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Que vaut la vitesse de chacune de ces particules un temps ", StyleBox["t", FontSlant->"Italic"], " apr\[EGrave]s qu'elles ont \[EAcute]t\[EAcute] l\[AHat]ch\[EAcute]es ?" }], "Text", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FormatType->TextForm], Cell[TextData[{ StyleBox["Corrig\[EAcute] 1", FontWeight->"Bold", FontSlant->"Italic"], "\nUne particule de charge ", StyleBox["q", FontSlant->"Italic"], " plac\[EAcute]e dans un champ \[EAcute]lectrique ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["E", FontSlant->"Italic"], FormBox["\[Rule]", "TextForm"]], TextForm]]], " subit une force ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["F", FontSlant->"Italic"], FormBox["\[Rule]", "TextForm"]], TextForm]]], "=", StyleBox["q", FontSlant->"Italic"], " ", Cell[BoxData[ FormBox[ OverscriptBox["E", FormBox["\[Rule]", "TextForm"]], TextForm]], FontSlant->"Italic"], ". Connaissant la masse m de la particule, on peut trouver son \ acc\[EAcute]l\[EAcute]ration ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["a", FontSlant->"Italic"], FormBox["\[Rule]", "TextForm"]], TextForm]]], "=", Cell[BoxData[ FormBox[ FractionBox[ FormBox[ OverscriptBox[ StyleBox["F", FontSlant->"Italic"], FormBox["\[Rule]", "TextForm"]], "TextForm"], StyleBox["m", FontSlant->"Italic"]], TextForm]]], ", puis la distance ", " ", StyleBox["d", FontSlant->"Italic"], "=", Cell[BoxData[ \(TextForm\`1\/2\)]], StyleBox["a", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["t", FontSlant->"Italic"], "2"], TextForm]]], " qu'elle parcourt en un temps ", StyleBox["t", FontSlant->"Italic"], ". Le travail ", StyleBox["qU", FontSlant->"Italic"], " = ", StyleBox["qEd", FontSlant->"Italic"], " effectu\[EAcute] par la force sur la particule est \[EAcute]gal \[AGrave] \ l'\[EAcute]nergie sa variation d'\[EAcute]nergie cin\[EAcute]tique, ce qui \ permet d'exprimer v en fonction des quantit\[EAcute]s donn\[EAcute]es ", StyleBox["v", FontSlant->"Italic"], "=", Cell[BoxData[ \(TextForm\`qE\/m\)], FontSlant->"Italic"], StyleBox["t", FontSlant->"Italic"] }], "Text", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FormatType->TextForm], Cell[BoxData[ \(v[champE_, \ t_, q_, m_] := q*champE/m*t\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(v[580, 4.8*10^\(-8\), e, me]\)\(\ \)\( (*\ vitesse\ de\ l' \[EAcute]lectron\ *) \)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(4.889840276634283`*^6\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(v[580, 4.8*10^\(-8\), e, mp]\)\( (*\ vitesse\ du\ proton\ *) \)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(2663.159153413847`\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Question 2", "Subsubsection", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FormatType->TextForm], Cell[TextData[{ "Un proton est projet\[EAcute] selon un axe ", StyleBox["Ox", FontSlant->"Italic"], " dans une r\[EAcute]gion o\[UGrave] r\[EGrave]gne un champ \ \[EAcute]lectrique uniforme ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["E", FontSlant->"Italic"], FormBox["\[Rule]", "TextForm"]], TextForm]]], " qui a m\[EHat]me direction que l'axe ", StyleBox["Ox", FontSlant->"Italic"], " mais qui est de sens oppos\[EAcute] au d\[EAcute]placement du proton. Le \ proton parcourt une distance ", StyleBox["d", FontSlant->"Italic"], " avant de s'immobiliser. \na) Que vaut l'acc\[EAcute]l\[EAcute]ration du \ proton ?\nb) Quelle est sa vitesse initiale ?\nc) Combien de temps ce \ freinage a-t-il dur\[EAcute] ?" }], "Text", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FormatType->TextForm], Cell[TextData[{ StyleBox["Corrig\[EAcute] 2", FontWeight->"Bold", FontSlant->"Italic"], "\na) La grandeur de l'acc\[EAcute]l\[EAcute]ration est donn\[EAcute]e ", StyleBox["a", FontSlant->"Italic"], "=", Cell[BoxData[ \(TextForm\`F\/m\)], FontSlant->"Italic"], "=", Cell[BoxData[ \(TextForm\`qE\/m\)], FontSlant->"Italic"], "\nb) Le travail de la force exerc\[EAcute]e sur le proton est \[EAcute]gal \ \[AGrave] sa variation d'\[EAcute]nergie cin\[EAcute]tique -", StyleBox["qEd", FontSlant->"Italic"], "=-", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"m", FormBox[\(v\^2\), "TextForm"]}], "2"], TextForm]], FontSlant->"Italic"], ", ce qui permet d'exprimer sa vitesse initiale ", StyleBox["v", FontSlant->"Italic"], "=", Cell[BoxData[ FormBox[ SqrtBox[ FormBox[ StyleBox[\(\(2 qEd\)\/m\), FontSlant->"Italic"], "TextForm"]], TextForm]]], "\nc) Le proton est anim\[EAcute] d'un MRUA. La dur\[EAcute]e du freinage \ est donn\[EAcute]e par", StyleBox[" t", FontSlant->"Italic"], "=", Cell[BoxData[ \(TextForm\`v\/a\)], FontSlant->"Italic"], "=", Cell[BoxData[ FormBox[ FractionBox[ SqrtBox[ FormBox[ StyleBox[\(\(2 qEd\)\/m\), FontSlant->"Italic"], "TextForm"]], StyleBox[\(qE\/m\), FontSlant->"Italic"]], TextForm]]], "=", Cell[BoxData[ FormBox[ SqrtBox[ StyleBox[\(\(2 dm\)\/qE\), FontSlant->"Italic"]], TextForm]]] }], "Text", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FormatType->TextForm], Cell[BoxData[ \(a[champE_, \ q_, \ m_] := q*champE/m\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(a[8*10^5, e, mp]\)\(\ \)\( (*\ acc\[EAcute]l\[EAcute]ration\ *) \)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(7.652756187970825`*^13\)], "Output"] }, Open ]], Cell[BoxData[ \(v[champE_, \ d_, q_, m_] := Sqrt[2 q*champE*d/m]\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(v[8*10^5, 0.07, e, mp]\)\( (*\ vitesse\ *) \)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(3.273203119752753`*^6\)], "Output"] }, Open ]], Cell[BoxData[ \(t[champE_, \ d_, q_, m_] := Sqrt[2 d*m/\((q*champE)\)]\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(t[8*10^5, 0.07, e, mp]\)\(\ \)\( (*\ dur\[EAcute]e\ du\ freinage\ *) \)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(4.277155889139417`*^-8\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Question 3", "Subsubsection", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FormatType->TextForm], Cell[TextData[{ "Un proton part de l'arr\[EHat]t et acc\[EAcute]l\[EGrave]re dans un champ \ \[EAcute]lectrique uniforme ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["E", FontSlant->"Italic"], FormBox["\[Rule]", "TextForm"]], TextForm]]], ". Un instant plus tard, sa vitesse - non relativiste car beaucoup plus \ petite que la vitesse de la lumi\[EGrave]re, vaut ", StyleBox["v", FontSlant->"Italic"], ".\na) Quelle est l'acc\[EAcute]l\[EAcute]ration de ce proton ?\nb) Quel \ temps faut-il au proton pour atteindre cette vitesse ?\nc) Quelle distance \ a-t-il parcourue lorsqu'il atteint cette vitesse ?\nd) Que vaut alors son \ \[EAcute]nergie cin\[EAcute]tique ?" }], "Text", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FormatType->TextForm], Cell[TextData[{ StyleBox["Corrig\[EAcute] 3", FontSlant->"Italic"], "\na) La grandeur de l'acc\[EAcute]l\[EAcute]ration est donn\[EAcute]e par \ ", Cell[BoxData[ FormBox[ StyleBox["a", FontSlant->"Italic"], TextForm]]], "=", Cell[BoxData[ FormBox[ FractionBox[ FormBox[ StyleBox["F", FontSlant->"Italic"], "TextForm"], StyleBox["m", FontSlant->"Italic"]], TextForm]]], "=", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ StyleBox["q", FontSlant->"Italic"], FormBox[ StyleBox["E", FontSlant->"Italic"], "TextForm"]}], StyleBox["m", FontSlant->"Italic"]], TextForm]]], "\nb) Le temps n\[EAcute]cessaire pour atteindre cette vitesse s'obtient \ par ", StyleBox["t", FontSlant->"Italic"], "=", Cell[BoxData[ FormBox[ FractionBox["v", StyleBox["a", FontSlant->"Italic"]], TextForm]]], "=", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["v", FontSlant->"Italic"], FractionBox[ RowBox[{ StyleBox["q", FontSlant->"Italic"], FormBox[ StyleBox["E", FontSlant->"Italic"], "TextForm"]}], StyleBox["m", FontSlant->"Italic"]]], TextForm]]], "=", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["mv", FontSlant->"Italic"], StyleBox["qE", FontSlant->"Italic"]], TextForm]]], "\nc) La distance parcourue (MRUA) est donn\[EAcute]e par ", StyleBox["d", FontSlant->"Italic"], "=", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["vt", FontSlant->"Italic"], "2"], TextForm]]], "=", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[ RowBox[{"m", FormBox[ SuperscriptBox["v", StyleBox["2", FontSlant->"Plain"]], "TextForm"]}], FontSlant->"Italic"], RowBox[{"2", StyleBox["qE", FontSlant->"Italic"]}]], TextForm]]], "\nd) Son \[EAcute]nergie cin\[EAcute]tique est donn\[EAcute]e par ", Cell[BoxData[ \(TextForm\`E\_cin\)], FontSlant->"Italic"], "=", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[ RowBox[{"m", FormBox[ SuperscriptBox["v", StyleBox["2", FontSlant->"Plain"]], "TextForm"]}], FontSlant->"Italic"], "2"], TextForm]]] }], "Text", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FormatType->TextForm], Cell[BoxData[ \(a[champE_, q_, m_] := q*champE/m\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(a[360, e, mp]\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(3.443740284586871`*^10\)], "Output"] }, Open ]], Cell[BoxData[ \(t[champE_, v_, q_, m_] := m*v/\((q*champE)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(t[360, 8*10^5, e, mp]\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(0.000023230555555555555`\)], "Output"] }, Open ]], Cell[BoxData[ \(d[champE_, v_, q_, m_] := m*v^2/\((2 q*champE)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(d[360, 8*10^5, e, mp]\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(9.292222222222222`\)], "Output"] }, Open ]], Cell[BoxData[ \(Ecin[v_, m_] := m*v^2/2\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(Ecin[8*10^5, mp]\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(5.35232`*^-16\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Question 4", "Subsubsection", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FormatType->TextForm], Cell[TextData[{ " Un proton se d\[EAcute]place \[AGrave] la vitesse ", StyleBox["v", FontSlant->"Italic"], " horizontalement. Il p\[EAcute]n\[EGrave]tre dans un champ \ \[EAcute]lectrique uniforme vertical ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["E", FontSlant->"Italic"], FormBox["\[Rule]", "TextForm"]], TextForm]]], ". \na) Quel temps faut-il \[AGrave] ce proton pour parcourir une distance \ ", StyleBox["x", FontSlant->"Italic"], " horizontalement ?\nb) Quel d\[EAcute]placement vertical ", StyleBox["y", FontSlant->"Italic"], " a-t-il subi apr\[EGrave]s avoir parcouru cette distance horizontale ", StyleBox["x ", FontSlant->"Italic"], "?", StyleBox["\n", FontSlant->"Italic"], "c) Que valent les composantes horizontale ", Cell[BoxData[ FormBox[ StyleBox[\(v\_x\), FontSlant->"Italic"], TextForm]]], " ", "et verticale ", Cell[BoxData[ FormBox[ StyleBox[\(v\_y\), FontSlant->"Italic"], TextForm]]], " ", "de sa vitesse lorsqu'il a parcouru cette distance horizontale ", StyleBox["x ", FontSlant->"Italic"], "?\n", StyleBox["N. B.", FontWeight->"Bold"], " ", "On n\[EAcute]glige tout effet gravitationnel dans ce probl\[EGrave]me." }], "Text", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FormatType->TextForm], Cell[TextData[{ StyleBox["Corrig\[EAcute] 4", FontSlant->"Italic"], "\na) Le mouvement horizontal du proton est une MRU. Le temps \ n\[EAcute]cessaire pour franchir une distance ", StyleBox["x", FontSlant->"Italic"], " vaut ", StyleBox["t", FontSlant->"Italic"], "=", Cell[BoxData[ FormBox[ FractionBox["x", StyleBox["v", FontSlant->"Italic"]], TextForm]]], "\nb) Le mouvement vertical du proton est une MRUA. La distance y franchie \ par le proton durant ce temps est donn\[EAcute]e par ", StyleBox["y", FontSlant->"Italic"], "=", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ StyleBox["1", FontSlant->"Plain"], StyleBox["2", FontSlant->"Plain"]], FontSlant->"Italic"], TextForm]]], StyleBox["a", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["t", FontSlant->"Italic"], "2"], TextForm]]], "=", Cell[BoxData[ \(TextForm\`1\/2\)]], Cell[BoxData[ FormBox[ FractionBox[ StyleBox["F", FontSlant->"Italic"], StyleBox["m", FontSlant->"Italic"]], TextForm]]], Cell[BoxData[ SuperscriptBox[ StyleBox["t", FontSlant->"Italic"], "2"]]], "=", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ StyleBox["qE", FontSlant->"Italic"], FormBox[ SuperscriptBox[ StyleBox["x", FontSlant->"Italic"], "2"], "TextForm"]}], RowBox[{"2", StyleBox["m", FontSlant->"Italic"], FormBox[ SuperscriptBox[ StyleBox["v", FontSlant->"Italic"], "2"], "TextForm"]}]], TextForm]]], "\nc) La ", "composante horizontale ", Cell[BoxData[ FormBox[ StyleBox[\(v\_x\), FontSlant->"Italic"], TextForm]]], " de la vitesse est constante et \[EAcute]gale ", StyleBox["v. ", FontSlant->"Italic"], "La ", "composante verticale ", "est donn\[EAcute]e par ", Cell[BoxData[ FormBox[ StyleBox[\(v\_y\), FontSlant->"Italic"], TextForm]]], "=at=", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["qEx", FontSlant->"Italic"], StyleBox["mv", FontSlant->"Italic"]], TextForm]]] }], "Text", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FormatType->TextForm], Cell[BoxData[ \(t[v_, x_] := x/v\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(t[6.4*10^5, 0.07]\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(1.0937500000000001`*^-7\)], "Output"] }, Open ]], Cell[BoxData[ \(y[v_, champE_, x_, q_, m_] := q*champE*x^2/\((2 m*v^2)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(y[6.4*10^5, 9.6*10^3, 0.07, e, mp]\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(0.005492945115389216`\)], "Output"] }, Open ]], Cell[BoxData[ \(vy[v_, champE_, x_, q_, m_] := q*champE*x/\((m*v)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(vy[6.4*10^5, 9.6*10^3, 0.07, e, mp]\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(100442.42496711707`\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Question 5", "Subsubsection", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FormatType->TextForm], Cell[TextData[{ "Un \[EAcute]lectron est projet\[EAcute] sous un angle \[Theta] par rapport \ \[AGrave] l'horizontale \[AGrave] une vitesse ", StyleBox["v ", FontSlant->"Italic"], "dans une r\[EAcute]gion de l'espace o\[UGrave] r\[EGrave]gne un champ \ \[EAcute]lectrique vertical ", Cell[BoxData[ OverscriptBox[ StyleBox["E", FontSlant->"Italic"], FormBox["\[Rule]", "TextForm"]]]], ".\na) Quel temps faut-il \[AGrave] cet \[EAcute]lectron pour retourner \ \[AGrave] sa hauteur initiale ?\nb) Quelle hauteur maximale atteint-il ?\nc) \ Que vaut son d\[EAcute]placement horizontal lorsqu'il atteint cette hauteur \ maximale ?\n", StyleBox["N. B.", FontWeight->"Bold"], " ", "On n\[EAcute]glige tout effet gravitationnel dans ce probl\[EGrave]me." }], "Text", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FormatType->TextForm], Cell[TextData[{ StyleBox["Corrig\[EAcute] 5", FontSlant->"Italic"], "\na) Le mouvement vertical de l'\[EAcute]lectron est une MRUA. Le temps n\ \[EAcute]cessaire pour retourner \[AGrave] sa hauteur initiale est le double \ du temps de mont\[EAcute]e. 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