(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 13.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 11209, 314] NotebookOptionsPosition[ 9299, 275] NotebookOutlinePosition[ 9698, 291] CellTagsIndexPosition[ 9655, 288] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Vitesse moyenne", "Title", CellChangeTimes->{{3.8510761026919327`*^9, 3.851076107454547*^9}},ExpressionUUID->"78d6083d-da2e-4420-9428-\ bdca1676e3fd"], Cell["\<\ Un \[EAcute]tudiant se rend tous les jours \[AGrave] la fac \[AGrave] v\ \[EAcute]lo . Il fait le trajet \[AGrave] 20 km/h de moyenne . \ \[CapitalAGrave] quelle vitesse doit il aller au retour pour que sa vitesse \ moyenne sur l' aller - retour soit de 40 km/h, s\[CloseCurlyQuote]il parcourt \ l\[CloseCurlyQuote]aller \[AGrave] 20 km/h?\ \>", "Text", CellChangeTimes->{{3.8510744611896067`*^9, 3.8510744972645693`*^9}},ExpressionUUID->"4a77a0b5-a0a6-4b51-aa13-\ d93065e343cd"], Cell["\<\ La vitesse moyenne s\[CloseCurlyQuote]obtient en divisant la distance totale \ par le temps total de parcours\ \>", "Text", CellChangeTimes->{{3.851074527674655*^9, 3.851074565046513*^9}},ExpressionUUID->"42ea4bbe-b752-4fbe-ae3e-\ f6522ff7531e"], Cell[BoxData[ RowBox[{"2", RowBox[{"d", "/", RowBox[{"(", RowBox[{"t1", "+", "t2"}], ")"}]}]}]], "Input", CellChangeTimes->{{3.851074594083644*^9, 3.851074609248535*^9}},ExpressionUUID->"d2f29bb3-d211-41ad-a213-\ 75af0956b353"], Cell[TextData[{ "Le temps pour l' aller vaut ", StyleBox["d", FontSlant->"Italic"], "/", StyleBox["v", FontSlant->"Italic"], "1 et pour le retour ", StyleBox["d", FontSlant->"Italic"], "/", StyleBox["v", FontSlant->"Italic"], "2. En substituant dans l\[CloseCurlyQuote]expression \ pr\[EAcute]c\[EAcute]dente et en simplifiant, on obtient, pour la vitesse \ moyenne ", StyleBox["v", FontSlant->"Italic"] }], "Text", CellChangeTimes->{{3.851074624222784*^9, 3.851074707945772*^9}, { 3.851074755961158*^9, 3.851074785234106*^9}},ExpressionUUID->"8b1bea0d-3ebb-4e7c-ba64-\ 526c5cdd57cd"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"2", RowBox[{"d", "/", RowBox[{"(", RowBox[{"t1", "+", "t2"}], ")"}]}]}], "/.", RowBox[{"{", RowBox[{ RowBox[{"t1", "->", RowBox[{"d", "/", "v1"}]}], ",", RowBox[{"t2", "->", RowBox[{"d", "/", "v2"}]}]}], "}"}]}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.8510747175194798`*^9, 3.851074750099942*^9}}, CellLabel->"In[89]:=",ExpressionUUID->"448a9219-e77e-4653-8876-6ed829b69a4a"], Cell[BoxData[ FractionBox[ RowBox[{"2", " ", "v1", " ", "v2"}], RowBox[{"v1", "+", "v2"}]]], "Output", CellChangeTimes->{ 3.851073429007427*^9, {3.851073835988529*^9, 3.851073847610545*^9}, { 3.851074738034563*^9, 3.851074750607747*^9}}, CellLabel->"Out[89]=",ExpressionUUID->"9062d64d-968d-4c7e-b808-b49f8ee16910"] }, Open ]], Cell[TextData[{ "En r\[EAcute]solvant par rapport \[AGrave] ", StyleBox["v", "Code", FontWeight->"Plain", FontSlant->"Italic"], "2, il vient" }], "Text", CellChangeTimes->{{3.85107483014501*^9, 3.851074855259286*^9}},ExpressionUUID->"5e108bf1-9297-4c2b-9c9b-\ 3210af1af59d"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{"v", "==", FractionBox[ RowBox[{"2", " ", "v1", " ", "v2"}], RowBox[{"v1", "+", "v2"}]]}], ",", "v2"}], "]"}]], "Input", CellChangeTimes->{{3.851074863592341*^9, 3.851074873930526*^9}}, CellLabel->"In[90]:=",ExpressionUUID->"8ce27114-1e2d-49a9-9572-f7abe550408b"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"v2", "\[Rule]", RowBox[{"-", FractionBox[ RowBox[{"v", " ", "v1"}], RowBox[{"v", "-", RowBox[{"2", " ", "v1"}]}]]}]}], "}"}], "}"}]], "Output", CellChangeTimes->{3.8510748762195*^9}, CellLabel->"Out[90]=",ExpressionUUID->"b5ab7663-6edf-4d74-b4b0-e6d744a29064"] }, Open ]], Cell["\<\ Si on admet que la vitesse \[AGrave] l' aller vaut 20 km/h, v2 s\ \[CloseCurlyQuote]exprime alors en fonction de v\ \>", "Text", CellChangeTimes->{{3.851074921079756*^9, 3.8510749476960487`*^9}, { 3.851075005286686*^9, 3.851075025953081*^9}},ExpressionUUID->"e8b75137-2401-4cd6-81f1-\ 08078222c6fe"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"v", " ", "v1"}], RowBox[{"v", "-", RowBox[{"2", " ", "v1"}]}]]}], "/.", RowBox[{"v1", "->", "20"}]}]], "Input", CellChangeTimes->{{3.851074259673436*^9, 3.851074260804223*^9}}, CellLabel->"In[85]:=",ExpressionUUID->"62226e42-64b7-461d-a936-43cbfdadca28"], Cell[BoxData[ RowBox[{"-", FractionBox[ RowBox[{"20", " ", "v"}], RowBox[{ RowBox[{"-", "40"}], "+", "v"}]]}]], "Output", CellChangeTimes->{3.8510742620480013`*^9}, CellLabel->"Out[85]=",ExpressionUUID->"a0db563d-2da9-4b24-bea3-ed87953a717b"] }, Open ]], Cell[TextData[{ "En reportant la vitesse ", StyleBox["v", FontSlant->"Italic"], "2 en fonction de la vitesse moyenne, on obtient" }], "Text", CellChangeTimes->{{3.8510750503089323`*^9, 3.8510751036104593`*^9}},ExpressionUUID->"5f1d938a-5d95-4826-b32d-\ 9769bb78d8d4"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{"vmax", "\[NotEqual]", "20"}], ",", RowBox[{"Plot", "[", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"20", " ", "v"}], RowBox[{ RowBox[{"-", "40"}], "+", "v"}]]}], ",", RowBox[{"{", RowBox[{"v", ",", "20", ",", "vmax"}], "}"}], ",", " ", RowBox[{"AxesLabel", "->", RowBox[{"{", RowBox[{ "\"\\"", ",", " ", "\"\\""}], "}"}]}], ",", RowBox[{"GridLines", "\[Rule]", "Automatic"}]}], "]"}], ",", RowBox[{"Text", "@", "\"\\""}]}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"vmax", ",", "20", ",", "\"\\""}], "}"}], ",", "20", ",", "40", ",", RowBox[{"Appearance", "->", "\"\\""}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.85107427488486*^9, 3.851074305540332*^9}, 3.851074343870349*^9, {3.85107511778622*^9, 3.851075207697076*^9}, { 3.8510752409525423`*^9, 3.8510753381878023`*^9}, {3.851075526829879*^9, 3.85107558300107*^9}, 3.851075673745084*^9, {3.851075723256159*^9, 3.851075924195004*^9}, {3.851076009403347*^9, 3.8510760410303793`*^9}, { 3.8510760746202717`*^9, 3.851076083046665*^9}}, CellLabel-> "In[108]:=",ExpressionUUID->"ddb1164f-341b-4cf7-8c5f-ee2f91eec8e1"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`vmax$$ = 28.62, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"sans titre\"", Typeset`specs$$ = {{{ Hold[$CellContext`vmax$$], 20, "vitesse moyenne"}, 20, 40}}, Typeset`size$$ = {540., {128., 135.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`vmax$$ = 20}, "ControllerVariables" :> {}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> If[$CellContext`vmax$$ != 20, Plot[-(20 $CellContext`v/(-40 + $CellContext`v)), {$CellContext`v, 20, $CellContext`vmax$$}, AxesLabel -> {"vitesse moyenne", "vitesse au retour"}, GridLines -> Automatic], Text["Choisissez une vitesse moyenne"]], "Specifications" :> {{{$CellContext`vmax$$, 20, "vitesse moyenne"}, 20, 40, Appearance -> "Labeled"}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{606., {196., 204.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.8510743077209187`*^9, 3.85107434465697*^9, 3.8510751651013517`*^9, 3.851075261450148*^9, {3.85107531131885*^9, 3.851075339402872*^9}, 3.851075492248946*^9, {3.85107552850773*^9, 3.851075591725281*^9}, 3.851075808249179*^9, {3.851075904404957*^9, 3.851075925406636*^9}, 3.851075957732677*^9, {3.851076027271551*^9, 3.8510760428464937`*^9}, 3.851076083845681*^9}, CellLabel-> "Out[108]=",ExpressionUUID->"0e9fca52-8152-43fc-b364-d9721f59b3ef"] }, Open ]] }, Open ]] }, WindowSize->{1371, 1090}, WindowMargins->{{20, Automatic}, {Automatic, 44}}, FrontEndVersion->"13.0 pour Mac OS X x86 (64-bit) (December 2, 2021)", StyleDefinitions->"Default.nb", ExpressionUUID->"477c10ee-8d82-461b-b28a-d387a052e459" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[580, 22, 160, 3, 146, "Title",ExpressionUUID->"78d6083d-da2e-4420-9428-bdca1676e3fd"], Cell[743, 27, 493, 9, 88, "Text",ExpressionUUID->"4a77a0b5-a0a6-4b51-aa13-d93065e343cd"], Cell[1239, 38, 259, 6, 53, "Text",ExpressionUUID->"42ea4bbe-b752-4fbe-ae3e-f6522ff7531e"], Cell[1501, 46, 244, 7, 46, "Input",ExpressionUUID->"d2f29bb3-d211-41ad-a213-75af0956b353"], Cell[1748, 55, 610, 22, 88, "Text",ExpressionUUID->"8b1bea0d-3ebb-4e7c-ba64-526c5cdd57cd"], Cell[CellGroupData[{ Cell[2383, 81, 488, 14, 46, "Input",ExpressionUUID->"448a9219-e77e-4653-8876-6ed829b69a4a"], Cell[2874, 97, 328, 7, 86, "Output",ExpressionUUID->"9062d64d-968d-4c7e-b808-b49f8ee16910"] }, Open ]], Cell[3217, 107, 285, 9, 53, "Text",ExpressionUUID->"5e108bf1-9297-4c2b-9c9b-3210af1af59d"], Cell[CellGroupData[{ Cell[3527, 120, 341, 8, 68, "Input",ExpressionUUID->"8ce27114-1e2d-49a9-9572-f7abe550408b"], Cell[3871, 130, 351, 10, 86, "Output",ExpressionUUID->"b5ab7663-6edf-4d74-b4b0-e6d744a29064"] }, Open ]], Cell[4237, 143, 315, 7, 53, "Text",ExpressionUUID->"e8b75137-2401-4cd6-81f1-08078222c6fe"], Cell[CellGroupData[{ Cell[4577, 154, 339, 9, 68, "Input",ExpressionUUID->"62226e42-64b7-461d-a936-43cbfdadca28"], Cell[4919, 165, 259, 7, 87, "Output",ExpressionUUID->"a0db563d-2da9-4b24-bea3-ed87953a717b"] }, Open ]], Cell[5193, 175, 276, 8, 53, "Text",ExpressionUUID->"5f1d938a-5d95-4826-b32d-9769bb78d8d4"], Cell[CellGroupData[{ Cell[5494, 187, 1486, 37, 215, "Input",ExpressionUUID->"ddb1164f-341b-4cf7-8c5f-ee2f91eec8e1"], Cell[6983, 226, 2288, 45, 445, "Output",ExpressionUUID->"0e9fca52-8152-43fc-b364-d9721f59b3ef"] }, Open ]] }, Open ]] } ] *)