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Concours national Deug</title></head><body style="background-color: rgb(255, 255, 255);" alink="#0099ff" vlink="#0099ff"> <center> <p> <table border="0"> <tbody> <tr> <td width="1373"> <center><span style="font-weight: bold; text-decoration: underline;"></span><span style="text-decoration: underline;"></span><span style="font-family: Times New Roman,Times,serif;"><span style="text-decoration: underline; font-weight: bold;"></span><span style="text-decoration: underline;"></span><span style="text-decoration: underline; font-weight: bold;"></span></span><span style="font-family: Times New Roman,Times,serif;"><span style="color: black;"><span style="font-weight: bold; text-decoration: underline;"></span></span></span><span style="font-weight: bold; text-decoration: underline;"></span><span style="text-decoration: underline; font-weight: bold;"></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"> </span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"><span style="text-decoration: underline;"></span></span><span style="font-family: Times New Roman,Times,serif;"><span style="color: black;"></span></span><font color="#009900"><span style="font-family: Times New Roman,Times,serif;"><span style="color: black;"></span></span></font> <hr style="height: 2px; width: 1%;" noshade="noshade"><span style="font-family: Times New Roman,Times,serif;"><span style="color: black;"></span></span><span style="font-family: Times New Roman,Times,serif;"><span style="color: black;"></span></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"></span><span style="font-family: Times New Roman,Times,serif;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></span><span style="font-size: 10pt; font-family: &quot;Arial&quot;,&quot;sans-serif&quot;;"><big><span style="font-family: Times New Roman,Times,serif;"></span></big><o:p></o:p></span><font color="#009900"><span style="color: black;"></span></font><span style="color: rgb(0, 153, 0);"><span style="color: black;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></span></span><span style="color: black;"></span><span style="font-weight: bold; color: rgb(0, 153, 0);"></span> <p><font color="#009900"> <table style="width: 1008px; height: 152px;" border="0"> <tbody> <tr> <td colspan="2" rowspan="1" style="vertical-align: top;"><br style="font-weight: bold;"> <div style="text-align: center;"><span style="font-family: Times New Roman; font-weight: bold;"></span><font color="#000000" face="Times New Roman" size="+2"><b>Repre de Frenet. Concours national Deug<br> </b></font><font color="#009900"><font color="#009900"><p><small style="color: black;"><small><em style="font-family: Georgia,&quot;Times New Roman&quot;,Times,serif; font-size: 20px; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; background-color: rgb(255, 255, 255);"><small>En poursuivant votre navigation sur ce site, vous acceptez l utilisation de <strong style="font-weight: normal;">Cookies</strong><strong style="font-weight: bold;"> </strong>vous proposant</small><small><strong style="font-weight: normal;"></strong><strong style="font-weight: bold;"> </strong><strong style="font-weight: normal;">des publicits adaptes vos centres d intrts.</strong></small> </em></small></small></p></font></font><br> <hr style="width: 10%; height: 2px;"></div> </td> </tr> <tr> <td colspan="2" rowspan="1" style="vertical-align: top;"> <div align="center"> 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</td> <td style="vertical-align: top;"> <div style="text-align: center;"><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"></span></font></font></font><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"><span style="text-decoration: underline; font-weight: bold;"></span><span style="font-family: Times New Roman;"><span style="font-weight: bold; text-decoration: underline;"></span></span></span></font></font><span style="font-weight: bold; text-decoration: underline;"></span><span style="text-decoration: underline;"></span><span style="font-weight: bold; color: rgb(0, 153, 0); text-decoration: underline;"><span style="color: black;"></span></span><span style="color: rgb(0, 153, 0);"><span style="color: black;"></span></span><span style="font-weight: bold; color: rgb(0, 153, 0); text-decoration: underline;"><span style="color: black;"></span></span><span style="font-weight: bold; text-decoration: underline;"></span><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; text-decoration: underline;"></span><span style="font-weight: bold; color: rgb(0, 153, 0);"></span><span style="font-weight: bold; text-decoration: underline;"></span><span style="font-style: italic; font-weight: bold;">Les vecteurs sont crits en gras et en bleu</span>.<br> Le rfrentiel R est considr comme galilen ; il est rapport au repre (O, <span style="font-weight: bold; color: rgb(51, 51, 255);">x</span>, <span style="font-weight: bold; color: rgb(51, 51, 255);">y</span><span style="color: rgb(51, 51, 255);"><span style="color: black;">,</span></span> <span style="font-weight: bold; color: rgb(51, 51, 255);">z</span> )<br> Un point matriel M se dplace sur une courbe dfinie par les quations paramtriques suivantes :<br> x = 2A e<sup><span style="font-family: Symbol;">a</span>t</sup> sin <span style="font-family: Symbol;">a</span>t<br> y = </span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">= 2A e<sup><span style="font-family: Symbol;">a</span>t</sup> cos <span style="font-family: Symbol;">a</span>t<br> z = </span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">A e<sup><span style="font-family: Symbol;">a</span>t</sup></span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">.<br> A et <span style="font-family: Symbol;">a</span> sont des constantes&nbsp;</span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">et x, y, z sont les coordonnes cartsiennes du point M l instant t.<br> On dsigne par ( M, <span style="font-family: Symbol; font-weight: bold; color: rgb(51, 51, 255);">t</span>, <span style="font-weight: bold; color: rgb(51, 51, 255);">n</span>, <span style="font-weight: bold; color: rgb(51, 51, 255);">b</span> )&nbsp; le repre de Frenet, <span style="font-weight: bold; color: rgb(51, 51, 255);"></span> tant le vecteur unitaire tangent en M latrajectoire, orient dans le sens du mouvement, <span style="font-weight: bold; color: rgb(51, 51, 255);">n</span> le vecteur unitaire normal en M <span style="font-weight: bold; color: rgb(51, 51, 255);"></span> dirig versle centre de courbure et <span style="font-weight: bold; color: rgb(51, 51, 255);">b</span> complte le repre afin qu il soit orthonorm direct.<br> On note s l abscisse curviligne du point M et R le rayon de courbure de la trajectoire au point M.<br> <span style="font-weight: bold; color: rgb(0, 153, 0);">Exprimer dans R la vitesse V du point M ainsi que sa norme en fonction de A, </span><span style="font-family: Symbol; font-weight: bold; color: rgb(0, 153, 0);">a</span><span style="font-weight: bold; color: rgb(0, 153, 0);"> et t</span>.<br> v<sub>x</sub> = dx/dt =</span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">2<span style="font-family: Symbol;">a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> sin <span style="font-family: Symbol;">a</span>t +</span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">2<span style="font-family: Symbol;">a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> cos <span style="font-family: Symbol;">a</span>t = </span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">2<span style="font-family: Symbol;">a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> (sin <span style="font-family: Symbol;">a</span>t +</span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">cos <span style="font-family: Symbol;">a</span>t).</span></span></span></font></font></font></font></font></font><br> <font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">v<sub>y</sub>&nbsp; </span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">= dy/dt =</span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">2<span style="font-family: Symbol;">a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> cos <span style="font-family: Symbol;">a</span>t -</span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">2<span style="font-family: Symbol;">a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> sin <span style="font-family: Symbol;">a</span>t </span></span></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">= </span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">2<span style="font-family: Symbol;">a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> (cos <span style="font-family: Symbol;">a</span>t - sin</span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> <span style="font-family: Symbol;">a</span>t).</span></span></span></font></font></font></font></font></font></font><br> <font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">v<sub>z</sub>&nbsp; </span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">= dz/dt =</span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup>.<br> v = (</span></span></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">v<sub>x</sub><sup>2</sup>+</span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">v<sub>y</sub><sup>2</sup>+</span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">v<sub>z</sub><sup>2</sup></span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">)<sup></sup> =</span></span></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">2a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> (</span></span></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">(sin <span style="font-family: Symbol;">a</span>t +</span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">cos <span style="font-family: Symbol;">a</span>t)<sup>2</sup>+</span></span></span></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">(cos <span style="font-family: Symbol;">a</span>t - sin</span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> <span style="font-family: Symbol;">a</span>t)<sup>2</sup>+0,25)<sup></sup>.<br> v=</span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">2a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> (</span></span></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">2sin<sup>2</sup> <span style="font-family: Symbol;">a</span>t +2</span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">cos<sup>2</sup> <span style="font-family: Symbol;">a</span>t</span></span></span></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">+0,25)<sup></sup>.<br> </span></span></span></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">v=</span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">2a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> (1-cos 2</span></span></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> </span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">a</span>t </span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">+1+</span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">cos 2</span></span></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> </span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">a</span>t</span></span></span></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">+0,25)<sup></sup>.<br> </span></span></span></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">Norme de la vitesse : v=</span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">3a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> .</span></span></span></font></font></font></font></font></font></font></font></font><br> <font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; color: rgb(0, 153, 0);">En dduire l expression de</span> <span style="font-weight: bold; color: rgb(51, 51, 255);"></span> dans le repre </span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">(O, <span style="font-weight: bold; color: rgb(51, 51, 255);">x</span>, <span style="font-weight: bold; color: rgb(51, 51, 255);">y</span><span style="color: rgb(51, 51, 255);"><span style="color: black;">,</span></span> <span style="font-weight: bold; color: rgb(51, 51, 255);">z</span> )</span></span></span></font></font></font></font>.<br> Les vecteurs vitesse et <span style="font-weight: bold; color: rgb(51, 51, 255);">t</span> sont colinaires et de mme sens.<br> <font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">v<sub>x</sub> /v = </span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">2<span style="font-family: Symbol;">/3</span> (sin <span style="font-family: Symbol;">a</span>t +</span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">cos <span style="font-family: Symbol;">a</span>t).</span></span></span></font></font></font></font></font></font><br> <font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">v<sub>y</sub> /v</span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> </span></span></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">= </span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">2<span style="font-family: Symbol;">/3</span> (cos <span style="font-family: Symbol;">a</span>t - sin</span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> <span style="font-family: Symbol;">a</span>t).</span></span></span></font></font></font></font></font></font></font><br> <font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">v<sub>z</sub>&nbsp; /v= 1/3</span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">.</span></span></span></font></font></font></font></font></font></font><br> <font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="color: black;"></span></span></span></span></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><span style="font-family: Times New Roman,Times,serif;"><span style="color: black;"></span></span></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></span></span></span></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; color: rgb(0, 153, 0);">Montrer que la vitesse</span> <span style="font-weight: bold; color: rgb(51, 51, 255);">v</span> <span style="font-weight: bold; color: rgb(0, 153, 0);">du point M fait un angle </span><span style="font-family: Symbol; font-weight: bold; color: rgb(0, 153, 0);">q</span><span style="font-weight: bold; color: rgb(0, 153, 0);"> constant avec l axe z</span>.<br> Faire le produit scalaire <span style="font-weight: bold; color: rgb(51, 51, 255);">v</span> . <span style="font-weight: bold; color: rgb(51, 51, 255);">u</span><sub style="font-weight: bold; color: rgb(51, 51, 255);">z</sub>= v cos <span style="font-family: Symbol;">q</span> =v<sub>x</sub> </span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; color: rgb(51, 51, 255);">u</span><sub style="font-weight: bold; color: rgb(51, 51, 255);">x</sub></span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">.</span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; color: rgb(51, 51, 255);">u</span><sub style="font-weight: bold; color: rgb(51, 51, 255);">z</sub></span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> +</span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">v<sub>y</sub> </span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; color: rgb(51, 51, 255);">u</span><sub style="font-weight: bold; color: rgb(51, 51, 255);">y</sub></span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">.</span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; color: rgb(51, 51, 255);">u</span><sub style="font-weight: bold; color: rgb(51, 51, 255);">z</sub></span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> +</span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">v<sub>z</sub> </span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; color: rgb(51, 51, 255);">u</span><sub style="font-weight: bold; color: rgb(51, 51, 255);">z</sub></span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">.</span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; color: rgb(51, 51, 255);">u</span><sub style="font-weight: bold; color: rgb(51, 51, 255);">z</sub></span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> = v<sub>z</sub> = </span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup>.</span></span></span></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><br> </span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">3a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> </span></span></span></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">cos <span style="font-family: Symbol;">q</span> =</span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> ; </span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">cos <span style="font-family: Symbol;">q</span> =1/3 ; <span style="font-family: Symbol;">q</span> =1,23 rad. </span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><br> </span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> <br> </span></span></span></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></span></span></span></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font><font color="#009900"><span style="color: black;"></span><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="font-weight: normal; color: black;" color="#009900"><font style="color: black;"><span style="font-family: Times New Roman;"></span></font></font></font></font></font><font color="#009900"><span style="color: black; font-weight: normal;"><span style="font-family: Times New Roman;"></span></span></font><font color="#009900"><span style="color: black; font-weight: normal;"><span style="font-family: Times New Roman;"> </span></span></font><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Times New Roman;"> </span></font><font style="color: black; font-style: italic;" color="#009900"><span style="font-family: Times New Roman;"> </span></font></font><span style="color: black; font-weight: bold;"></span> <span style="font-weight: bold; color: rgb(0, 153, 0);"></span></div> 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style="font-weight: normal; color: black;"><span style="font-family: Times New Roman,Times,serif;"><span style="font-weight: bold; text-decoration: underline;"></span></span></span><font color="#009900"><span style="font-weight: normal; color: black;"><span style="text-decoration: underline; font-weight: bold;"></span></span></font><span style="font-weight: normal; color: black;"><span style="font-family: Times New Roman,Times,serif;"><span style="font-weight: bold; text-decoration: underline;"></span></span></span><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="font-weight: bold;"></span></span></span></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font style="font-weight: normal; color: black;" color="#009900"><font style="color: black;"><span style="font-family: Times New Roman;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span><span style="font-weight: bold; text-decoration: underline;"></span><span style="text-decoration: underline;"></span></span></font></font></font><font color="#009900"><font color="#009900"><font style="font-weight: normal; color: black;" color="#009900"><font style="color: black;"><span style="font-family: Times New Roman;"></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="font-weight: normal; color: black;" color="#009900"><font style="color: black;"><span style="font-family: Times New Roman;"></span></font></font></font></font></font><font style="font-weight: normal; color: black;" color="#009900"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"></span></font></font></font></font><span style="color: black; font-weight: normal;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></span><span style="color: black; font-weight: normal;"></span><span style="color: black; font-weight: normal;"><span style="color: rgb(0, 153, 0);"><span style="color: black;"></span></span></span><font color="#009900"><span style="color: black;"></span></font><font style="color: black; font-weight: normal;" color="#009900"><span style="font-weight: normal;"><span style="font-family: Times New Roman;"><span style="font-weight: bold;"></span><span style="font-weight: bold; text-decoration: underline;"></span></span></span></font><font style="color: black; font-weight: normal;" color="#009900"><span style="font-weight: normal;"><span style="font-family: Times New Roman;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span><span style="text-decoration: underline; font-weight: bold;"></span></span></span></font><font color="#009900"><font color="#009900"><span style="color: black; font-weight: normal;"></span></font></font> <font style="color: black; font-weight: normal;" color="#009900"><span style="font-weight: normal;"><span style="font-family: Times New Roman;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></span></span></font><span style="font-weight: normal; color: black;"><span style="color: black;"><span style="font-weight: bold; text-decoration: underline;"></span><span style="text-decoration: underline;"></span><span style="font-weight: bold; text-decoration: underline;"></span></span></span><span style="color: black; font-weight: normal;"><span style="font-weight: bold; text-decoration: underline;"></span></span><font style="font-weight: normal;" color="#000000"><span style="color: black;"><span style="font-weight: bold; text-decoration: underline;"></span></span></font><font style="font-weight: normal;" color="#000000"><span style="font-weight: bold; text-decoration: underline;"></span></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; color: rgb(0, 153, 0);">Exprimer dans R l acclration du point M ainsi que sa norme en fonction de A, </span><span style="font-family: Symbol; font-weight: bold; color: rgb(0, 153, 0);">a</span><span style="font-weight: bold; color: rgb(0, 153, 0);"> et t</span>.<br> </span></span></span></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">a<sub>x</sub> = dv<sub>x</sub>/dt =</span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> </span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">2<span style="font-family: Symbol;">a<sup>2</sup></span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> (sin <span style="font-family: Symbol;">a</span>t +</span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">cos <span style="font-family: Symbol;">a</span>t)+</span></span></span></font></font></font></font></font></font></font><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">2a<sup>2</sup></span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> (</span></span></span></font></font></font></font></font></font></font><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">cos <span style="font-family: Symbol;">a</span>t -sin</span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> <span style="font-family: Symbol;">a</span>t)</span></span></span></font></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> =</span></span></span></font></font></font></font></font></font></font><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">4<span style="font-family: Symbol;">a<sup>2</sup></span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> cos <span style="font-family: Symbol;">a</span>t</span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">.</span></span></span></font></font></font></font></font></font><br> <font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">a<sub>y</sub>&nbsp; </span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">= d</span></span></span></font></font></font></font></font><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">v<sub>y</sub></span></span></span></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">/dt </span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">= </span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">2<span style="font-family: Symbol;"></span></span></span></span></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">a<sup>2</sup></span></span></span></span></font></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;"></span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> (cos <span style="font-family: Symbol;">a</span>t - sin</span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> <span style="font-family: Symbol;">a</span>t)+</span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> </span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">2<span style="font-family: Symbol;"></span></span></span></span></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">a<sup>2</sup></span></span></span></span></font></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;"></span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> (-sin <span style="font-family: Symbol;">a</span>t - cos</span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> <span style="font-family: Symbol;">a</span>t) =</span></span></span></font></font></font></font></font></font></font></font></font><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">-4<span style="font-family: Symbol;"></span></span></span></span></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">a<sup>2</sup></span></span></span></span></font></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;"></span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> </span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">sin <span style="font-family: Symbol;">a</span>t</span></span></span></font></font></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">.</span></span></span></font></font></font></font></font></font></font><br> <font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">a<sub>z</sub>&nbsp; </span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">= dv<sub>z</sub>/dt =</span></span></span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">a</span><sup>2</sup>A e<sup><span style="font-family: Symbol;">a</span>t</sup>.<br> Norme de l'acclration : a </span></span></span></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">= (</span></span></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">a<sub>x</sub><sup>2</sup>+</span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">a<sub>y</sub><sup>2</sup>+</span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">a<sub>z</sub><sup>2</sup></span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">)<sup></sup> =</span></span></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;"></span></span></span></span></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">4<span style="font-family: Symbol;">a<sup>2</sup></span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> </span></span></span></font></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">(</span></span></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">cos<sup>2</sup> <span style="font-family: Symbol;">a</span>t+</span></span></span></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> sin<sup>2</sup></span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"> <span style="font-family: Symbol;">a</span>t+1/16)<sup></sup>.<br> a=</span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;"></span></span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;"></span></span></span></span></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">4<span style="font-family: Symbol;">a<sup>2</sup></span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> </span></span></span></font></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">(</span></span></span></font></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;"></span> 17 / 16</span></span></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">)<sup></sup> =17<sup></sup></span></span></span></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">a<sup>2</sup></span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> </span></span></span></font></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;"></span>.</span></span></span></font></font></font></font></font></font></font></font></font><br> <font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; color: rgb(0, 153, 0);">Dterminer la norme a</span><sub style="font-weight: bold; color: rgb(0, 153, 0);">t</sub><span style="font-weight: bold; color: rgb(0, 153, 0);"> de l acclration tangentielle du point M en fonction de A,</span><span style="font-family: Symbol; font-weight: bold; color: rgb(0, 153, 0);">a</span><span style="font-weight: bold; color: rgb(0, 153, 0);"> et t</span>.<br> a<sub>t</sub> = dv/dt = </span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol; font-weight: normal;">3a<sup>2</sup></span><span style="font-weight: normal;">A e</span><sup style="font-weight: normal;"><span style="font-family: Symbol;">a</span>t</sup><span style="font-weight: normal;"> .</span></span></span></span></font></font></font></font></font></font></font></font></font></font><br> <font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; color: rgb(0, 153, 0);">&nbsp;En dduire la norme a</span><sub style="font-weight: bold; color: rgb(0, 153, 0);">n</sub><span style="font-weight: bold; color: rgb(0, 153, 0);"> de l acclration normale du point M</span>.<br> a<sup>2</sup> = a<sup>2</sup><sub>t</sub> + a<sup>2</sup><sub>n</sub> ; </span></span></span></font></font></font></font><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">a<sup>2</sup><sub>n</sub>&nbsp; =</span></span></span></font></font></font></font></font><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">a<sup>2</sup> - a<sup>2</sup><sub>t</sub>&nbsp; =</span></span></span></font></font></font></font></font><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">17</span></span></span></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">a<sup>4</sup></span>A<sup>2</sup> e<sup><span style="font-family: Symbol;">2a</span>t</sup> </span></span></span></font></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;"></span>-</span></span></span></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol; font-weight: normal;">9a<sup>4</sup></span><span style="font-weight: normal;">A<sup>2</sup> e</span><sup style="font-weight: normal;"><span style="font-family: Symbol;">2a</span>t</sup><span style="font-weight: normal;">&nbsp; =8</span></span></span></span></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol; font-weight: normal;">a<sup>4</sup></span><span style="font-weight: normal;">A<sup>2</sup> e</span><sup style="font-weight: normal;"><span style="font-family: Symbol;">2a</span>t</sup><span style="font-weight: normal;">&nbsp; ; a<sub>n</sub> = 8<sup></sup></span></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol; font-weight: normal;">a<sup>2</sup></span><span style="font-weight: normal;">A e</span><sup style="font-weight: normal;"><span style="font-family: Symbol;">a</span>t</sup><span style="font-weight: normal;"> .</span></span></span></span></font></font></font></font></font></font></font></font></font></font></font><br> <font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: bold; color: rgb(0, 153, 0);">En dduire le rayon de courbure R de la trajectoire au point M</span>.<br> </span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: normal;"> a<sub>n</sub> =v<sup>2</sup> /<span style="font-family: Times New Roman;">R</span> ; </span></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: normal;"><span style="font-family: Times New Roman;">R</span> = </span></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: normal;">v<sup>2</sup> /</span></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: normal;"> a<sub>n</sub> =(</span></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font></font></font><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol;">3a</span>A e<sup><span style="font-family: Symbol;">a</span>t</sup> </span></span></span></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: normal;">)<sup>2</sup>/ (</span></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: normal;">8<sup></sup></span></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-family: Symbol; font-weight: normal;">a<sup>2</sup></span><span style="font-weight: normal;">A e</span><sup style="font-weight: normal;"><span style="font-family: Symbol;">a</span>t</sup><span style="font-weight: normal;"> </span></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: normal;">) =9/8<sup></sup> </span></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: normal;">A e</span><sup style="font-weight: normal;"><span style="font-family: Symbol;">a</span>t</sup><span style="font-weight: normal;"> = </span></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: normal;">9/8<sup></sup> </span></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: normal;">z.</span></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font></font></font><br> <font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;">R est proportionnel la coordonne z du point M.</span></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"><span style="font-weight: normal;"></span></span></span></span></font></font></font></font></font></font><br> <font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="color: black;"></span></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-family: Times New Roman,Times,serif;"><span style="color: black;"><span style="font-weight: normal;"></span></span></span></font></font></font><span style="color: black; font-weight: normal;"></span><font color="#009900"><font color="#009900"><span style="color: black; font-weight: normal;"></span></font></font><font color="#009900"><span style="color: black; font-weight: normal;"> </span></font><font color="#009900"><font color="#009900"><span style="color: black; font-weight: normal;"></span></font></font><font color="#009900"><font color="#009900"><font color="#009900"><span style="color: black; font-weight: normal;"></span></font></font></font><span style="color: black; font-weight: normal;"></span><span style="color: black; font-weight: normal;"></span><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="color: black; font-weight: normal;"></span></font></font></font></font><font color="#009900"><span style="color: black; font-weight: normal;"></span></font><font color="#009900"><span style="color: black; font-weight: normal;"></span></font><font color="#009900"><span style="font-weight: normal; color: black;"></span></font><span style="font-weight: normal; color: black;"></span><font color="#009900"><font style="color: black; font-weight: normal;" color="#009900"><span style="font-weight: normal;"><span style="font-family: Times New Roman;"></span></span></font></font><span style="color: black; font-weight: normal;"></span> <font color="#009900"><font color="#009900"><font color="#009900"><font style="font-weight: normal; color: black;" color="#009900"><font style="color: black;"><span style="font-family: Times New Roman;"></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="font-weight: normal; color: black;" color="#009900"><font style="color: black;"><span style="font-family: Times New Roman;"></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"></span></span></font></font></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="color: black; font-weight: normal;"> </span></font></font></font></font></font></font></font><font color="#009900"><span style="font-weight: normal; color: black;"></span></font><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"></span></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"></span></font></font></font></font></font><span style="font-weight: normal; color: black;"> </span><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"><span style="color: black;"></span></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></span></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"></span></font></font></font></font></font></font><font color="#009900"><span style="font-weight: normal; color: black;"></span></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"></span></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"><span style="font-family: Symbol;"></span></span></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"></span></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"></span></span></span></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"></span></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"> </span></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"></span></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"></span></font></font></font><font color="#009900"><span style="font-weight: normal; color: black;"></span></font><font color="#009900"><font color="#009900"><span style="font-weight: normal; color: black;"></span></font></font></td> </tr> </tbody> </table> <span style="color: black;"><span style="font-family: Times New Roman,Times,serif;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></span></span></font><span style="font-family: Times New Roman,Times,serif;"><span style="text-decoration: underline; font-weight: bold;"></span></span><span style="font-family: Times New Roman,Times,serif;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></span><font color="#009900"><span style="color: black;"><span style="font-family: Times New Roman,Times,serif;"></span></span></font></p> <span style="font-weight: bold; color: rgb(0, 153, 0);"></span><span style="color: black;"><span style="text-decoration: underline; 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153, 0);"></span></font><font color="#000000"><span style="font-weight: bold; color: black; text-decoration: underline;"></span></font><b style=""><span style="color: rgb(0, 153, 0);"><o:p></o:p></span></b> <center> </center> <span style="font-weight: bold; text-decoration: underline;"></span> <table style="width: 1279px; height: 32px; text-align: left; margin-left: auto; margin-right: auto;" border="0" cellpadding="2" cellspacing="2"> <tbody> <tr> <td style="vertical-align: top;"> <div align="center"><span style="text-decoration: underline;"> </span> <center> <font color="#000000"> <script language="" type="text/javascript"><!-- google_ad_client = "pub-0015053057299262"; google_ad_width = 728; google_ad_height = 15; google_ad_format = "728x15_0ads_al"; google_ad_channel = ""; google_color_border = "2BA94F"; google_color_bg = "FFFFFF"; google_color_link = "CC0000"; google_color_text = "000000"; google_color_url = "CC0000"; //--></script> <script language="" type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"></script></font></center> <br> </div> </td> </tr> <tr> <td style="width: 1016px; vertical-align: top; text-align: center;"><font color="#009900"><span style="color: black; font-weight: normal;"><span style="font-weight: bold; color: black; text-decoration: underline;"></span><span style="font-weight: bold; text-decoration: underline;"></span></span></font><font color="#000000"><span style="color: black;"><span style="font-family: Times New Roman;"><span style="font-weight: bold; text-decoration: underline;"></span><span style="font-weight: bold; color: rgb(0, 153, 0);"></span><span style="color: black;"></span></span></span></font><font color="#000000"><span style="color: black;"><span style="font-family: Times New Roman;"></span></span></font><span style="font-weight: bold;"></span><span style="text-decoration: underline;"><span style="font-weight: bold;"></span></span><span style="text-decoration: underline;"></span><span style="font-weight: bold; text-decoration: underline;"></span><span style="text-decoration: underline;"></span><span style="font-weight: bold; text-decoration: underline;"></span><font style="text-decoration: underline; font-weight: bold;" color="#009900"><font style="font-weight: normal;" color="#000000"><span style="font-family: Times New Roman;"><span style="font-weight: bold;"></span></span></font></font><font style="font-weight: bold;" color="#009900"><font style="font-weight: normal;" color="#000000"><span style="font-family: Times New Roman;"></span></font></font> <center><font color="#000000" face="Times New Roman" size="+1">vitesse, acclration, repre de Frenet :<img src="../../ifrance/Images/fiches-bac/accelere130.gif" x-claris-useimagewidth="" x-claris-useimageheight="" align="middle" height="195" width="245"></font></center> <p>On donne les quations paramtriques de la trajectoire plane d'un point mobile par rapport un rfrentiel : x= 2t et y= 4t-4t.</p> <span style="font-weight: bold; color: rgb(0, 153, 0);">Dterminer l'quation de la trajectoire</span>.<br> Eliminer le temps entre x et y : t= x ; report dans y<br> y= x-2x trajectoire parabolique.<br> <span style="font-weight: bold; color: rgb(0, 153, 0);">Calculer la vitesse du mobile</span>.<br> vecteur vitesse : drive par rapport au temps du vecteur position <font color="#0033ff"><b>v </b></font>(dx/dt = 2 ; dy/dt = 8t-4)<br> valeur (norme) de la vitesse : v = v<sub>x</sub>+v<sub>y</sub> =<font color="#ff0000"><u>4+(8t-4)</u></font> = 64t-64t-20 en m s<sup>-1</sup>.<br> <span style="font-weight: bold; color: rgb(0, 153, 0);">Montrer que son acclration est constante</span>.<br> vecteur acclration : drive par rapport au temps du vecteur vitesse <font color="#0033ff"><b>a </b></font>(dv<sub>x</sub>/dt = 0 ; dv<sub>y</sub>/dt = 8)<br> valeur (norme) de l'acclration : a = a<sub>x</sub>+a<sub>y</sub> =0+8 soit a = <font color="#ff0000"><u>8 m s</u><sup>-2</sup></font>, indpendante du temps.<br> <span style="font-weight: bold; color: rgb(0, 153, 0);">Dterminer les composantes normale et tangentielle de l'acclration dans un repre de Frenet</span>.<br> vecteur acclration tangentielle, colinaire au vecteur vitesse :<br> a<sub>T</sub>=dv/dt avec v = [4+(8t-4)]<sup></sup>.<br> a<sub>T</sub>= dv/dt = 16(8 t -4)** [4+(8t-4)]<sup>-</sup>=8(8 t -4) [4+(8t-4)]<sup>-</sup>.<br> a = a<sub>T</sub>+ a<sub>N</sub> soit a<sub>N</sub> = a-a<sub>T</sub>= 8 -8(8 t -4) [4+(8t-4)]<sup>-1</sup>.<br> rduire au mme dnominateur [4+(8t-4)] :<br> a<sub>N</sub> =8*4 [4+(8t-4)]<sup>-1</sup> soit a<sub>N</sub>= <font color="#ff0000"><u>16[4+(8t-4)]</u><sup>-</sup></font>.<br> <span style="font-weight: bold; color: rgb(0, 153, 0);">En dduire le rayon de courbure</span>. <p>&nbsp; Le rayon de courbure <font face="symbol">r</font> est lie l'acclration normale a<sub>N </sub>et la vitesse v : a<sub>N </sub>= v/<font face="symbol">r</font> .<br> <font face="symbol">r </font>= v / a<sub>N </sub>= <font color="#ff0000">[4+(8t-4)]<sup>-3/2 </sup>/16</font>. </p> <br> <font color="#009900"><font style="font-weight: normal;" color="#000000"><span style="font-family: Times New Roman;"></span></font></font><font color="#009900"><font style="font-weight: normal;" color="#000000"><span style="font-family: Times New Roman;"></span></font></font><font color="#009900"><font style="font-weight: normal;" color="#000000"><span style="font-family: Times New Roman;"></span></font></font><br> <font color="#009900"><font style="font-weight: normal;" color="#000000"><span style="font-family: Times New Roman;"></span></font></font><span style="color: black;"></span><span style="color: black;"><br> </span><font color="#000000"><span style="color: black;"><span style="font-family: Times New Roman;"><span style="color: rgb(0, 153, 0);"><span style="color: black;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></span></span></span></span></font><span style="color: black;"></span><span 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normal;"></span><span style="font-weight: normal;"></span><span style="font-weight: normal;"></span><span style="font-weight: normal;"></span><span style="font-weight: normal;"></span><span style="font-weight: normal;"></span><span style="font-weight: normal;"></span><font style="font-weight: normal;" color="#009900"><span style="font-family: Times New Roman,Times,serif;"><span style="color: black;"><span style="font-weight: normal;"></span></span></span></font><span style="font-weight: normal;"><span style="font-weight: bold; text-decoration: underline;"></span></span><span style="font-weight: bold; color: rgb(0, 153, 0);"></span><span style="color: rgb(0, 153, 0); font-weight: bold;"></span><span style="font-weight: bold; color: rgb(0, 153, 0);"></span><span style="font-weight: normal;"></span><span style="font-weight: normal;"></span><span style="font-weight: normal;"></span><span style="font-weight: normal;"></span><span style="font-weight: normal;"></span> <span style="font-weight: normal;"></span><span style="font-family: Times New Roman,Times,serif; font-weight: normal;"><span style="font-weight: bold; text-decoration: underline;"></span><span style="font-weight: bold; color: black; text-decoration: underline;"></span></span><span style="font-family: Times New Roman,Times,serif; font-weight: normal;"><span style="font-weight: bold; text-decoration: underline;"></span></span><span style="font-weight: normal;"><span style="font-weight: bold; text-decoration: underline;"></span></span><span style="font-weight: normal;"><span style="font-family: Times New Roman;"></span></span><font color="#009900"><font color="#009900"><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span><span style="text-decoration: underline;"></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New 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color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"></span></span></font></font></font></font></font></font></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="font-weight: bold; text-decoration: underline;"></span><span style="font-weight: bold; color: rgb(0, 153, 0);"></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font color="#009900"><font style="font-weight: normal;" color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font color="#009900"><font style="color: black;" color="#009900"><span style="font-family: Symbol;"><span style="font-family: Times New Roman;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span><span style="font-weight: bold; text-decoration: underline;"></span></span></span></font></font></font></font></font></font></font></font></font></font></font></font></font></font></font></font></font><font color="#009900"><font style="font-weight: normal; color: black;" color="#009900"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span><span style="font-weight: bold; color: black; text-decoration: underline;"></span><span style="font-weight: bold; text-decoration: underline;"></span></font></font><span style="color: black; font-weight: normal;"><span style="font-weight: bold; color: rgb(0, 153, 0);"></span><span style="font-weight: bold; text-decoration: underline;"></span></span><font color="#000000"><span style="color: black;"><span style="font-family: Times New Roman;"><span style="color: rgb(0, 153, 0);"><span style="color: black;"><span style="font-weight: normal;"></span></span></span></span></span></font><font color="#000000"><span style="color: black;"><span style="font-family: Times New Roman;"><span 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