(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 7401, 214]*) (*NotebookOutlinePosition[ 8179, 241]*) (* CellTagsIndexPosition[ 8135, 237]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Saut \[AGrave] l'\[EAcute]lastique. Solution", "H2", FontFamily->"Verdana"], Cell[TextData[{ "On souhaite \[EAcute]tudier un saut \[AGrave] \ l\[CloseCurlyQuote]\[EAcute]lastique depuis le barrage-vo\[UHat]te hydro\ \[EAcute]lectrique de Contra, haut de 220 m et situ\[EAcute] dans le val \ Verzasca au Tessin. Un homme de masse ", StyleBox["m", FontSlant->"Italic"], " saute du haut du barrage. Il est soumis \[AGrave] la force de pesanteur \ et \[AGrave] une force de frottement proportionnelle \[AGrave] la vitesse ", StyleBox["Ffrott", FontSlant->"Italic"], "=-", StyleBox["bv", FontSlant->"Italic"], ". On fixe l\[CloseCurlyQuote]origine ", StyleBox["z", FontSlant->"Italic"], "=0 de l\[CloseCurlyQuote]axe orient\[EAcute] vers le bas \[AGrave] la \ hauteur du tremplin. Une force de rappel ", StyleBox["Frappel", FontSlant->"Italic"], "=-", StyleBox["k", FontSlant->"Italic"], "(", StyleBox["z", FontSlant->"Italic"], "-", StyleBox["z", FontSlant->"Italic"], "0)) commence \[AGrave] agir d\[EGrave]s que \ l\[CloseCurlyQuote]\[EAcute]lastique est tendu. ", StyleBox["z", FontSlant->"Italic"], "0 d\[EAcute]signe la position du sauteur lorsque la force de rappel \ commence \[AGrave] agir.\n\nApr\[EGrave]s une chute de 5 secondes, l\ \[CloseCurlyQuote]\[EAcute]lastique est tendu et la force de rappel commence \ \[AGrave] agir. Sachant que le sauteur quitte le tremplin avec une vitesse \ initiale nulle;\n\[Bullet] \[EAcute]crivez les \[EAcute]quations \ r\[EAcute]gissant le mouvement :\n- durant les 5 premi\[EGrave]res secondes\n\ - apr\[EGrave]s 5 secondes\n\[Bullet] donnez les solutions g\[EAcute]n\ \[EAcute]rales de ces deux \[EAcute]quations \n\[Bullet] repr\[EAcute]sentez \ l\[CloseCurlyQuote]horaire du sauteur pour ", StyleBox["t", FontSlant->"Italic"], " compr", StyleBox["i", FontSlant->"Italic"], "s entre 0 et 40 s." }], "Body", FormatType->TextForm], Cell[TextData[{ "Cherchons la solution g\[EAcute]n\[EAcute]rale ", Cell[BoxData[ \(TextForm\`z\_1\)]], "(", StyleBox["t", FontSlant->"Italic"], ") lorsque le sauteur n'est soumis qu'\[AGrave] son poids et \[AGrave] une \ force de frottement proportionnelle \[AGrave] sa vitesse (premi\[EGrave]re \ \[EAcute]quation) pour une vitesse initiale nulle (deuxi\[EGrave]me \ \[EAcute]quation) et une position initiale \[EAcute]gale \[AGrave] 0 (troisi\ \[EGrave]me \[EAcute]quation):" }], "Body"], Cell[BoxData[ \(sol1 = DSolve[{\(z''\)[t] \[Equal] g - b*\(z'\)[t]/m, \(z'\)[0] \[Equal] 0, z[0] \[Equal] 0}, z[t], t]\)], "Input"], Cell["\<\ Introduisons les donn\[EAcute]es num\[EAcute]riques dans la \ solution et calculons la position du sauteur apr\[EGrave]s 5 secondes:\ \>", \ "Body", FormatType->TextForm], Cell[BoxData[{ \(\(dn = {b \[Rule] 14, g \[Rule] 10, k \[Rule] 70, m \[Rule] 70};\)\), "\[IndentingNewLine]", \(z0 = \(sol1[\([1, 1, 2]\)] /. dn\) /. t \[Rule] 5\), "\[IndentingNewLine]", \(% // N\)}], "Input"], Cell["Repr\[EAcute]sentons la position du sauteur en fonction du temps:", \ "Body", FormatType->TextForm], Cell[BoxData[{ \(\(fs = 10;\)\), "\[IndentingNewLine]", \(\(sol1[\([1, 1, 2]\)] /. dn;\)\), "\[IndentingNewLine]", \(\(h1 = Plot[%, {t, 0, 5}, PlotStyle \[Rule] {Hue[1]}, AxesLabel \[Rule] {"\", "\"}, TextStyle \[Rule] {FontFamily \[Rule] Verdana, FontSize \[Rule] fs, FontSlant \[Rule] Italic}];\)\)}], "Input"], Cell["Calculons la vitesse du sauteur apr\[EGrave]s 5 secondes:", "Body"], Cell[BoxData[{ \(v0 = \(D[sol1[\([1, 1, 2]\)], t] /. dn\) /. t \[Rule] 5\), "\[IndentingNewLine]", \(% // N\)}], "Input"], Cell[TextData[{ "Cherchons la solution g\[EAcute]n\[EAcute]rale ", Cell[BoxData[ \(TextForm\`z\_2\)]], "(", StyleBox["t", FontSlant->"Italic"], ") lorsque le sauteur est soumis \[AGrave] son poids, \[AGrave] une force \ de frottement proportionnelle \[AGrave] sa vitesse et \[AGrave] une force de \ rappel proportionnelle \[AGrave] l'\[EAcute]longation de \ l'\[EAcute]lastique:" }], "Body", FormatType->TextForm], Cell[BoxData[ \(sol2 = DSolve[{\(z''\)[t] \[Equal] g - \((k \((z[t] - z0)\) + b*\(z'\)[t])\)/m, \(z'\)[0] \[Equal] v0, z[0] \[Equal] z0}, z[t], t]\)], "Input"], Cell["Repr\[EAcute]sentons l'horaire du sauteur:", "Body", FormatType->TextForm], Cell[BoxData[{ \(\(sol2[\([1, 1, 2]\)] /. dn;\)\), "\[IndentingNewLine]", \(\(Plot[%, {t, 0, 40}, TextStyle \[Rule] {FontFamily \[Rule] Verdana, FontSize \[Rule] fs, FontSlant \[Rule] Italic}];\)\)}], "Input"], Cell["\<\ Cette courbe doit \[EHat]tre d\[EAcute]cal\[EAcute]e vers la droite \ de 5 s car la force de rappel n'agit qu'apr\[EGrave]s ce laps de temps. \ Translatons la courbe de 5 seconde vers la droite et redessinons-la:\ \>", \ "Body"], Cell[BoxData[{ \(\(\(sol2[\([1, 1, 2]\)] /. dn\) /. t \[Rule] t - 5;\)\), "\[IndentingNewLine]", \(\(h2 = Plot[%, {t, 5, 40}, PlotStyle \[Rule] {Hue[2/3]}, TextStyle \[Rule] {FontFamily \[Rule] Verdana, FontSize \[Rule] fs, FontSlant \[Rule] Italic}];\)\)}], "Input"], Cell["Assemblons les deux horaires sur un m\[EHat]me graphique:", "Body", FormatType->TextForm], Cell[BoxData[ \(\(Show[h1, h2, PlotRange \[Rule] All, TextStyle \[Rule] {FontFamily \[Rule] Verdana, FontSize \[Rule] fs, FontSlant \[Rule] Italic}];\)\)], "Input"] }, Open ]] }, FrontEndVersion->"5.2 for Macintosh", ScreenRectangle->{{0, 993}, {0, 746}}, WindowToolbars->{"RulerBar", "EditBar"}, WindowSize->{870, 613}, WindowMargins->{{20, Automatic}, {Automatic, 21}}, ShowCellLabel->False, CellLabelAutoDelete->True, Magnification->1.25, StyleDefinitions -> "HTML.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 83, 1, 62, "H2"], Cell[1862, 56, 1890, 50, 311, "Body"], Cell[3755, 108, 509, 12, 74, "Body"], Cell[4267, 122, 153, 3, 26, "Input"], Cell[4423, 127, 182, 5, 54, "Body"], Cell[4608, 134, 244, 5, 53, "Input"], Cell[4855, 141, 107, 2, 34, "Body"], Cell[4965, 145, 380, 7, 67, "Input"], Cell[5348, 154, 73, 0, 34, "Body"], Cell[5424, 156, 139, 3, 39, "Input"], Cell[5566, 161, 435, 12, 74, "Body"], Cell[6004, 175, 196, 4, 26, "Input"], Cell[6203, 181, 82, 1, 34, "Body"], Cell[6288, 184, 240, 4, 53, "Input"], Cell[6531, 190, 238, 5, 54, "Body"], Cell[6772, 197, 320, 6, 53, "Input"], Cell[7095, 205, 97, 1, 34, "Body"], Cell[7195, 208, 190, 3, 39, "Input"] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)