", "*", "V1Final", "*", "V1Final"}], " ", "+", " ", RowBox[{"mass2", "*", "V2Final", "*", "V2Final"}]}]}], FontColor->RGBColor[0, 0, 1]]}], StyleBox["}", FontColor->RGBColor[0, 0, 1]]}], StyleBox[",", FontColor->RGBColor[0, 0, 1]], StyleBox[ RowBox[{"{", RowBox[{"V1Final", ",", " ", "V2Final", ",", "\[Phi]"}], "}"}], FontColor->RGBColor[0, 0, 1]]}], StyleBox["]", FontColor->RGBColor[0, 0, 1]]}]], "Input", CellChangeTimes->{{3.4123756273541107`*^9, 3.412375641800735*^9}, { 3.412375672327794*^9, 3.412375840219401*^9}, {3.412375905457551*^9, 3.412376051393561*^9}, {3.412376088393512*^9, 3.4123761193142157`*^9}, { 3.412376711897766*^9, 3.41237672629526*^9}, {3.412376809111554*^9, 3.412376952756043*^9}, 3.412377006045718*^9, 3.41237707349649*^9, { 3.412377132346696*^9, 3.412377205730919*^9}, {3.41237727517283*^9, 3.412377284050343*^9}, 3.412377334453252*^9, {3.412377413675239*^9, 3.412377439618927*^9}, {3.412377597377164*^9, 3.412377605262925*^9}, { 3.412377792332828*^9, 3.412377794957539*^9}, {3.412378007773806*^9, 3.41237804472155*^9}}, FontSize->18], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{"V1Final", "\[Rule]", "279522"}], ",", RowBox[{"V2Final", "\[Rule]", "210635"}], ",", RowBox[{"\[Phi]", "\[Rule]", "0.92502"}]}], FontColor->RGBColor[0, 0, 1]]], "Input", CellChangeTimes->{ 3.412377619606287*^9, 3.412377729101602*^9, 3.4123777993114223`*^9, { 3.412378056349031*^9, 3.412378070395791*^9}, {3.4123781313470488`*^9, 3.41237818805879*^9}}, FontSize->18], Cell[TextData[StyleBox["The \[Phi] angle in radians can be converted to 53 \ degrees. It is a general result that when one mass collides with a second \ equal mass which is at rest, then the two masses have final velocities which \ are perpendicular to each other, that is 90 degrees apart.", FontSize->18, FontColor->RGBColor[0, 0, 1]]], "Text", CellChangeTimes->{{3.412378226450862*^9, 3.4123782569999228`*^9}, { 3.412378393195846*^9, 3.412378459686504*^9}, {3.412378506055327*^9, 3.412378509908442*^9}, {3.41237855833816*^9, 3.412378727330768*^9}}, TextAlignment->Left, TextJustification->1.] }, WindowToolbars->"EditBar", WindowSize->{852, 815}, WindowMargins->{{22, Automatic}, {Automatic, 28}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, FrontEndVersion->"6.0 for Mac OS X PowerPC (32-bit) (June 19, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[568, 21, 254, 6, 32, "Text"], Cell[825, 29, 2209, 52, 392, "Text"], Cell[3037, 83, 2645, 95, 104, "Text"], Cell[5685, 180, 1213, 31, 187, "Text"], Cell[6901, 213, 955, 31, 92, "Input"], Cell[7859, 246, 449, 14, 32, "Text"], Cell[8311, 262, 2490, 65, 149, "Input"], Cell[10804, 329, 434, 11, 33, "Input"], Cell[11241, 342, 606, 10, 74, "Text"] } ] *) (* End of internal cache information *)