Finesse (aérodynamique)

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class="vector-toc-link" href="#Valeurs_typiques"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Valeurs typiques</span> </div> </a> <ul id="toc-Valeurs_typiques-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Équivalence_entre_les_définitions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Équivalence_entre_les_définitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Équivalence entre les définitions</span> </div> </a> <ul id="toc-Équivalence_entre_les_définitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finesse_air_et_finesse_sol" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Finesse_air_et_finesse_sol"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Finesse air et finesse sol</span> </div> </a> <ul id="toc-Finesse_air_et_finesse_sol-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calcul_de_la_finesse_maximale" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Calcul_de_la_finesse_maximale"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Calcul de la finesse maximale</span> </div> </a> <button aria-controls="toc-Calcul_de_la_finesse_maximale-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Calcul de la finesse maximale</span> </button> <ul id="toc-Calcul_de_la_finesse_maximale-sublist" class="vector-toc-list"> <li id="toc-Relation_entre_la_traînée_induite_et_la_traînée_parasite" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_entre_la_traînée_induite_et_la_traînée_parasite"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Relation entre la traînée induite et la traînée parasite</span> </div> </a> <ul id="toc-Relation_entre_la_traînée_induite_et_la_traînée_parasite-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Démonstration_simplifiée_pour_un_planeur" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Démonstration_simplifiée_pour_un_planeur"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Démonstration simplifiée pour un planeur</span> </div> </a> <ul id="toc-Démonstration_simplifiée_pour_un_planeur-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vitesse_optimale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vitesse_optimale"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Vitesse optimale</span> </div> </a> <ul id="toc-Vitesse_optimale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Détermination_des_coefficients_de_traînée_et_d'Oswald" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Détermination_des_coefficients_de_traînée_et_d'Oswald"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Détermination des coefficients de traînée et d'Oswald</span> </div> </a> <ul id="toc-Détermination_des_coefficients_de_traînée_et_d'Oswald-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calcul_de_la_finesse_maximale_(d'un_planeur)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Calcul_de_la_finesse_maximale_(d'un_planeur)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Calcul de la finesse maximale (d'un planeur)</span> </div> </a> <ul id="toc-Calcul_de_la_finesse_maximale_(d'un_planeur)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Effet_de_la_masse_sur_la_vitesse_optimale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Effet_de_la_masse_sur_la_vitesse_optimale"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Effet de la masse sur la vitesse optimale</span> </div> </a> <ul id="toc-Effet_de_la_masse_sur_la_vitesse_optimale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polaire_des_vitesses" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polaire_des_vitesses"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7</span> <span>Polaire des vitesses</span> </div> </a> <ul id="toc-Polaire_des_vitesses-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vitesse_de_chute_à_finesse_maximale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vitesse_de_chute_à_finesse_maximale"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.8</span> <span>Vitesse de chute à finesse maximale</span> </div> </a> <ul id="toc-Vitesse_de_chute_à_finesse_maximale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vitesse_de_chute_minimum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vitesse_de_chute_minimum"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.9</span> <span>Vitesse de chute minimum</span> </div> </a> <ul id="toc-Vitesse_de_chute_minimum-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Application_au_planeur_ASW_27" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Application_au_planeur_ASW_27"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Application au planeur ASW 27</span> </div> </a> <ul id="toc-Application_au_planeur_ASW_27-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Autres_domaines" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Autres_domaines"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Autres domaines</span> </div> </a> <ul id="toc-Autres_domaines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Généralisation_de_la_notion_de_finesse_à_tous_les_modes_de_transport" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Généralisation_de_la_notion_de_finesse_à_tous_les_modes_de_transport"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Généralisation de la notion de finesse à tous les modes de transport</span> </div> </a> <ul id="toc-Généralisation_de_la_notion_de_finesse_à_tous_les_modes_de_transport-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Articles_connexes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Articles_connexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Articles connexes</span> </div> </a> <ul id="toc-Articles_connexes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_et_références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_et_références"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Notes et références</span> </div> </a> <button aria-controls="toc-Notes_et_références-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Notes et références</span> </button> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Références" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Références"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>Références</span> </div> </a> <ul id="toc-Références-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bibliographie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliographie"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Bibliographie</span> </div> </a> <ul id="toc-Bibliographie-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Sommaire" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Basculer la table des matières" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Basculer la table des matières</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Finesse (aérodynamique)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Aller à un article dans une autre langue. Disponible en 25 langues." > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-25" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">25 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B3%D8%A8%D8%A9_%D8%A7%D9%84%D8%B1%D9%81%D8%B9_%D8%A5%D9%84%D9%89_%D8%A7%D9%84%D8%B3%D8%AD%D8%A8" title="نسبة الرفع إلى السحب – arabe" lang="ar" hreflang="ar" data-title="نسبة الرفع إلى السحب" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D1%8D%D1%80%D0%B0%D0%B4%D1%8B%D0%BD%D0%B0%D0%BC%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D1%8F%D0%BA%D0%B0%D1%81%D1%86%D1%8C" title="Аэрадынамічная якасць – biélorusse" lang="be" hreflang="be" data-title="Аэрадынамічная якасць" data-language-autonym="Беларуская" data-language-local-name="biélorusse" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%B5%D1%80%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D1%87%D0%BD%D0%BE_%D0%BA%D0%B0%D1%87%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Аеродинамично качество – bulgare" lang="bg" hreflang="bg" data-title="Аеродинамично качество" data-language-autonym="Български" data-language-local-name="bulgare" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Relaci%C3%B3_sustentaci%C3%B3/arrossegament" title="Relació sustentació/arrossegament – catalan" lang="ca" hreflang="ca" data-title="Relació sustentació/arrossegament" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Klouzavost" title="Klouzavost – tchèque" lang="cs" hreflang="cs" data-title="Klouzavost" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Glidetal" title="Glidetal – danois" lang="da" hreflang="da" data-title="Glidetal" data-language-autonym="Dansk" data-language-local-name="danois" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gleitzahl_(Flugzeug)" title="Gleitzahl (Flugzeug) – allemand" lang="de" hreflang="de" data-title="Gleitzahl (Flugzeug)" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Lift-to-drag_ratio" title="Lift-to-drag ratio – anglais" lang="en" hreflang="en" data-title="Lift-to-drag ratio" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Relaci%C3%B3n_entre_sustentaci%C3%B3n_y_resistencia" title="Relación entre sustentación y resistencia – espagnol" lang="es" hreflang="es" data-title="Relación entre sustentación y resistencia" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%99%D7%97%D7%A1_%D7%92%D7%9C%D7%99%D7%A9%D7%94" title="יחס גלישה – hébreu" lang="he" hreflang="he" data-title="יחס גלישה" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Finesa" title="Finesa – croate" lang="hr" hreflang="hr" data-title="Finesa" data-language-autonym="Hrvatski" data-language-local-name="croate" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sikl%C3%B3sz%C3%A1m" title="Siklószám – hongrois" lang="hu" hreflang="hu" data-title="Siklószám" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Efficienza_aerodinamica" title="Efficienza aerodinamica – italien" lang="it" hreflang="it" data-title="Efficienza aerodinamica" data-language-autonym="Italiano" data-language-local-name="italien" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%8F%9A%E6%8A%97%E6%AF%94" title="揚抗比 – japonais" lang="ja" hreflang="ja" data-title="揚抗比" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Glijgetal" title="Glijgetal – néerlandais" lang="nl" hreflang="nl" data-title="Glijgetal" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Glidetall" title="Glidetall – norvégien bokmål" lang="nb" hreflang="nb" data-title="Glidetall" data-language-autonym="Norsk bokmål" data-language-local-name="norvégien bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Doskona%C5%82o%C5%9B%C4%87_aerodynamiczna" title="Doskonałość aerodynamiczna – polonais" lang="pl" hreflang="pl" data-title="Doskonałość aerodynamiczna" data-language-autonym="Polski" data-language-local-name="polonais" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Raz%C3%A3o_de_planeio" title="Razão de planeio – portugais" lang="pt" hreflang="pt" data-title="Razão de planeio" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D1%8D%D1%80%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BA%D0%B0%D1%87%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Аэродинамическое качество – russe" lang="ru" hreflang="ru" data-title="Аэродинамическое качество" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/K%C4%BAzavos%C5%A5" title="Kĺzavosť – slovaque" lang="sk" hreflang="sk" data-title="Kĺzavosť" data-language-autonym="Slovenčina" data-language-local-name="slovaque" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Jadralno_%C5%A1tevilo" title="Jadralno število – slovène" lang="sl" hreflang="sl" data-title="Jadralno število" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Glidtal" title="Glidtal – suédois" lang="sv" hreflang="sv" data-title="Glidtal" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kald%C4%B1rma-s%C3%BCr%C3%BCkleme_oran%C4%B1" title="Kaldırma-sürükleme oranı – turc" lang="tr" hreflang="tr" data-title="Kaldırma-sürükleme oranı" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%B5%D1%80%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D1%96%D1%87%D0%BD%D0%B0_%D1%8F%D0%BA%D1%96%D1%81%D1%82%D1%8C" title="Аеродинамічна якість – ukrainien" lang="uk" hreflang="uk" data-title="Аеродинамічна якість" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8D%87%E9%98%BB%E6%AF%94" title="升阻比 – chinois" lang="zh" hreflang="zh" data-title="升阻比" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q845727#sitelinks-wikipedia" title="Modifier les liens interlangues" class="wbc-editpage">Modifier les 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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Apparence</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">masquer</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Un article de Wikipédia, l'encyclopédie libre.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><div class="bandeau-container metadata homonymie hatnote"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Aide:Homonymie" title="Aide:Homonymie"><img alt="Page d’aide sur l’homonymie" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/20px-Logo_disambig.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/30px-Logo_disambig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/40px-Logo_disambig.svg.png 2x" data-file-width="512" data-file-height="375" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p>Pour les articles homonymes, voir <a href="/wiki/Finesse" class="mw-disambig" title="Finesse">finesse</a>. </p> </div></div> <p>La <b>finesse</b> est une caractéristique <a href="/wiki/A%C3%A9rodynamique" title="Aérodynamique">aérodynamique</a> définie comme le <a href="/wiki/Rapport_(math%C3%A9matiques)" title="Rapport (mathématiques)">rapport</a> entre la <a href="/wiki/Portance_(a%C3%A9rodynamique)" title="Portance (aérodynamique)">portance</a> et la <a href="/wiki/Tra%C3%AEn%C3%A9e" title="Traînée">traînée</a>. </p><p>Elle est parfois désignée par le terme de langue anglaise <span class="citation">« L/D ratio »</span> signifiant <span class="citation not_fr_quote" lang="en">« <span class="italique">Lift/Drag ratio</span> »</span>, c'est-à-dire rapport portance/traînée en français. </p><p>On peut aussi définir de manière équivalente la finesse comme le rapport des coefficients de portance et de traînée <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{z} \over C_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{z} \over C_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee57a3139ce3f88fa4a0662c47c5a93319dc784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:3.67ex; height:5.843ex;" alt="{\displaystyle C_{z} \over C_{x}}"></span>, à condition que ces deux coefficients soient rapportés à la même surface. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Définition"><span id="D.C3.A9finition"></span>Définition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=1" title="Modifier la section : Définition" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=1" title="Modifier le code source de la section : Définition"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Angles_en_a%C3%A9ronautique.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Angles_en_a%C3%A9ronautique.png/220px-Angles_en_a%C3%A9ronautique.png" decoding="async" width="220" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Angles_en_a%C3%A9ronautique.png/330px-Angles_en_a%C3%A9ronautique.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Angles_en_a%C3%A9ronautique.png/440px-Angles_en_a%C3%A9ronautique.png 2x" data-file-width="846" data-file-height="582" /></a><figcaption>Schéma définissant assiette, incidence et pente.</figcaption></figure> <p>La finesse d'un <a href="/wiki/A%C3%A9rodyne" title="Aérodyne">aérodyne</a> à voilure fixe est le rapport entre sa <a href="/wiki/Portance_(a%C3%A9rodynamique)" title="Portance (aérodynamique)">portance</a> et sa <a href="/wiki/Tra%C3%AEn%C3%A9e" title="Traînée">traînée</a> aérodynamique. En vol plané (sans force de traction/propulsion) à <a href="/wiki/Vitesse_vraie" class="mw-redirect" title="Vitesse vraie">vitesse vraie</a> (vitesse de l'aéronef par rapport à la masse d'air dans laquelle il se déplace) constante, et donc à pente constante, elle est égale au rapport entre la distance horizontale parcourue et la hauteur de chute ou encore au rapport entre la vitesse horizontale et la vitesse verticale (<a href="/wiki/Taux_de_chute" title="Taux de chute">taux de chute</a>). Bien sûr, cette définition est à adapter suivant l'objet étudié : <a href="/wiki/Voile_(navire)" title="Voile (navire)">voile</a> de bateau, profil de <a href="/wiki/Car%C3%A8ne_(bateau)" title="Carène (bateau)">carène</a>… </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {finesse}}={P \over T}={{\rm {distance~horizontale~parcourue}} \over {\rm {hauteur\ perdue}}}={v_{\mathrm {horizontale} } \over v_{\mathrm {verticale} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mtext> </mtext> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">z</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mtext> </mtext> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">r</mi> <mtext> </mtext> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">z</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {finesse}}={P \over T}={{\rm {distance~horizontale~parcourue}} \over {\rm {hauteur\ perdue}}}={v_{\mathrm {horizontale} } \over v_{\mathrm {verticale} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3689b17295efa660336bb66de3ba51b390018973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:60.343ex; height:5.843ex;" alt="{\displaystyle {\rm {finesse}}={P \over T}={{\rm {distance~horizontale~parcourue}} \over {\rm {hauteur\ perdue}}}={v_{\mathrm {horizontale} } \over v_{\mathrm {verticale} }}}"></span> </p><p>Pour un aérodyne donné, la finesse varie en fonction de l'incidence de l'aile. Cependant, comme le coefficient de portance varie aussi avec l'angle d'incidence, pour obtenir une portance équivalente au poids, il faut adapter la vitesse. C'est pourquoi la finesse varie avec la vitesse. </p><p>Dans le cas d'un <a href="/wiki/Planeur" title="Planeur">planeur</a>, la finesse varie en fonction de la vitesse sur trajectoire en suivant une courbe qu'on appelle la <i><a href="/wiki/Polaire_des_vitesses" title="Polaire des vitesses">polaire des vitesses</a></i>.<br /> Cette courbe représente le <a href="/wiki/Taux_de_chute" title="Taux de chute">taux de chute</a> en fonction de la vitesse sur trajectoire (ou « vitesse indiquée »). Elle est croissante entre la valeur de la vitesse de décrochage jusqu'à la valeur de la vitesse correspondant au taux de chute minimal, puis décroissante au-delà. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Glide_ratio.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/4/43/Glide_ratio.gif" decoding="async" width="300" height="80" class="mw-file-element" data-file-width="300" data-file-height="80" /></a><figcaption></figcaption></figure> <p>À vitesse constante, la <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\rm {pente}}|=\arctan \left({1 \over {\rm {finesse}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\rm {pente}}|=\arctan \left({1 \over {\rm {finesse}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f63db9ddf6cc3d0fe01505f8240f955ed0b7c70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.372ex; height:6.176ex;" alt="{\displaystyle |{\rm {pente}}|=\arctan \left({1 \over {\rm {finesse}}}\right)}"></span> </p><p>Par exemple, une finesse de 7 correspond à un angle de plané de ~8<abbr class="abbr" title="degré"><a href="/wiki/Degr%C3%A9_(angle)" title="Degré (angle)">°</a></abbr>. </p> <div class="mw-heading mw-heading2"><h2 id="Valeurs_typiques">Valeurs typiques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=2" title="Modifier la section : Valeurs typiques" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=2" title="Modifier le code source de la section : Valeurs typiques"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les avions ont généralement des finesses comprises entre 8 et 20 : les avions de ligne ont des finesses comprises entre 16 et 18, l'<a href="/wiki/Airbus_A320" title="Airbus A320">Airbus A320</a> a une finesse de 17, le <a href="/wiki/Boeing_747" title="Boeing 747">Boeing 747</a> de 17,7<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>. Le <a href="/wiki/Concorde_(avion)" title="Concorde (avion)">Concorde</a> avait une finesse de 4 au décollage, 12 à <a href="/wiki/Nombre_de_Mach" title="Nombre de Mach">Mach</a> 0,95 et 7,5 à Mach 2<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup> </p><p>Les derniers prototypes de « <a href="/wiki/Wingsuit" class="mw-redirect" title="Wingsuit">wingsuit</a> » permettent d'atteindre une finesse de 3. Les <a href="/wiki/Parapente" title="Parapente">parapentes</a> modernes ont une finesse comprise entre 9 et 13<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup>. Les <a href="/wiki/Deltaplane" title="Deltaplane">deltaplanes</a> « souples » modernes ont une finesse comprise entre 14 et 16, et les <a href="/wiki/Deltaplane#Évolution_des_ailes" title="Deltaplane">deltaplanes « rigides »</a> modernes ont une finesse comprise entre 18 et 22. Les <a href="/wiki/Planeur" title="Planeur">planeurs</a> de construction en bois et toile de 27 à 32 et les planeurs plastiques ont commencé à 30 et sont à plus de 60 actuellement. </p><p>Typiquement, sur un planeur moderne : </p> <ul><li>la vitesse de finesse maximale se situe entre 80 et <span title="33,333 36 m/s" style="cursor:help">120</span> <abbr class="abbr" title="kilomètre par heure">km/h</abbr> selon le modèle et la charge alaire<sup id="cite_ref-cumulus-soaring_polars.htm_4-0" class="reference"><a href="#cite_note-cumulus-soaring_polars.htm-4"><span class="cite_crochet">[</span>4<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-ASW28-18_polar_5-0" class="reference"><a href="#cite_note-ASW28-18_polar-5"><span class="cite_crochet">[</span>5<span class="cite_crochet">]</span></a></sup> ;</li> <li>la vitesse de taux de chute minimal est de l'ordre de <span title="22,222 24 m/s" style="cursor:help">80</span> <abbr class="abbr" title="kilomètre par heure">km/h</abbr> et le taux de chute correspondant de l'ordre de 0,8 à <span title="1,8 km/h" style="cursor:help">0,5</span> <abbr class="abbr" title="mètre par seconde">m/s</abbr> ;</li> <li>la vitesse de décrochage est de l'ordre de <span title="19,444 46 m/s" style="cursor:help">70</span> <abbr class="abbr" title="kilomètre par heure">km/h</abbr>.</li></ul> <p>Un avion à <a href="/wiki/Propulsion_humaine" title="Propulsion humaine">propulsion humaine</a> permettant de voler en pédalant a un meilleur rapport portance/traînée de 30<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite_crochet">[</span>6<span class="cite_crochet">]</span></a></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Équivalence_entre_les_définitions"><span id=".C3.89quivalence_entre_les_d.C3.A9finitions"></span>Équivalence entre les définitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=3" title="Modifier la section : Équivalence entre les définitions" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=3" title="Modifier le code source de la section : Équivalence entre les définitions"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Forces_en_vol_plan%C3%A9.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Forces_en_vol_plan%C3%A9.png/220px-Forces_en_vol_plan%C3%A9.png" decoding="async" width="220" height="164" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Forces_en_vol_plan%C3%A9.png/330px-Forces_en_vol_plan%C3%A9.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Forces_en_vol_plan%C3%A9.png/440px-Forces_en_vol_plan%C3%A9.png 2x" data-file-width="977" data-file-height="727" /></a><figcaption>Forces en vol plané</figcaption></figure> <p><b>Système :</b> avion </p><p><b>Référentiel :</b> terrestre supposé galiléen </p><p><b>Bilan des <a href="/wiki/Forces_int%C3%A9rieures_et_forces_ext%C3%A9rieures" title="Forces intérieures et forces extérieures">forces extérieures</a> au système :</b> </p> <ul><li>Portance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c660ae75328e3f1c0a8e10f216bd68e48b3a11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.773ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{z}}"></span> perpendiculaire à la vitesse de déplacement de l'avion</li> <li>Traînée <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf341353fa9161d93c2d922b7ac8f17ef83b9d01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.943ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{x}}"></span> opposée à la vitesse de déplacement de l'avion</li> <li>Poids <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\vec {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\vec {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e302f443675b1752a1abbb888f2abc47b3fdc69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.215ex; height:2.676ex;" alt="{\displaystyle m{\vec {g}}}"></span></li></ul> <p>D'après la <a href="/wiki/Lois_du_mouvement_de_Newton" title="Lois du mouvement de Newton">deuxième loi de Newton</a> on a : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{d{\vec {V}} \over dt}={\vec {F}}_{x}+{\vec {F_{z}}}+m{\vec {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{d{\vec {V}} \over dt}={\vec {F}}_{x}+{\vec {F_{z}}}+m{\vec {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20ab60098637f59a9ad8b37982fc1f0894ff555a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.313ex; height:6.176ex;" alt="{\displaystyle m{d{\vec {V}} \over dt}={\vec {F}}_{x}+{\vec {F_{z}}}+m{\vec {g}}}"></span> </p><p>On suppose que l'aéronef est en mouvement non accéléré et l'on a donc : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {0}}=m{d{\vec {V}} \over dt}={\vec {F}}_{x}+{\vec {F_{z}}}+m{\vec {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {0}}=m{d{\vec {V}} \over dt}={\vec {F}}_{x}+{\vec {F_{z}}}+m{\vec {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1807ce8fd902f1b96284768692241761a9646ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.574ex; height:6.176ex;" alt="{\displaystyle {\vec {0}}=m{d{\vec {V}} \over dt}={\vec {F}}_{x}+{\vec {F_{z}}}+m{\vec {g}}}"></span> </p><p>Soient <i>C</i><sub>z</sub> le <a href="/wiki/Coefficient_de_portance" title="Coefficient de portance">coefficient de portance</a> et <i>C</i><sub>x</sub> le <a href="/wiki/Coefficient_de_tra%C3%AEn%C3%A9e" title="Coefficient de traînée">coefficient de traînée</a>. On note que le coefficient de portance est en première approximation proportionnel à l'<a href="/wiki/Incidence_(a%C3%A9rodynamique)" title="Incidence (aérodynamique)">angle d'incidence</a>. </p><p>Cela se traduit donc en projetant sur chacun des axes par : </p> <ul><li>sur O<i>x</i> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=-{1 \over 2}\rho V^{2}SC_{x}+mg\sin \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>m</mi> <mi>g</mi> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=-{1 \over 2}\rho V^{2}SC_{x}+mg\sin \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e5bb148eb83fafc37af6924cd9e438146b91d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.463ex; height:5.176ex;" alt="{\displaystyle 0=-{1 \over 2}\rho V^{2}SC_{x}+mg\sin \gamma }"></span></li> <li>sur O<i>z</i> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0={1 \over 2}\rho V^{2}SC_{z}-mg\cos \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mi>g</mi> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0={1 \over 2}\rho V^{2}SC_{z}-mg\cos \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e901c53369acc3199814e7635ad91e3e2a85842c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.74ex; height:5.176ex;" alt="{\displaystyle 0={1 \over 2}\rho V^{2}SC_{z}-mg\cos \gamma }"></span></li></ul> <p>Et donc, pour un vol plané à <a href="/wiki/Vitesse_vraie" class="mw-redirect" title="Vitesse vraie">vitesse vraie</a> constante : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {finesse}}={1 \over \tan |\gamma |}={{\rm {distance~horizontale~parcourue}} \over {\rm {hauteur~perdue}}}={v_{horizontale} \over v_{verticale}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mtext> </mtext> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">z</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mtext> </mtext> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">r</mi> <mtext> </mtext> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mi>o</mi> <mi>r</mi> <mi>i</mi> <mi>z</mi> <mi>o</mi> <mi>n</mi> <mi>t</mi> <mi>a</mi> <mi>l</mi> <mi>e</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>e</mi> <mi>r</mi> <mi>t</mi> <mi>i</mi> <mi>c</mi> <mi>a</mi> <mi>l</mi> <mi>e</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {finesse}}={1 \over \tan |\gamma |}={{\rm {distance~horizontale~parcourue}} \over {\rm {hauteur~perdue}}}={v_{horizontale} \over v_{verticale}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fee245f62d3c99f7110d72f08b75bbbbb06addb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:65.273ex; height:6.176ex;" alt="{\displaystyle {\rm {finesse}}={1 \over \tan |\gamma |}={{\rm {distance~horizontale~parcourue}} \over {\rm {hauteur~perdue}}}={v_{horizontale} \over v_{verticale}}}"></span> </p><p>Et donc : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={1 \over \tan \gamma }={C_{z} \over C_{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f={1 \over \tan \gamma }={C_{z} \over C_{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9d5f7c081ea4f38871d46829dcdf302e35995e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.991ex; height:5.843ex;" alt="{\displaystyle f={1 \over \tan \gamma }={C_{z} \over C_{x}}}"></span> </p><p>Pour un <a href="/wiki/Planeur" title="Planeur">planeur</a>, on pourra aisément écrire que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \gamma \approx \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> <mo>≈</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \gamma \approx \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d9c9a0964a4ca565c811a1d3cb6b41eef42be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.37ex; height:2.509ex;" alt="{\displaystyle \tan \gamma \approx \gamma }"></span> (si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> est exprimé en <a href="/wiki/Radian" title="Radian">radians</a>). Cependant cela ne sera pas correct pour un <a href="/wiki/Wingsuit" class="mw-redirect" title="Wingsuit">wingsuit</a> qu'on pourrait presque assimiler à un « fer à repasser ». </p> <div class="mw-heading mw-heading2"><h2 id="Finesse_air_et_finesse_sol">Finesse air et finesse sol</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=4" title="Modifier la section : Finesse air et finesse sol" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=4" title="Modifier le code source de la section : Finesse air et finesse sol"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <b>finesse air</b> d'un <a href="/wiki/A%C3%A9ronef" title="Aéronef">aéronef</a> est donnée par rapport à la masse d'air dans laquelle il se déplace. C'est souvent celle que le constructeur annonce car elle est indépendante du vent. </p><p>La <b>finesse sol</b>, elle, est calculée par rapport au sol. C'est souvent la plus intéressante car c'est celle qui détermine si un parcours jusqu'à un but est possible ou non. Cette finesse doit tenir compte du déplacement de l'air (du vent) par rapport au sol. </p><p>Quand l'aéronef se déplace dans la direction et le sens du vent, la finesse sol augmente, et inversement s'il se déplace dans le sens inverse. Par vent fort de face, l'aéronef peut avoir une vitesse sol et une finesse sol faibles ou négatives, ce qui sera d'ailleurs souvent une raison suffisante pour annuler le vol. </p><p>La finesse air et la finesse sol sont égales lorsque l'air est calme et ne subit aucun mouvement vertical ni horizontal. </p> <div class="mw-heading mw-heading2"><h2 id="Calcul_de_la_finesse_maximale">Calcul de la finesse maximale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=5" title="Modifier la section : Calcul de la finesse maximale" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=5" title="Modifier le code source de la section : Calcul de la finesse maximale"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Relation_entre_la_traînée_induite_et_la_traînée_parasite"><span id="Relation_entre_la_tra.C3.AEn.C3.A9e_induite_et_la_tra.C3.AEn.C3.A9e_parasite"></span>Relation entre la traînée induite et la traînée parasite</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=6" title="Modifier la section : Relation entre la traînée induite et la traînée parasite" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=6" title="Modifier le code source de la section : Relation entre la traînée induite et la traînée parasite"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Drag.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Drag.jpg/220px-Drag.jpg" decoding="async" width="220" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Drag.jpg/330px-Drag.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Drag.jpg/440px-Drag.jpg 2x" data-file-width="543" data-file-height="564" /></a><figcaption>Courbes montrant les trainée induite, parasite ainsi que la trainée combinée par rapport à la vitesse de l'air</figcaption></figure> <p>On va montrer qu'un aéronef atteint sa finesse maximale lorsque la <a href="/wiki/Tra%C3%AEn%C3%A9e_induite" title="Traînée induite">traînée induite</a> est égale à la traînée parasite. </p><p>La traînée parasite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {p} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {p} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/524bad2afae2e96646d93918f86cc2987bc11804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.91ex; height:2.843ex;" alt="{\displaystyle R_{\mathrm {p} }}"></span> causée par la résistance de l'air peut s'écrire sous la forme </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {p} }=qSC_{x,\mathrm {p} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>q</mi> <mi>S</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {p} }=qSC_{x,\mathrm {p} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/009b5f1389ec1f6fe8a86da3fc7b2b145be72bcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.783ex; height:2.843ex;" alt="{\displaystyle R_{\mathrm {p} }=qSC_{x,\mathrm {p} }}"></span> </p><p>où <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{xp}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{xp}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5596073c241e28768323ca0a519fcb69355529a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.661ex; height:2.843ex;" alt="{\displaystyle C_{xp}}"></span> est le coefficient de traînée parasite et on a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{xp}=cte}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mi>c</mi> <mi>t</mi> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{xp}=cte}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bfa5e948d2dfc6b8c64b848882321c633f2a64f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.69ex; height:2.843ex;" alt="{\displaystyle C_{xp}=cte}"></span>. Soit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> l'envergure de l'aile et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> sa corde moyenne (~ largeur moyenne de l'aile). <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q={1 \over 2}\rho V^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q={1 \over 2}\rho V^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3befbb5faff1a88ad207c08068e85bde1025493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.34ex; height:5.176ex;" alt="{\displaystyle q={1 \over 2}\rho V^{2}}"></span> est la <a href="/wiki/Pression_dynamique" title="Pression dynamique">pression dynamique</a>. </p><p>On pose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={b \over c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={b \over c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36975b4a7a6d56f667e18439583e850d4a9bf5dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.297ex; height:5.343ex;" alt="{\displaystyle \lambda ={b \over c}}"></span> l'allongement de l'aile. On rappelle que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={b^{2} \over \lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>λ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={b^{2} \over \lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c4053f4aa9d24decfcfa48b6cf0ac6becc7889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.486ex; height:5.843ex;" alt="{\displaystyle S={b^{2} \over \lambda }}"></span> </p><p>On note <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> la <a href="/wiki/Masse_volumique_de_l%27air" title="Masse volumique de l'air">masse volumique de l'air</a>. On obtient : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {p} }={1 \over 2}\rho V^{2}SC_{x,\mathrm {p} }={1 \over 2}{\rho b^{2}V^{2}C_{x,\mathrm {p} } \over \lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> <mi>λ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {p} }={1 \over 2}\rho V^{2}SC_{x,\mathrm {p} }={1 \over 2}{\rho b^{2}V^{2}C_{x,\mathrm {p} } \over \lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7853f1e8ae28e2fccd11aaeef19e5907365353c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.249ex; height:6.009ex;" alt="{\displaystyle R_{\mathrm {p} }={1 \over 2}\rho V^{2}SC_{x,\mathrm {p} }={1 \over 2}{\rho b^{2}V^{2}C_{x,\mathrm {p} } \over \lambda }}"></span> </p><p>La traînée induite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49003ef922492fa2e2e5fa63dfdfc76b556edc99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.454ex; height:2.509ex;" alt="{\displaystyle R_{\mathrm {i} }}"></span> s'exprime comme suit : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {i} }=2{F_{z}^{2} \over b^{2}\rho V^{2}\pi e}={1 \over 2}\rho V^{2}SC_{x,\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {i} }=2{F_{z}^{2} \over b^{2}\rho V^{2}\pi e}={1 \over 2}\rho V^{2}SC_{x,\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a16e2bb2eed412da318245760050e120884d0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.71ex; height:6.176ex;" alt="{\displaystyle R_{\mathrm {i} }=2{F_{z}^{2} \over b^{2}\rho V^{2}\pi e}={1 \over 2}\rho V^{2}SC_{x,\mathrm {i} }}"></span> avec <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x,\mathrm {i} }={C_{z}^{2} \over \lambda \pi e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mi>λ</mi> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x,\mathrm {i} }={C_{z}^{2} \over \lambda \pi e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b22a12c8e4759b8743248a17e591fec4530272a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.454ex; height:5.676ex;" alt="{\displaystyle C_{x,\mathrm {i} }={C_{z}^{2} \over \lambda \pi e}}"></span> </p><p>où <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cab2da6853c5be26eaccb249e553e45ff9b03ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.496ex; height:2.509ex;" alt="{\displaystyle F_{z}}"></span> est la portance, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> est la vitesse de l'aéronef et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> est le coefficient d'Oswald. Cette dernière formule provient de la <a href="/wiki/Th%C3%A9orie_des_profils_minces" title="Théorie des profils minces">théorie des profils minces</a>. </p><p>Lorsqu'un <a href="/wiki/Avion" title="Avion">avion</a> ou <a href="/wiki/Planeur" title="Planeur">planeur</a> est en vol, la traînée induite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {i} }(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {i} }(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f53cc349d1f3acb65979e26ebe348f156cfa18a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.05ex; height:2.843ex;" alt="{\displaystyle R_{\mathrm {i} }(V)}"></span> et la traînée parasite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {p} }(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {p} }(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daba26af583c87dcf703a259b0ed34706b1a7ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.507ex; height:3.009ex;" alt="{\displaystyle R_{\mathrm {p} }(V)}"></span> s'ajoutent et constituent la traînée totale : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(V)={1 \over 2}\rho V^{2}SC_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(V)={1 \over 2}\rho V^{2}SC_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfc187880bf44edfb901b208362b6f5658622e02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.964ex; height:5.176ex;" alt="{\displaystyle R(V)={1 \over 2}\rho V^{2}SC_{x}}"></span> avec <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x}=C_{x,\mathrm {p} }+C_{x,\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x}=C_{x,\mathrm {p} }+C_{x,\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce70ab44d5f6aaae146299f29b1c249bfbc3afd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.728ex; height:2.843ex;" alt="{\displaystyle C_{x}=C_{x,\mathrm {p} }+C_{x,\mathrm {i} }}"></span> </p><p>Pour ne pas alourdir les calculs avec des racines carrés dans la suite on exprimera non pas la finesse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, mais la finesse au carré et on a alors : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{2}={C_{z}^{2} \over C_{x}^{2}}={\lambda \pi eC_{x,\mathrm {i} } \over C_{x}^{2}}={\lambda \pi e}{C_{x}-C_{x,\mathrm {p} } \over C_{x}^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>λ</mi> <mi>π</mi> <mi>e</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mrow> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>π</mi> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{2}={C_{z}^{2} \over C_{x}^{2}}={\lambda \pi eC_{x,\mathrm {i} } \over C_{x}^{2}}={\lambda \pi e}{C_{x}-C_{x,\mathrm {p} } \over C_{x}^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04266d9d17cba55f7a18716eaf3d57df4235564b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.201ex; height:6.343ex;" alt="{\displaystyle f^{2}={C_{z}^{2} \over C_{x}^{2}}={\lambda \pi eC_{x,\mathrm {i} } \over C_{x}^{2}}={\lambda \pi e}{C_{x}-C_{x,\mathrm {p} } \over C_{x}^{2}}}"></span> </p><p>On dérive par rapport à <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46c7c57eb7d5bafc9dbb07668033f00118f5e48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.834ex; height:2.509ex;" alt="{\displaystyle C_{x}}"></span> : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2f{df \over dC_{x}}=\lambda \pi e{-C_{x}^{2}+2C_{x,\mathrm {p} }C_{x} \over C_{x}^{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>f</mi> </mrow> <mrow> <mi>d</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>λ</mi> <mi>π</mi> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2f{df \over dC_{x}}=\lambda \pi e{-C_{x}^{2}+2C_{x,\mathrm {p} }C_{x} \over C_{x}^{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d6603b73c9efb0819b2c7aab8211797abd44f9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.736ex; height:6.509ex;" alt="{\displaystyle 2f{df \over dC_{x}}=\lambda \pi e{-C_{x}^{2}+2C_{x,\mathrm {p} }C_{x} \over C_{x}^{4}}}"></span> </p><p>Pour que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> soit maximale il faut que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {df \over dC_{x}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>f</mi> </mrow> <mrow> <mi>d</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {df \over dC_{x}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/561a45d80c91d2e006e9a941325a2f268b08228d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.147ex; height:5.843ex;" alt="{\displaystyle {df \over dC_{x}}=0}"></span> ce qui revient ici à déterminer les racines d'un polynôme du second degré en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46c7c57eb7d5bafc9dbb07668033f00118f5e48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.834ex; height:2.509ex;" alt="{\displaystyle C_{x}}"></span>. </p><p>On obtient donc que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {max} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {max} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57daba828fa06ec43f9579b84052378bda2fd440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.43ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {max} }}"></span> est atteinte quand <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {C_{x}=2C_{x,\mathrm {p} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {C_{x}=2C_{x,\mathrm {p} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1936cb963842a6849742a7c8bf24e73aa2f065d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.301ex; height:2.843ex;" alt="{\displaystyle {C_{x}=2C_{x,\mathrm {p} }}}"></span> c'est-à-dire : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x,\mathrm {p} }=C_{x,\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x,\mathrm {p} }=C_{x,\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9a4c6fe28f99eec1e4b885e8d65351b6e14646" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.053ex; height:2.843ex;" alt="{\displaystyle C_{x,\mathrm {p} }=C_{x,\mathrm {i} }}"></span> et donc <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {p} }=R_{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {p} }=R_{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6407a0b3d9c171b1f2aff153ea9f742e00a73b01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.462ex; height:2.843ex;" alt="{\displaystyle R_{\mathrm {p} }=R_{\mathrm {i} }}"></span> </p><p>Ce qui signifie que la traînée induite est égale à la traînée parasite. </p> <div class="mw-heading mw-heading3"><h3 id="Démonstration_simplifiée_pour_un_planeur"><span id="D.C3.A9monstration_simplifi.C3.A9e_pour_un_planeur"></span>Démonstration simplifiée pour un planeur</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=7" title="Modifier la section : Démonstration simplifiée pour un planeur" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=7" title="Modifier le code source de la section : Démonstration simplifiée pour un planeur"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tout ce qui suit appliqué aux planeurs a été présenté dans l'ouvrage de Frank Irving <i>The Paths of Soaring Flight</i><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite_crochet">[</span>7<span class="cite_crochet">]</span></a></sup>. </p><p>Dans les cours d'aérodynamique pour pilotes, il est souvent affirmé sans justification que la traînée induite est proportionnelle à 1/V<sup>2</sup> et que la traînée parasite est proportionnelle à V<sup>2</sup>. Dans ces conditions, la démonstration du théorème ci-dessus devient triviale qui est alors un simple <a href="/wiki/Corollaire" class="mw-disambig" title="Corollaire">corollaire</a> des <a href="/wiki/Postulats" class="mw-redirect" title="Postulats">postulats</a> énoncés ci-dessus. Dans ce qui suit, les postulats vont être démontrés et l'on va en conclure le théorème ci-dessus. </p><p>Les planeurs ont des angles de plané qui sont très petits et l'on peut donc supposer que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{z}=mg}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{z}=mg}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eb0a4157c926f9e94d68b3982868e206a9380ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.751ex; height:2.509ex;" alt="{\displaystyle F_{z}=mg}"></span> </p><p>La traînée induite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {i} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {i} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49003ef922492fa2e2e5fa63dfdfc76b556edc99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.454ex; height:2.509ex;" alt="{\displaystyle R_{\mathrm {i} }}"></span> s'exprime comme suit : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{i}=2{F_{z}^{2} \over b^{2}\rho V^{2}\pi e}=2{m^{2}g^{2} \over b^{2}\rho V^{2}\pi e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{i}=2{F_{z}^{2} \over b^{2}\rho V^{2}\pi e}=2{m^{2}g^{2} \over b^{2}\rho V^{2}\pi e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf1e50e597f51a324572199c78e7ec4fb6d313d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.039ex; height:6.343ex;" alt="{\displaystyle R_{i}=2{F_{z}^{2} \over b^{2}\rho V^{2}\pi e}=2{m^{2}g^{2} \over b^{2}\rho V^{2}\pi e}}"></span> </p><p>La traînée parasite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {p} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {p} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/524bad2afae2e96646d93918f86cc2987bc11804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.91ex; height:2.843ex;" alt="{\displaystyle R_{\mathrm {p} }}"></span> causée par la résistance de l'air peut s'écrire sous la forme </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {p} }={1 \over 2}\rho V^{2}SC_{x,\mathrm {p} }={1 \over 2}{\rho b^{2}V^{2}C_{x,\mathrm {p} } \over \lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> <mi>λ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {p} }={1 \over 2}\rho V^{2}SC_{x,\mathrm {p} }={1 \over 2}{\rho b^{2}V^{2}C_{x,\mathrm {p} } \over \lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7853f1e8ae28e2fccd11aaeef19e5907365353c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.249ex; height:6.009ex;" alt="{\displaystyle R_{\mathrm {p} }={1 \over 2}\rho V^{2}SC_{x,\mathrm {p} }={1 \over 2}{\rho b^{2}V^{2}C_{x,\mathrm {p} } \over \lambda }}"></span> </p><p>Lorsqu'un <a href="/wiki/Planeur" title="Planeur">planeur</a> est en vol, la traînée induite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {i} }(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {i} }(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f53cc349d1f3acb65979e26ebe348f156cfa18a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.05ex; height:2.843ex;" alt="{\displaystyle R_{\mathrm {i} }(V)}"></span> et la traînée parasite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {p} }(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {p} }(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daba26af583c87dcf703a259b0ed34706b1a7ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.507ex; height:3.009ex;" alt="{\displaystyle R_{\mathrm {p} }(V)}"></span> s'ajoutent et constituent la traînée totale <i>R</i>(<i>V</i>). La finesse d'un planeur sera optimale lorsque la traînée totale <i>R</i>(<i>V</i>) est minimale. On résout donc l'équation </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathrm {d} R(V) \over \mathrm {d} V}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {d} R(V) \over \mathrm {d} V}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b01c142345e9e3b78386d5c18589edeb09bb6728" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.75ex; height:5.843ex;" alt="{\displaystyle {\mathrm {d} R(V) \over \mathrm {d} V}=0}"></span> </p><p>On définit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> tels que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={1 \over 2}{\rho b^{2}C_{x,\mathrm {p} } \over \lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> <mi>λ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={1 \over 2}{\rho b^{2}C_{x,\mathrm {p} } \over \lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0e330f80b8d1d9e500b3c136bebe2bc2378a034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.88ex; height:6.009ex;" alt="{\displaystyle \alpha ={1 \over 2}{\rho b^{2}C_{x,\mathrm {p} } \over \lambda }}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ={2m^{2}g^{2} \over b^{2}\rho \pi e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ={2m^{2}g^{2} \over b^{2}\rho \pi e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/437c11c188fd53d5f8cb44358873a3419e074cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.696ex; height:6.343ex;" alt="{\displaystyle \beta ={2m^{2}g^{2} \over b^{2}\rho \pi e}}"></span>. On peut écrire symboliquement : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {p} }(V)=\alpha V^{2}\qquad R_{\mathrm {i} }(V)={\beta \over V^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="2em" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {p} }(V)=\alpha V^{2}\qquad R_{\mathrm {i} }(V)={\beta \over V^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd18f8e2e87b56c4af6933523d7ed71f8e6c1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:31.665ex; height:5.676ex;" alt="{\displaystyle R_{\mathrm {p} }(V)=\alpha V^{2}\qquad R_{\mathrm {i} }(V)={\beta \over V^{2}}}"></span> </p><p>Après avoir calculé la dérivée de <i>R</i>(<i>V</i>), on résout donc : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\alpha V-2{\beta \over V^{3}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>α</mi> <mi>V</mi> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\alpha V-2{\beta \over V^{3}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8681122797c56e318ce12698ecb2da42533c7376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.508ex; height:5.676ex;" alt="{\displaystyle 2\alpha V-2{\beta \over V^{3}}=0}"></span> </p><p>Et donc en multipliant la relation ci-dessus par <i>V</i>, on obtient : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha V^{2}={\beta \over V^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha V^{2}={\beta \over V^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/912ef53783cb04ffc3c897866df4935ee03fd733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.365ex; height:5.676ex;" alt="{\displaystyle \alpha V^{2}={\beta \over V^{2}}}"></span> </p><p>ce qui signifie que la traînée induite est égale à la traînée parasite. </p> <div class="mw-heading mw-heading3"><h3 id="Vitesse_optimale">Vitesse optimale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=8" title="Modifier la section : Vitesse optimale" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=8" title="Modifier le code source de la section : Vitesse optimale"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On pose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={1 \over 2}{\rho b^{2}C_{x,\mathrm {p} } \over \lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> <mi>λ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={1 \over 2}{\rho b^{2}C_{x,\mathrm {p} } \over \lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0e330f80b8d1d9e500b3c136bebe2bc2378a034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.88ex; height:6.009ex;" alt="{\displaystyle \alpha ={1 \over 2}{\rho b^{2}C_{x,\mathrm {p} } \over \lambda }}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ={2F_{z}^{2} \over b^{2}\rho \pi e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ={2F_{z}^{2} \over b^{2}\rho \pi e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d12c62b2335f54d5351a905aa11f9c47dba28b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.936ex; height:6.176ex;" alt="{\displaystyle \beta ={2F_{z}^{2} \over b^{2}\rho \pi e}}"></span>. On a alors : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{p}(V)=\alpha V^{2}\qquad R_{i}(V)={\beta \over V^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="2em" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{p}(V)=\alpha V^{2}\qquad R_{i}(V)={\beta \over V^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02451938011856eb086c9e6b89a2e076acf92441" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:31.688ex; height:5.676ex;" alt="{\displaystyle R_{p}(V)=\alpha V^{2}\qquad R_{i}(V)={\beta \over V^{2}}}"></span></dd></dl> <p>Le planeur atteindra sa finesse maximale en air calme lorsque <i>la traînée induite sera égale à la traînée parasite</i>, c'est-à-dire<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite_crochet">[</span>8<span class="cite_crochet">]</span></a></sup> : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha V^{2}={\beta \over V^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha V^{2}={\beta \over V^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/912ef53783cb04ffc3c897866df4935ee03fd733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.365ex; height:5.676ex;" alt="{\displaystyle \alpha V^{2}={\beta \over V^{2}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{f}=\left({\beta \over \alpha }\right)^{1 \over 4}={{\sqrt {2}} \over (\pi e)^{1 \over 4}b}\times {\sqrt {F_{z} \over \rho }}\times \left({\lambda \over C_{x,\mathrm {p} }}\right)^{1 \over 4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mi>α</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mi>e</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>ρ</mi> </mfrac> </msqrt> </mrow> <mo>×</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{f}=\left({\beta \over \alpha }\right)^{1 \over 4}={{\sqrt {2}} \over (\pi e)^{1 \over 4}b}\times {\sqrt {F_{z} \over \rho }}\times \left({\lambda \over C_{x,\mathrm {p} }}\right)^{1 \over 4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f54771e7a55da9f2424d4a04aa8156575b83529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:45.499ex; height:8.509ex;" alt="{\displaystyle V_{f}=\left({\beta \over \alpha }\right)^{1 \over 4}={{\sqrt {2}} \over (\pi e)^{1 \over 4}b}\times {\sqrt {F_{z} \over \rho }}\times \left({\lambda \over C_{x,\mathrm {p} }}\right)^{1 \over 4}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Détermination_des_coefficients_de_traînée_et_d'Oswald"><span id="D.C3.A9termination_des_coefficients_de_tra.C3.AEn.C3.A9e_et_d.27Oswald"></span>Détermination des coefficients de traînée et d'Oswald</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=9" title="Modifier la section : Détermination des coefficients de traînée et d'Oswald" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=9" title="Modifier le code source de la section : Détermination des coefficients de traînée et d'Oswald"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si l'on connaît la vitesse à laquelle la finesse maximale est connue, on peut en déduire le coefficient de traînée parasite et le coefficient d'Oswald. Ces coefficients valent : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x,\mathrm {p} }={P \over f\rho V_{f}^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <mrow> <mi>f</mi> <mi>ρ</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x,\mathrm {p} }={P \over f\rho V_{f}^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf9ed532dccb7daa84b2e9c14a74f0ed3a0e590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:13.592ex; height:6.676ex;" alt="{\displaystyle C_{x,\mathrm {p} }={P \over f\rho V_{f}^{2}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e={4fP \over \pi \lambda \rho V_{f}^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>f</mi> <mi>P</mi> </mrow> <mrow> <mi>π</mi> <mi>λ</mi> <mi>ρ</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={4fP \over \pi \lambda \rho V_{f}^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f58fbe3137c23ecec0e83c48c8a38094fa4b9c33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:11.878ex; height:6.843ex;" alt="{\displaystyle e={4fP \over \pi \lambda \rho V_{f}^{2}}}"></span></dd></dl> <p><i>P</i> est la <a href="/wiki/Charge_alaire" title="Charge alaire">charge alaire</a> et <i>λ</i> est l'allongement de l'aile. </p> <div class="NavFrame" style="clear:both; margin-bottom:1em; border-style:solid; background-color:var(--background-color-base, #fff);" title="[afficher]/[masquer]"> <div class="NavHead" style="text-align:center; min-height:1.6em; color:var(--color-emphasized, #000);">Démonstration des formules</div> <div class="NavContent" style="margin:0; color:var(--color-base, #202122); display:block; text-align:left;"> <p>On a à finesse maximum : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{p}=\alpha V_{f}^{2}={\beta \over V_{f}^{2}}=R_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mi>α</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{p}=\alpha V_{f}^{2}={\beta \over V_{f}^{2}}=R_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/332dfdc9329adc2eb98dd4cd19fe64f3dc783536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.948ex; height:6.843ex;" alt="{\displaystyle R_{p}=\alpha V_{f}^{2}={\beta \over V_{f}^{2}}=R_{i}}"></span></dd></dl> <p>Si <i>R</i> est la traînée <i>totale</i>, on a donc : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=R_{p}+R_{i}=2R_{p}=2R_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=R_{p}+R_{i}=2R_{p}=2R_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eddac6e8988a268c8f5adcf51d7c75cc108f33e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.998ex; height:2.843ex;" alt="{\displaystyle R=R_{p}+R_{i}=2R_{p}=2R_{i}}"></span></dd></dl> <p>On suppose connue la finesse maximale <i>f</i> (publiée par le constructeur). Soit <i>W</i> le poids (en tant que force) du planeur. On a alors à l'équilibre </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {W \over R}=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <mi>R</mi> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {W \over R}=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a11134c05ec00a96d4d202f2f13ee29f24e730" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.648ex; height:5.343ex;" alt="{\displaystyle {W \over R}=f}"></span></dd></dl> <p>Donc : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {W \over 2R_{i}}=f\qquad {W \over 2R_{p}}=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <mrow> <mn>2</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <mrow> <mn>2</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {W \over 2R_{i}}=f\qquad {W \over 2R_{p}}=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f54b5ab069593d149e8b4641416b72014cd951b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.783ex; height:5.843ex;" alt="{\displaystyle {W \over 2R_{i}}=f\qquad {W \over 2R_{p}}=f}"></span></dd></dl> <p>Donc, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {W \over 2\alpha V_{f}^{2}}={W \over 2R_{p}}=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <mrow> <mn>2</mn> <mi>α</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <mrow> <mn>2</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {W \over 2\alpha V_{f}^{2}}={W \over 2R_{p}}=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69dfa75585a37aceeeb25269b7f5e9201086f822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.755ex; height:6.676ex;" alt="{\displaystyle {W \over 2\alpha V_{f}^{2}}={W \over 2R_{p}}=f}"></span></dd></dl> <p>On substitue : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {W \over \displaystyle 2\times {1 \over 2}\times {\rho b^{2}C_{x,\mathrm {p} } \over \lambda }V_{f}^{2}}=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> <mi>λ</mi> </mfrac> </mrow> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {W \over \displaystyle 2\times {1 \over 2}\times {\rho b^{2}C_{x,\mathrm {p} } \over \lambda }V_{f}^{2}}=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3180f1530bda0d16b4c9ba934984b9ddf6190f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:25.322ex; height:9.009ex;" alt="{\displaystyle {W \over \displaystyle 2\times {1 \over 2}\times {\rho b^{2}C_{x,\mathrm {p} } \over \lambda }V_{f}^{2}}=f}"></span></dd></dl> <p>Donc, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {W\lambda \over \rho b^{2}C_{x,\mathrm {p} }V_{f}^{2}}=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>W</mi> <mi>λ</mi> </mrow> <mrow> <mi>ρ</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {W\lambda \over \rho b^{2}C_{x,\mathrm {p} }V_{f}^{2}}=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7e398b285890c2ad9c79b9a98f8ee2253477c71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.644ex; height:6.843ex;" alt="{\displaystyle {W\lambda \over \rho b^{2}C_{x,\mathrm {p} }V_{f}^{2}}=f}"></span></dd></dl> <p>Donc, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x,\mathrm {p} }={W\lambda \over f\rho b^{2}V_{f}^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>W</mi> <mi>λ</mi> </mrow> <mrow> <mi>f</mi> <mi>ρ</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x,\mathrm {p} }={W\lambda \over f\rho b^{2}V_{f}^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c569ff92ebcaca0c684e57e5f0baf5f882ab6c97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.644ex; height:6.843ex;" alt="{\displaystyle C_{x,\mathrm {p} }={W\lambda \over f\rho b^{2}V_{f}^{2}}}"></span></dd></dl> <p>On remarque que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={b^{2} \over S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={b^{2} \over S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b194527e34a34db13dfdde3ff72d245bef7a5a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.342ex; height:5.843ex;" alt="{\displaystyle \lambda ={b^{2} \over S}}"></span> et donc : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x,\mathrm {p} }={W \over S}\times {1 \over f\rho V_{f}^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <mi>S</mi> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>f</mi> <mi>ρ</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x,\mathrm {p} }={W \over S}\times {1 \over f\rho V_{f}^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb389220f0642b9b822ee2f09bdbacd33c68bb32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.704ex; height:6.676ex;" alt="{\displaystyle C_{x,\mathrm {p} }={W \over S}\times {1 \over f\rho V_{f}^{2}}}"></span></dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W/S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W/S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bc991f4bdbe51abcd0f18a56d070dbd1c07cff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.097ex; height:2.843ex;" alt="{\displaystyle W/S}"></span> est la charge alaire notée <i>P</i> qui a la dimension d'une pression. Le coefficient de traînée parasite s'exprime comme suit : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x,\mathrm {p} }={P \over f\rho V_{f}^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <mrow> <mi>f</mi> <mi>ρ</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x,\mathrm {p} }={P \over f\rho V_{f}^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf9ed532dccb7daa84b2e9c14a74f0ed3a0e590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:13.592ex; height:6.676ex;" alt="{\displaystyle C_{x,\mathrm {p} }={P \over f\rho V_{f}^{2}}}"></span></dd></dl> <p>De même, on a : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {W \over \displaystyle 2{\beta \over V_{f}^{2}}}={W \over 2R_{p}}=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mstyle> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <mrow> <mn>2</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {W \over \displaystyle 2{\beta \over V_{f}^{2}}}={W \over 2R_{p}}=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d746ef6ab8eb1a546b15c005a9d2f4ce33ebcc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:18.103ex; height:10.009ex;" alt="{\displaystyle {W \over \displaystyle 2{\beta \over V_{f}^{2}}}={W \over 2R_{p}}=f}"></span> On substitue : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {W \over \displaystyle 2{2W^{2} \over b^{2}\rho \pi eV_{f}^{2}}}=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <mi>π</mi> <mi>e</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {W \over \displaystyle 2{2W^{2} \over b^{2}\rho \pi eV_{f}^{2}}}=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb14704cbb8d254b91e29c2df77098e19738ab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:15.852ex; height:10.009ex;" alt="{\displaystyle {W \over \displaystyle 2{2W^{2} \over b^{2}\rho \pi eV_{f}^{2}}}=f}"></span></dd></dl> <p>Donc, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {b^{2}\rho \pi eV_{f}^{2} \over 4W}=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <mi>π</mi> <mi>e</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>4</mn> <mi>W</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {b^{2}\rho \pi eV_{f}^{2} \over 4W}=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e567974245724e17e1386862a169e83762c40e7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.854ex; height:6.509ex;" alt="{\displaystyle {b^{2}\rho \pi eV_{f}^{2} \over 4W}=f}"></span></dd></dl> <p>Donc, le coefficient d'Oswald <i>e</i> vaut (il est supposé être compris entre 0 et 1) : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e={4fW \over \pi \rho b^{2}V_{f}^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>f</mi> <mi>W</mi> </mrow> <mrow> <mi>π</mi> <mi>ρ</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={4fW \over \pi \rho b^{2}V_{f}^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59c75011aa5e0a14a63f8ab559d6e20dc55396cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:12.575ex; height:6.843ex;" alt="{\displaystyle e={4fW \over \pi \rho b^{2}V_{f}^{2}}}"></span></dd></dl> <p>Si l'on se ramène à la charge alaire : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e={4fP \over \pi \lambda \rho V_{f}^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>f</mi> <mi>P</mi> </mrow> <mrow> <mi>π</mi> <mi>λ</mi> <mi>ρ</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={4fP \over \pi \lambda \rho V_{f}^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f58fbe3137c23ecec0e83c48c8a38094fa4b9c33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:11.878ex; height:6.843ex;" alt="{\displaystyle e={4fP \over \pi \lambda \rho V_{f}^{2}}}"></span></dd></dl></div> <div class="NavEnd"> </div> </div> <div class="mw-heading mw-heading3"><h3 id="Calcul_de_la_finesse_maximale_(d'un_planeur)"><span id="Calcul_de_la_finesse_maximale_.28d.27un_planeur.29"></span>Calcul de la finesse maximale (d'un planeur)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=10" title="Modifier la section : Calcul de la finesse maximale (d'un planeur)" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=10" title="Modifier le code source de la section : Calcul de la finesse maximale (d'un planeur)"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:M%C3%A9canique_du_vol_plan%C3%A9.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/M%C3%A9canique_du_vol_plan%C3%A9.jpg/220px-M%C3%A9canique_du_vol_plan%C3%A9.jpg" decoding="async" width="220" height="123" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/M%C3%A9canique_du_vol_plan%C3%A9.jpg/330px-M%C3%A9canique_du_vol_plan%C3%A9.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/M%C3%A9canique_du_vol_plan%C3%A9.jpg/440px-M%C3%A9canique_du_vol_plan%C3%A9.jpg 2x" data-file-width="1181" data-file-height="659" /></a><figcaption> Schéma des forces appliquées sur un planeur en vol rectiligne uniforme.</figcaption></figure> <p>Un <a href="/wiki/Planeur" title="Planeur">planeur</a> n'a pas de moteur ; il est « propulsé » par la composante sur trajectoire de son propre poids<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite_crochet">[</span>Note 1<span class="cite_crochet">]</span></a></sup> (voir le schéma ci-contre). </p><p>Soit <i>f(V)</i> la finesse du planeur défini par le rapport de la vitesse horizontale à la vitesse verticale. Soit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> l'angle de plané en <a href="/wiki/Radian" title="Radian">radians</a>. Comme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> est petit, on peut écrire que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \approx \tan \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>≈</mo> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \approx \tan \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3de20d36fc7bc57a38cedb71005726fedfab2a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.37ex; height:2.509ex;" alt="{\displaystyle \gamma \approx \tan \gamma }"></span> et donc que : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \approx {1 \over f(V)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \approx {1 \over f(V)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde976d20640e462f0212122bee0167e1e006046" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.072ex; height:6.009ex;" alt="{\displaystyle \gamma \approx {1 \over f(V)}}"></span> </p><p>Quand le planeur est en équilibre, en mouvement non accéléré, on a : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(V)=\tan(\gamma F_{z})\approx \gamma F_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≈</mo> <mi>γ</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(V)=\tan(\gamma F_{z})\approx \gamma F_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa98a32a4e452554b48436bcc042574895710bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.244ex; height:2.843ex;" alt="{\displaystyle R(V)=\tan(\gamma F_{z})\approx \gamma F_{z}}"></span> </p><p>De plus, <i>la finesse maximale est une caractéristique de l'aéronef et est donc constante</i> (tant que les caractéristiques de l'aéronef sont inchangées). </p><p>Dans ce qui suit, on démontre cette assertion qui ne semble pas évidente. On rappelle que lorsque la planeur atteint sa finesse maximale la traînée induite est égale à la traînée parasite. On obtient donc : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={R_{i}(V)+R_{p}(V) \over F_{z}}={2R_{p}(V) \over F_{z}}={\rho C_{x,\mathrm {p} }b^{2}V^{2} \over \lambda F_{z}}={\rho C_{x,\mathrm {p} }b^{2} \over \lambda F_{z}}\times \left({{\sqrt {2}} \over (\pi e)^{1 \over 4}b}\times {\sqrt {F_{z} \over \rho }}\times \left({\lambda \over C_{x,\mathrm {p} }}\right)^{1 \over 4}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>λ</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>λ</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>×</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mi>e</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>ρ</mi> </mfrac> </msqrt> </mrow> <mo>×</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={R_{i}(V)+R_{p}(V) \over F_{z}}={2R_{p}(V) \over F_{z}}={\rho C_{x,\mathrm {p} }b^{2}V^{2} \over \lambda F_{z}}={\rho C_{x,\mathrm {p} }b^{2} \over \lambda F_{z}}\times \left({{\sqrt {2}} \over (\pi e)^{1 \over 4}b}\times {\sqrt {F_{z} \over \rho }}\times \left({\lambda \over C_{x,\mathrm {p} }}\right)^{1 \over 4}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d337d8e401110186d16ceafa399f99be38ae4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:95.184ex; height:9.343ex;" alt="{\displaystyle \gamma ={R_{i}(V)+R_{p}(V) \over F_{z}}={2R_{p}(V) \over F_{z}}={\rho C_{x,\mathrm {p} }b^{2}V^{2} \over \lambda F_{z}}={\rho C_{x,\mathrm {p} }b^{2} \over \lambda F_{z}}\times \left({{\sqrt {2}} \over (\pi e)^{1 \over 4}b}\times {\sqrt {F_{z} \over \rho }}\times \left({\lambda \over C_{x,\mathrm {p} }}\right)^{1 \over 4}\right)^{2}}"></span> </p><p>Et donc : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =2{\sqrt {C_{x,\mathrm {p} } \over \lambda \pi e}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mrow> <mi>λ</mi> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =2{\sqrt {C_{x,\mathrm {p} } \over \lambda \pi e}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52cabbfd88fadb63b7c9f3ecea7da001b6395879" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.889ex; height:7.509ex;" alt="{\displaystyle \gamma =2{\sqrt {C_{x,\mathrm {p} } \over \lambda \pi e}}}"></span> </p><p>Et donc : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over \gamma }=f={1 \over 2}{\sqrt {\lambda \pi e \over C_{x,\mathrm {p} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>γ</mi> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>λ</mi> <mi>π</mi> <mi>e</mi> </mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over \gamma }=f={1 \over 2}{\sqrt {\lambda \pi e \over C_{x,\mathrm {p} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82a78af0e48b23ef2927e95f4ef06d48a60c5b7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.938ex; height:7.509ex;" alt="{\displaystyle {1 \over \gamma }=f={1 \over 2}{\sqrt {\lambda \pi e \over C_{x,\mathrm {p} }}}}"></span> </p><p>Comme annoncé ci-dessus, la finesse maximale ne dépend pas de la masse du planeur et ni de la densité de l'air environnant. Elle dépend uniquement de l'aérodynamisme du planeur et de sa géométrie (allongement) : la <i>finesse maximale</i> est une caractéristique de l'aéronef et est donc <i>constante</i>. Ceci justifie <i>a posteriori</i> que la vitesse de chute du planeur augmentera en même temps que sa masse. Donc, lorsque les conditions aérologiques sont moins favorables<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite_crochet">[</span>Note 2<span class="cite_crochet">]</span></a></sup>, il est préférable de minimiser la masse du planeur pour minimiser la vitesse de chute et donc de ne pas ajouter d'eau dans les ailes ou, si l'on est déjà en vol, de vidanger les ailes. </p><p>De plus, plus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> est grand, plus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \approx \tan \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>≈</mo> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \approx \tan \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3de20d36fc7bc57a38cedb71005726fedfab2a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.37ex; height:2.509ex;" alt="{\displaystyle \gamma \approx \tan \gamma }"></span> sera petit. Donc, les planeurs ayant des grandes ailes, pour une surface alaire équivalente, aura un plus petit angle de plané et donc une plus grande finesse. Ceci est la raison pour laquelle certains planeurs de compétition en classe libre peuvent avoir jusqu'à 30 mètres d'envergure. </p> <div class="mw-heading mw-heading3"><h3 id="Effet_de_la_masse_sur_la_vitesse_optimale">Effet de la masse sur la vitesse optimale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=11" title="Modifier la section : Effet de la masse sur la vitesse optimale" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=11" title="Modifier le code source de la section : Effet de la masse sur la vitesse optimale"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cette section suppose que l'aéronef a une finesse suffisante pour que l'on puisse supposer que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \approx \tan \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>≈</mo> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \approx \tan \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3de20d36fc7bc57a38cedb71005726fedfab2a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.37ex; height:2.509ex;" alt="{\displaystyle \gamma \approx \tan \gamma }"></span>. </p><p>On considère un planeur de masse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> volant à sa vitesse de finesse maximale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adfdbc929f16cb00bb43289c223651b41f7b9f80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{1}}"></span>. Le poids du planeur est donné par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \gamma F_{z}=mg}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \gamma F_{z}=mg}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff9a34074590f4471e2f8a0efb1bb8940db12ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.512ex; height:2.676ex;" alt="{\displaystyle \cos \gamma F_{z}=mg}"></span>. Pour simplifier la discussion, on suppose que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \gamma \approx 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> <mo>≈</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \gamma \approx 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b95522bd84cb8cfbd914232405a9a1f88cf0463" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.021ex; height:2.676ex;" alt="{\displaystyle \cos \gamma \approx 1}"></span>. On a donc : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x,\mathrm {p} }={4 \over \lambda \pi e}~{m^{2}g^{2} \over \rho ^{2}S^{2}{V_{1}}^{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mi>λ</mi> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x,\mathrm {p} }={4 \over \lambda \pi e}~{m^{2}g^{2} \over \rho ^{2}S^{2}{V_{1}}^{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c76c4dbeac7e678b68ed1efb1389e915f6808512" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.624ex; height:6.676ex;" alt="{\displaystyle C_{x,\mathrm {p} }={4 \over \lambda \pi e}~{m^{2}g^{2} \over \rho ^{2}S^{2}{V_{1}}^{4}}}"></span> </p><p>On considère maintenant le même planeur auquel on a ajouté de l'eau et qui a une masse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> et une vitesse de finesse maximale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceaa689a894f5020a7b46177d201cbce2d41122b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{2}}"></span>. On a alors : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {4 \over \lambda \pi e}~{m^{2}g^{2} \over \rho ^{2}S^{2}{V_{1}}^{4}}=C_{x,\mathrm {p} }={4 \over \lambda \pi e}~{M^{2}g^{2} \over \rho ^{2}S^{2}{V_{2}}^{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mi>λ</mi> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mi>λ</mi> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {4 \over \lambda \pi e}~{m^{2}g^{2} \over \rho ^{2}S^{2}{V_{1}}^{4}}=C_{x,\mathrm {p} }={4 \over \lambda \pi e}~{M^{2}g^{2} \over \rho ^{2}S^{2}{V_{2}}^{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05d8d7263ec76a80749e0d29aa0bb24175f3ad8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:39.041ex; height:6.676ex;" alt="{\displaystyle {4 \over \lambda \pi e}~{m^{2}g^{2} \over \rho ^{2}S^{2}{V_{1}}^{4}}=C_{x,\mathrm {p} }={4 \over \lambda \pi e}~{M^{2}g^{2} \over \rho ^{2}S^{2}{V_{2}}^{4}}}"></span> </p><p>Donc, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {m^{2} \over {V_{1}}^{4}}={M^{2} \over {V_{2}}^{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {m^{2} \over {V_{1}}^{4}}={M^{2} \over {V_{2}}^{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f34757c9f67962975da172a107a6f2b782a13e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.787ex; height:6.509ex;" alt="{\displaystyle {m^{2} \over {V_{1}}^{4}}={M^{2} \over {V_{2}}^{4}}}"></span> </p><p>Donc, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({V_{2} \over V_{1}}\right)^{4}=\left({M \over m}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <mi>m</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({V_{2} \over V_{1}}\right)^{4}=\left({M \over m}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14954b5a79af1c742b119a01d82e0793befb2747" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.573ex; height:6.676ex;" alt="{\displaystyle \left({V_{2} \over V_{1}}\right)^{4}=\left({M \over m}\right)^{2}}"></span> </p><p>et donc : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {V_{2} \over V_{1}}={\sqrt {M \over m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>M</mi> <mi>m</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {V_{2} \over V_{1}}={\sqrt {M \over m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a642f138ff6233e5d11f26f345ce04b2ed3c70db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.946ex; height:6.176ex;" alt="{\displaystyle {V_{2} \over V_{1}}={\sqrt {M \over m}}}"></span>. </p><p>On constate que <i>la vitesse optimale varie donc comme la racine carrée de la masse du planeur.</i> </p><p>En augmentant la masse, on augmente donc aussi la vitesse de finesse maximale mais la valeur de la finesse maximale elle reste constante. La finesse maximale étant indépendante de la masse de l'aéronef, ceci signifie que le même planeur auquel on ajoute de l'eau aura la même portée, mais volera plus vite pour maintenir la même portée. C'est pourquoi lorsque les conditions météorologiques sont très favorables (puissantes ascendances), les planeurs de compétition sont remplis d'eau dans les ailes. </p> <div class="mw-heading mw-heading3"><h3 id="Polaire_des_vitesses">Polaire des vitesses</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=12" title="Modifier la section : Polaire des vitesses" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=12" title="Modifier le code source de la section : Polaire des vitesses"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La polaire des vitesses peut se mettre sous la forme<sup id="cite_ref-Irving18_11-0" class="reference"><a href="#cite_note-Irving18-11"><span class="cite_crochet">[</span>9<span class="cite_crochet">]</span></a></sup> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z}=AV^{3}+B{1 \over V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mi>A</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>V</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z}=AV^{3}+B{1 \over V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc97847110521d23341e8115b9d9a1a47a66638b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.397ex; height:5.343ex;" alt="{\displaystyle V_{z}=AV^{3}+B{1 \over V}}"></span></dd></dl> <p>où <i>A</i> et <i>B</i> sont des constantes à déterminer<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite_crochet">[</span>Note 3<span class="cite_crochet">]</span></a></sup>. </p><p>On évalue maintenant la vitesse de chute en fonction de la vitesse horizontale pour n'importe quelle vitesse. On a : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \gamma ={R_{i}(V)+R_{p}(V) \over F_{z}}={2F_{z} \over b^{2}\rho V^{2}\pi e}+{\rho C_{x,\mathrm {p} }b^{2}V^{2} \over 2\lambda F_{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>λ</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \gamma ={R_{i}(V)+R_{p}(V) \over F_{z}}={2F_{z} \over b^{2}\rho V^{2}\pi e}+{\rho C_{x,\mathrm {p} }b^{2}V^{2} \over 2\lambda F_{z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f70db20567f65e910c34776f17bb9350ebf340f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.046ex; height:6.509ex;" alt="{\displaystyle \tan \gamma ={R_{i}(V)+R_{p}(V) \over F_{z}}={2F_{z} \over b^{2}\rho V^{2}\pi e}+{\rho C_{x,\mathrm {p} }b^{2}V^{2} \over 2\lambda F_{z}}}"></span> </p><p>La <a href="/wiki/Polaire_des_vitesses" title="Polaire des vitesses">polaire des vitesses</a> exprime la vitesse de chute <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96e11ccf23a1ff91086633817a622a93dc70c4c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.357ex; height:2.509ex;" alt="{\displaystyle V_{z}}"></span> en fonction de la vitesse horizontale. Etant donné que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> est très petit, on a : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \gamma \approx \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> <mo>≈</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \gamma \approx \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d9c9a0964a4ca565c811a1d3cb6b41eef42be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.37ex; height:2.509ex;" alt="{\displaystyle \tan \gamma \approx \gamma }"></span> </p><p>On peut donc considérer que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z}=\gamma V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mi>γ</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z}=\gamma V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/909cf197bcc5451f4a1db948e29d9bcf77fb1fc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.505ex; height:2.676ex;" alt="{\displaystyle V_{z}=\gamma V}"></span>. Donc, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z}=\left({2F_{z} \over b^{2}\rho V^{2}\pi e}+{\rho C_{x,\mathrm {p} }b^{2}V^{2} \over 2\lambda F_{z}}\right)V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>λ</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z}=\left({2F_{z} \over b^{2}\rho V^{2}\pi e}+{\rho C_{x,\mathrm {p} }b^{2}V^{2} \over 2\lambda F_{z}}\right)V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f79f1a6b72b2fa16b1cf7bf6217efc77c3cfc30a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.895ex; height:7.509ex;" alt="{\displaystyle V_{z}=\left({2F_{z} \over b^{2}\rho V^{2}\pi e}+{\rho C_{x,\mathrm {p} }b^{2}V^{2} \over 2\lambda F_{z}}\right)V}"></span> </p><p>Cette formule exprime la polaire des vitesses. On constate que pour <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> grand, la finesse décroît comme le carré de la vitesse horizontale. </p><p>On notera que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {F_{z}\lambda \over b^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>λ</mi> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F_{z}\lambda \over b^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/207f04745ba7eafeeaf67d64a409f3ec7b2b8025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:4.688ex; height:5.676ex;" alt="{\displaystyle {F_{z}\lambda \over b^{2}}}"></span> est la charge alaire qui est souvent exprimée en <abbr class="abbr" title="décanewton par mètre carré">daN/m<sup>2</sup></abbr> ou plus incorrectement en <abbr class="abbr" title="kilogramme-force par mètre carré">kgf/m<sup>2</sup></abbr>. Si l'on appelle <i>P</i> cette charge alaire (qui est homogène à une pression), on obtient : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z}=\left({P \over \rho V^{2}\lambda \pi e}+{\rho C_{x,\mathrm {p} }V^{2} \over 2P}\right)V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <mrow> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>λ</mi> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>P</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z}=\left({P \over \rho V^{2}\lambda \pi e}+{\rho C_{x,\mathrm {p} }V^{2} \over 2P}\right)V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0171a98f29e301013277030dddc58302b44aa61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.146ex; height:7.509ex;" alt="{\displaystyle V_{z}=\left({P \over \rho V^{2}\lambda \pi e}+{\rho C_{x,\mathrm {p} }V^{2} \over 2P}\right)V}"></span> </p><p>et donc : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\rho C_{x,\mathrm {p} } \over 2P}\qquad B={P \over \rho \lambda \pi e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi>P</mi> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <mrow> <mi>ρ</mi> <mi>λ</mi> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\rho C_{x,\mathrm {p} } \over 2P}\qquad B={P \over \rho \lambda \pi e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e975e7abfa1e5d6b74e51f35d82dcc51c7c397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.402ex; height:6.176ex;" alt="{\displaystyle A={\rho C_{x,\mathrm {p} } \over 2P}\qquad B={P \over \rho \lambda \pi e}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Vitesse_de_chute_à_finesse_maximale"><span id="Vitesse_de_chute_.C3.A0_finesse_maximale"></span>Vitesse de chute à finesse maximale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=13" title="Modifier la section : Vitesse de chute à finesse maximale" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=13" title="Modifier le code source de la section : Vitesse de chute à finesse maximale"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On a : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z,f}=\left({\alpha \over V_{f}^{2}}+\beta V_{f}^{2}\right){V_{f} \over F_{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>+</mo> <mi>β</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z,f}=\left({\alpha \over V_{f}^{2}}+\beta V_{f}^{2}\right){V_{f} \over F_{z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1402834babe1d62c2f24354e2a76ba6d75e321f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.169ex; height:7.676ex;" alt="{\displaystyle V_{z,f}=\left({\alpha \over V_{f}^{2}}+\beta V_{f}^{2}\right){V_{f} \over F_{z}}}"></span> </p><p>Comme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha /V^{2}=\beta V^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>β</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha /V^{2}=\beta V^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e22719d75b9bf030f79d003bcfce7589b4f71e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.023ex; height:3.176ex;" alt="{\displaystyle \alpha /V^{2}=\beta V^{2}}"></span> à finesse maximale, on obtient donc : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z,f}=2\beta V_{f}^{2}{V_{f} \over F_{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>β</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z,f}=2\beta V_{f}^{2}{V_{f} \over F_{z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f67805b625ffb9e37ee1de8eb36482601dfe7a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.615ex; height:5.843ex;" alt="{\displaystyle V_{z,f}=2\beta V_{f}^{2}{V_{f} \over F_{z}}}"></span> </p><p>En substituant β, on obtient </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z,f}=\rho {C_{x,\mathrm {p} } \over \lambda }b^{2}{V_{f}^{3} \over F_{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mi>λ</mi> </mfrac> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z,f}=\rho {C_{x,\mathrm {p} } \over \lambda }b^{2}{V_{f}^{3} \over F_{z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2fe117d814f5b264afde415d3e43fa7359e1cfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.92ex; height:6.676ex;" alt="{\displaystyle V_{z,f}=\rho {C_{x,\mathrm {p} } \over \lambda }b^{2}{V_{f}^{3} \over F_{z}}}"></span> </p><p>On remplace <i>Vf</i> et donc, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z,f}=\rho {C_{x,\mathrm {p} } \over \lambda }{b^{2} \over F_{z}}\left[{{\sqrt {2}} \over (\pi e)^{1 \over 4}b}\times {\sqrt {F_{z} \over \rho }}\times \left({\lambda \over C_{x,\mathrm {p} }}\right)^{1 \over 4}\right]^{3}=\left({C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{2{\sqrt {2}} \over (\pi e)^{3 \over 4}}{1 \over b}{\sqrt {F_{z} \over \rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mi>λ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> <msup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mi>e</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>ρ</mi> </mfrac> </msqrt> </mrow> <mo>×</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mi>λ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mi>e</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>ρ</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z,f}=\rho {C_{x,\mathrm {p} } \over \lambda }{b^{2} \over F_{z}}\left[{{\sqrt {2}} \over (\pi e)^{1 \over 4}b}\times {\sqrt {F_{z} \over \rho }}\times \left({\lambda \over C_{x,\mathrm {p} }}\right)^{1 \over 4}\right]^{3}=\left({C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{2{\sqrt {2}} \over (\pi e)^{3 \over 4}}{1 \over b}{\sqrt {F_{z} \over \rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be2eba8e7ef0b7bb94f3022a569ab54317348ca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:77.626ex; height:9.343ex;" alt="{\displaystyle V_{z,f}=\rho {C_{x,\mathrm {p} } \over \lambda }{b^{2} \over F_{z}}\left[{{\sqrt {2}} \over (\pi e)^{1 \over 4}b}\times {\sqrt {F_{z} \over \rho }}\times \left({\lambda \over C_{x,\mathrm {p} }}\right)^{1 \over 4}\right]^{3}=\left({C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{2{\sqrt {2}} \over (\pi e)^{3 \over 4}}{1 \over b}{\sqrt {F_{z} \over \rho }}}"></span> </p><p>On note que : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x,\mathrm {p} }={\lambda \pi e \over 4f^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>λ</mi> <mi>π</mi> <mi>e</mi> </mrow> <mrow> <mn>4</mn> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x,\mathrm {p} }={\lambda \pi e \over 4f^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22877feecf4aa8565a15f01f677c299aad8b7f07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.911ex; height:6.009ex;" alt="{\displaystyle C_{x,\mathrm {p} }={\lambda \pi e \over 4f^{2}}}"></span> </p><p>En substituant, on obtient : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z,f}=\left({\pi e \over 4f^{2}}\right)^{1 \over 4}{2{\sqrt {2}} \over (\pi e)^{3 \over 4}}{1 \over b}{\sqrt {F_{z} \over \rho }}={2 \over b}{\sqrt {F_{z} \over f\pi e\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>e</mi> </mrow> <mrow> <mn>4</mn> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mi>e</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>ρ</mi> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>b</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow> <mi>f</mi> <mi>π</mi> <mi>e</mi> <mi>ρ</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z,f}=\left({\pi e \over 4f^{2}}\right)^{1 \over 4}{2{\sqrt {2}} \over (\pi e)^{3 \over 4}}{1 \over b}{\sqrt {F_{z} \over \rho }}={2 \over b}{\sqrt {F_{z} \over f\pi e\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e92ccfc35cb88e98ca98ee42858682a616b3749f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:43.951ex; height:8.509ex;" alt="{\displaystyle V_{z,f}=\left({\pi e \over 4f^{2}}\right)^{1 \over 4}{2{\sqrt {2}} \over (\pi e)^{3 \over 4}}{1 \over b}{\sqrt {F_{z} \over \rho }}={2 \over b}{\sqrt {F_{z} \over f\pi e\rho }}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Vitesse_de_chute_minimum">Vitesse de chute minimum</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=14" title="Modifier la section : Vitesse de chute minimum" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=14" title="Modifier le code source de la section : Vitesse de chute minimum"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En reprenant les notations ci-dessus, on a : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z}={\alpha \over F_{z}V}+{\beta V^{3} \over F_{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>β</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z}={\alpha \over F_{z}V}+{\beta V^{3} \over F_{z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4133b0577107c3e7d4666a1cd3ad644f6bff7b8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.555ex; height:6.176ex;" alt="{\displaystyle V_{z}={\alpha \over F_{z}V}+{\beta V^{3} \over F_{z}}}"></span> </p><p>On appelle vitesse minimale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40836783b20c374654ab03ddd6b01586ad9b4fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.03ex; height:2.509ex;" alt="{\displaystyle V_{m}}"></span> la vitesse horizontale pour laquelle le taux de chute minimal est atteint. Elle est atteinte lorsque <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {dV_{z} \over dV}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dV_{z} \over dV}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/664265a971bab5917ca17bbe4957199e05f852d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.67ex; height:5.509ex;" alt="{\displaystyle {dV_{z} \over dV}=0}"></span>. On obtient donc : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\alpha \over F_{z}V_{m}^{2}}+3{\beta V_{m}^{2} \over F_{z}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>β</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\alpha \over F_{z}V_{m}^{2}}+3{\beta V_{m}^{2} \over F_{z}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a73f61017937a95415d7eaa088070fc29fa66a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.633ex; height:6.343ex;" alt="{\displaystyle -{\alpha \over F_{z}V_{m}^{2}}+3{\beta V_{m}^{2} \over F_{z}}=0}"></span> </p><p>Soit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc0d1604bbf75069df35d14afa9fac3e883be3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.492ex; height:2.843ex;" alt="{\displaystyle V_{f}}"></span> la vitesse à finesse maximale. Donc<sup id="cite_ref-Irving20_14-0" class="reference"><a href="#cite_note-Irving20-14"><span class="cite_crochet">[</span>11<span class="cite_crochet">]</span></a></sup>, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{m}=\left({\alpha \over 3\beta }\right)^{1 \over 4}={\left({1 \over 3}\right)}^{1 \over 4}~V_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mn>3</mn> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mtext> </mtext> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{m}=\left({\alpha \over 3\beta }\right)^{1 \over 4}={\left({1 \over 3}\right)}^{1 \over 4}~V_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c44a90d3ad98da98cec3696661ebde484f1f8c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.942ex; height:7.176ex;" alt="{\displaystyle V_{m}=\left({\alpha \over 3\beta }\right)^{1 \over 4}={\left({1 \over 3}\right)}^{1 \over 4}~V_{f}}"></span> </p><p>On obtient donc : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{m}\approx 0.76\times V_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>≈</mo> <mn>0.76</mn> <mo>×</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{m}\approx 0.76\times V_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92c7ea11b68449c63687ef1e0037c8f1f458f96f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.595ex; height:2.843ex;" alt="{\displaystyle V_{m}\approx 0.76\times V_{f}}"></span> </p><p>On a : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z,m}=\left({\alpha \over V_{m}^{2}}+\beta V_{m}^{2}\right){V_{m} \over F_{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>+</mo> <mi>β</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z,m}=\left({\alpha \over V_{m}^{2}}+\beta V_{m}^{2}\right){V_{m} \over F_{z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ea108963d5815507135a08fee4401e64ea9352f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.099ex; height:6.176ex;" alt="{\displaystyle V_{z,m}=\left({\alpha \over V_{m}^{2}}+\beta V_{m}^{2}\right){V_{m} \over F_{z}}}"></span> </p><p>On a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{m}=\left({\alpha \over 3\beta }\right)^{1 \over 4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mn>3</mn> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{m}=\left({\alpha \over 3\beta }\right)^{1 \over 4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f476c8faceb7738c20c69a4408ea9d20cc8bcfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.616ex; height:7.176ex;" alt="{\displaystyle V_{m}=\left({\alpha \over 3\beta }\right)^{1 \over 4}}"></span> et donc <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{m}^{2}={\sqrt {\alpha \over 3\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>α</mi> <mrow> <mn>3</mn> <mi>β</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{m}^{2}={\sqrt {\alpha \over 3\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec40b1ec22dcd6d9526e24ed650ea9ebb126b917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.783ex; height:6.176ex;" alt="{\displaystyle V_{m}^{2}={\sqrt {\alpha \over 3\beta }}}"></span> que l'on substitue et donc, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z,m}={V_{m} \over F_{z}}\left({\alpha \over {\sqrt {\alpha \over 3\beta }}}+\beta {\sqrt {\alpha \over 3\beta }}\right)={V_{m} \over F_{z}}{\sqrt {\alpha \beta }}\left({\sqrt {3}}+{1 \over {\sqrt {3}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>α</mi> <mrow> <mn>3</mn> <mi>β</mi> </mrow> </mfrac> </msqrt> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>α</mi> <mrow> <mn>3</mn> <mi>β</mi> </mrow> </mfrac> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α</mi> <mi>β</mi> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z,m}={V_{m} \over F_{z}}\left({\alpha \over {\sqrt {\alpha \over 3\beta }}}+\beta {\sqrt {\alpha \over 3\beta }}\right)={V_{m} \over F_{z}}{\sqrt {\alpha \beta }}\left({\sqrt {3}}+{1 \over {\sqrt {3}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cea7f50fddf440e48b60b7ba8c9ebaa4423ce1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:57.052ex; height:10.509ex;" alt="{\displaystyle V_{z,m}={V_{m} \over F_{z}}\left({\alpha \over {\sqrt {\alpha \over 3\beta }}}+\beta {\sqrt {\alpha \over 3\beta }}\right)={V_{m} \over F_{z}}{\sqrt {\alpha \beta }}\left({\sqrt {3}}+{1 \over {\sqrt {3}}}\right)}"></span> </p><p>On substitue <i>V</i><sub>m</sub> et donc, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z,m}={1 \over F_{z}}\left({\alpha \over 3\beta }\right)^{1 \over 4}{\sqrt {\alpha \beta }}{4 \over {\sqrt {3}}}={\alpha ^{3 \over 4}\beta ^{1 \over 4} \over 3^{1 \over 4}}\times {1 \over F_{z}}\times {4 \over {\sqrt {3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mn>3</mn> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α</mi> <mi>β</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </msup> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z,m}={1 \over F_{z}}\left({\alpha \over 3\beta }\right)^{1 \over 4}{\sqrt {\alpha \beta }}{4 \over {\sqrt {3}}}={\alpha ^{3 \over 4}\beta ^{1 \over 4} \over 3^{1 \over 4}}\times {1 \over F_{z}}\times {4 \over {\sqrt {3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb937b9a8d92deeface3f6e47b2b684777c8dcfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:51.432ex; height:8.176ex;" alt="{\displaystyle V_{z,m}={1 \over F_{z}}\left({\alpha \over 3\beta }\right)^{1 \over 4}{\sqrt {\alpha \beta }}{4 \over {\sqrt {3}}}={\alpha ^{3 \over 4}\beta ^{1 \over 4} \over 3^{1 \over 4}}\times {1 \over F_{z}}\times {4 \over {\sqrt {3}}}}"></span> </p><p>On substitue maintenant α et β et donc, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z,m}=\left({2F_{z}^{2} \over b^{2}\rho \pi e}\right)^{3 \over 4}\left({1 \over 2}{\rho b^{2}C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{1 \over 3^{1 \over 4}}\times {1 \over F_{z}}\times {4 \over {\sqrt {3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> <mi>λ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z,m}=\left({2F_{z}^{2} \over b^{2}\rho \pi e}\right)^{3 \over 4}\left({1 \over 2}{\rho b^{2}C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{1 \over 3^{1 \over 4}}\times {1 \over F_{z}}\times {4 \over {\sqrt {3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c9bc1be2f15e2ef73292659bf62e59641e8a42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:51.411ex; height:8.843ex;" alt="{\displaystyle V_{z,m}=\left({2F_{z}^{2} \over b^{2}\rho \pi e}\right)^{3 \over 4}\left({1 \over 2}{\rho b^{2}C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{1 \over 3^{1 \over 4}}\times {1 \over F_{z}}\times {4 \over {\sqrt {3}}}}"></span> </p><p>On obtient donc<sup id="cite_ref-Irving20_14-1" class="reference"><a href="#cite_note-Irving20-14"><span class="cite_crochet">[</span>11<span class="cite_crochet">]</span></a></sup> : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z,m}={4{\sqrt {2}} \over 3^{3 \over 4}}{1 \over (\pi e)^{3 \over 4}}\left({C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{1 \over b}{\sqrt {F_{z} \over \rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mi>e</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mi>λ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>ρ</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z,m}={4{\sqrt {2}} \over 3^{3 \over 4}}{1 \over (\pi e)^{3 \over 4}}\left({C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{1 \over b}{\sqrt {F_{z} \over \rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62d1b355171d5fe82d4edc6aa35537dd7aec6f9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:37.103ex; height:8.676ex;" alt="{\displaystyle V_{z,m}={4{\sqrt {2}} \over 3^{3 \over 4}}{1 \over (\pi e)^{3 \over 4}}\left({C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{1 \over b}{\sqrt {F_{z} \over \rho }}}"></span> </p><p>Le rapport entre la vitesse de chute minimale et la vitesse de chute à finesse maximale est : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {V_{z,m} \over V_{z,f}}={{4{\sqrt {2}} \over {\sqrt {3}}b}{1 \over (\pi e)^{3 \over 4}}\left({C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{1 \over b}{\sqrt {F_{z} \over \rho }} \over \left({C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{2{\sqrt {2}} \over (\pi e)^{3 \over 4}}{1 \over b}{\sqrt {F_{z} \over \rho }}}={4{\sqrt {2}} \over 3^{3 \over 4}}\times {1 \over 2{\sqrt {2}}}={2 \over 3^{3 \over 4}}\approx 0.88}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mi>b</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mi>e</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mi>λ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>ρ</mi> </mfrac> </msqrt> </mrow> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mi>λ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mi>e</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>ρ</mi> </mfrac> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </msup> </mfrac> </mrow> <mo>≈</mo> <mn>0.88</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {V_{z,m} \over V_{z,f}}={{4{\sqrt {2}} \over {\sqrt {3}}b}{1 \over (\pi e)^{3 \over 4}}\left({C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{1 \over b}{\sqrt {F_{z} \over \rho }} \over \left({C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{2{\sqrt {2}} \over (\pi e)^{3 \over 4}}{1 \over b}{\sqrt {F_{z} \over \rho }}}={4{\sqrt {2}} \over 3^{3 \over 4}}\times {1 \over 2{\sqrt {2}}}={2 \over 3^{3 \over 4}}\approx 0.88}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1118b257d304d351a7ac7d564de945d2049b06a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:63.793ex; height:15.843ex;" alt="{\displaystyle {V_{z,m} \over V_{z,f}}={{4{\sqrt {2}} \over {\sqrt {3}}b}{1 \over (\pi e)^{3 \over 4}}\left({C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{1 \over b}{\sqrt {F_{z} \over \rho }} \over \left({C_{x,\mathrm {p} } \over \lambda }\right)^{1 \over 4}{2{\sqrt {2}} \over (\pi e)^{3 \over 4}}{1 \over b}{\sqrt {F_{z} \over \rho }}}={4{\sqrt {2}} \over 3^{3 \over 4}}\times {1 \over 2{\sqrt {2}}}={2 \over 3^{3 \over 4}}\approx 0.88}"></span> </p><p>On constate donc que la vitesse de chute minimale n'est que 12 % inférieure à la vitesse de chute à finesse maximale. </p> <div class="mw-heading mw-heading2"><h2 id="Application_au_planeur_ASW_27">Application au planeur ASW 27</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=15" title="Modifier la section : Application au planeur ASW 27" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=15" title="Modifier le code source de la section : Application au planeur ASW 27"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On considère le planeur <a href="/wiki/Alexander_Schleicher_ASW_27" title="Alexander Schleicher ASW 27">Alexander Schleicher ASW 27</a><sup id="cite_ref-ASW27_15-0" class="reference"><a href="#cite_note-ASW27-15"><span class="cite_crochet">[</span>12<span class="cite_crochet">]</span></a></sup>. </p><p>Le constructeur affirme que son planeur a une finesse de 48. Les chiffres officiels sont les suivants : </p> <ul><li><i>λ</i> = 25 ;</li> <li><i>e</i> = 0,85 ;</li> <li><i>b</i> = 15 <abbr class="abbr" title="mètre">m</abbr> ;</li> <li><i>C</i><sub>x,p</sub> = 0,0072 (ajusté pour satisfaire la finesse déclarée).</li></ul> <p>On obtient alors : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over \gamma }={1 \over 2}{\sqrt {25\times \pi \times 0,85 \over 0,0072}}=48,1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>γ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>25</mn> <mo>×</mo> <mi>π</mi> <mo>×</mo> <mn>0</mn> <mo>,</mo> <mn>85</mn> </mrow> <mrow> <mn>0</mn> <mo>,</mo> <mn>0072</mn> </mrow> </mfrac> </msqrt> </mrow> <mo>=</mo> <mn>48</mn> <mo>,</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over \gamma }={1 \over 2}{\sqrt {25\times \pi \times 0,85 \over 0,0072}}=48,1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94429ab75f76b77ba032d356405d66b4c981b516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.834ex; height:7.509ex;" alt="{\displaystyle {1 \over \gamma }={1 \over 2}{\sqrt {25\times \pi \times 0,85 \over 0,0072}}=48,1}"></span> </p><p>La masse à vide du planeur est 245 kilogrammes. On considère un pilote de masse 65 kilogrammes volant aux <a href="/wiki/Conditions_normales_de_temp%C3%A9rature_et_de_pression" title="Conditions normales de température et de pression">conditions normales de température et de pression</a>. On a alors : </p> <ul><li>ρ = 1,225 ;</li> <li>m = 310 <abbr class="abbr" title="kilogramme">kg</abbr>.</li></ul> <p>La vitesse à laquelle la finesse maximale est atteinte est </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{m}={{\sqrt {2}} \over (\pi \times 0,85)^{1 \over 4}\times 15}\times {\sqrt {310\times 9,8 \over 1,225}}\times \left({25 \over 0,0072}\right)^{1 \over 4}=28,19~\mathrm {m/s} =101,5~\mathrm {km/h} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mo>×</mo> <mn>0</mn> <mo>,</mo> <mn>85</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mo>×</mo> <mn>15</mn> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>310</mn> <mo>×</mo> <mn>9</mn> <mo>,</mo> <mn>8</mn> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mn>225</mn> </mrow> </mfrac> </msqrt> </mrow> <mo>×</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>25</mn> <mrow> <mn>0</mn> <mo>,</mo> <mn>0072</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mn>28</mn> <mo>,</mo> <mn>19</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">s</mi> </mrow> <mo>=</mo> <mn>101</mn> <mo>,</mo> <mn>5</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">h</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{m}={{\sqrt {2}} \over (\pi \times 0,85)^{1 \over 4}\times 15}\times {\sqrt {310\times 9,8 \over 1,225}}\times \left({25 \over 0,0072}\right)^{1 \over 4}=28,19~\mathrm {m/s} =101,5~\mathrm {km/h} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e07a55ac310ce52b86c55c7aaba3511cc217f7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:84.094ex; height:8.509ex;" alt="{\displaystyle V_{m}={{\sqrt {2}} \over (\pi \times 0,85)^{1 \over 4}\times 15}\times {\sqrt {310\times 9,8 \over 1,225}}\times \left({25 \over 0,0072}\right)^{1 \over 4}=28,19~\mathrm {m/s} =101,5~\mathrm {km/h} }"></span> </p><p>Le constructeur affirme que la finesse maximale est atteinte à <span title="27,777 8 m/s" style="cursor:help">100</span> <abbr class="abbr" title="kilomètre par heure">km/h</abbr> ce qui fait que le modèle n'engendre une erreur que de moins de 2 %. </p><p>Donc, la vitesse horizontale de chute minimale sera<sup id="cite_ref-ASW27_15-1" class="reference"><a href="#cite_note-ASW27-15"><span class="cite_crochet">[</span>12<span class="cite_crochet">]</span></a></sup>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{m}=98,7\times 0,76=77~\mathrm {km/h} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mn>98</mn> <mo>,</mo> <mn>7</mn> <mo>×</mo> <mn>0</mn> <mo>,</mo> <mn>76</mn> <mo>=</mo> <mn>77</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">h</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{m}=98,7\times 0,76=77~\mathrm {km/h} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a830492d714d99fce991e88d77f3dab096ef63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.634ex; height:2.843ex;" alt="{\displaystyle V_{m}=98,7\times 0,76=77~\mathrm {km/h} }"></span> </p><p>En examinant la polaire des vitesses, on constate que la vitesse de chute minimum est à <span title="21,388 906 m/s" style="cursor:help">77</span> <abbr class="abbr" title="kilomètre par heure">km/h</abbr> ce qui correspond donc à la formule ci-dessus. </p><p>Le taux de chute minimal est </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{z,m}={28,19 \over 48}\times 0,88=0,52}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>28</mn> <mo>,</mo> <mn>19</mn> </mrow> <mn>48</mn> </mfrac> </mrow> <mo>×</mo> <mn>0</mn> <mo>,</mo> <mn>88</mn> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>52</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{z,m}={28,19 \over 48}\times 0,88=0,52}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61714b67dbe05941820068edd063389b22d80902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.857ex; height:5.509ex;" alt="{\displaystyle V_{z,m}={28,19 \over 48}\times 0,88=0,52}"></span> </p><p>Le constructeur affirme que le taux de chute minimal est <span title="1,872 km/h" style="cursor:help">0,52</span> <abbr class="abbr" title="mètre par seconde">m/s</abbr>. </p><p>On constate que dans le cas du planeur ASW-27, la <a href="/wiki/Th%C3%A9orie_des_profils_minces" title="Théorie des profils minces">théorie des profils minces</a> peut représenter la polaire des vitesses et les caractéristiques du planeur à moins de 2 % près. </p> <div class="mw-heading mw-heading2"><h2 id="Autres_domaines">Autres domaines</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=16" title="Modifier la section : Autres domaines" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=16" title="Modifier le code source de la section : Autres domaines"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Une voile est aussi un profil. La notion de finesse s'applique donc aussi à ce profil, mais de plusieurs façons. Voir <a href="/wiki/Effort_sur_une_voile#Finesses" title="Effort sur une voile">finesse d'une voile de bateau</a>.</li></ul> <ul><li>Une hélice aquatique est composée de plusieurs pales, chacune ayant un profil. La définition de finesse est identique à la finesse aérodynamique, le fluide étant de l'eau.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Généralisation_de_la_notion_de_finesse_à_tous_les_modes_de_transport"><span id="G.C3.A9n.C3.A9ralisation_de_la_notion_de_finesse_.C3.A0_tous_les_modes_de_transport"></span>Généralisation de la notion de finesse à tous les modes de transport</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=17" title="Modifier la section : Généralisation de la notion de finesse à tous les modes de transport" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=17" title="Modifier le code source de la section : Généralisation de la notion de finesse à tous les modes de transport"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Gabrielli-von_Karman,_update2004.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Gabrielli-von_Karman%2C_update2004.png/220px-Gabrielli-von_Karman%2C_update2004.png" decoding="async" width="220" height="258" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Gabrielli-von_Karman%2C_update2004.png/330px-Gabrielli-von_Karman%2C_update2004.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Gabrielli-von_Karman%2C_update2004.png/440px-Gabrielli-von_Karman%2C_update2004.png 2x" data-file-width="756" data-file-height="887" /></a><figcaption>Mise à jour 2004 du diagramme de Gabrielli-von Karman donnant la finesse « totale » des différents modes de transport.</figcaption></figure> <p>Plus généralement, la notion de finesse peut s'appliquer avantageusement à tous les modes de transport (de marchandise ou de passagers) pour permettre l'évaluation de leur rendement énergétique. En effet, le rendement de chaque véhicule est le quotient du poids de ce véhicule sur les forces de traînée qui le freinent (diagramme de <a href="/wiki/Giuseppe_Gabrielli" title="Giuseppe Gabrielli">Gabrielli</a>-<a href="/wiki/Theodore_von_K%C3%A1rm%C3%A1n" title="Theodore von Kármán">von Kármán</a><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite_crochet">[</span>13<span class="cite_crochet">]</span></a></sup> ci-contre). En dressant ce diagramme<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite_crochet">[</span>14<span class="cite_crochet">]</span></a></sup>, après avoir pris acte de l'impossibilité de mesurer la valeur que chaque homme accorde à la vitesse de ses déplacements, Karman et Gabrielli ont posé les fondations d'un système de mesure de l'économie des déplacements (de marchandise ou d'humains), ce système de mesure demeurant valide plus de 70 ans après sa création<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite_crochet">[</span>15<span class="cite_crochet">]</span></a></sup>. </p><p>Pour un vélo, par exemple, dont le coefficient de <a href="/wiki/R%C3%A9sistance_au_roulement#Coefficient_de_résistance_au_roulement" title="Résistance au roulement">résistance au roulement</a> va de 0,0022 à 0,005, la finesse à basse vitesse ira de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{0,0022}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>0</mn> <mo>,</mo> <mn>0022</mn> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{0,0022}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb779eaddbb4dd1e22022977e5454dd9c6e0b026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:5.403ex; height:3.843ex;" alt="{\displaystyle \textstyle {\frac {1}{0,0022}}}"></span> (soit 454) à 200 (si la traînée aérodynamique est négligée). Autre exemple : Pour une berline la traînée est la somme de sa <a href="/wiki/A%C3%A9rodynamique_automobile" title="Aérodynamique automobile">traînée aérodynamique</a> et de sa <a href="/wiki/R%C3%A9sistance_au_roulement" title="Résistance au roulement">résistance au roulement</a>)<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite_crochet">[</span>16<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite_crochet">[</span>17<span class="cite_crochet">]</span></a></sup>. Le coefficient de <a href="/wiki/R%C3%A9sistance_au_roulement#Coefficient_de_résistance_au_roulement" title="Résistance au roulement">résistance au roulement</a> des meilleurs pneumatiques pour berlines s'abaisse jusqu'à 0,006. La finesse d'une telle berline en ville est donc plus faible<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite_crochet">[</span>18<span class="cite_crochet">]</span></a></sup> que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{0,006}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>0</mn> <mo>,</mo> <mn>006</mn> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{0,006}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/800fe1309cfdf7c43a4de41f6253adedbe3d6a47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:4.581ex; height:3.843ex;" alt="{\displaystyle \textstyle {\frac {1}{0,006}}}"></span>, soit 166. Cependant, il suffit de pousser un tel véhicule pour constater que, malgré cette excellente finesse, la traînée de roulement est très forte (donc la perte d'énergie par roulement très forte également). Cela suffit à suggérer que la finesse ne soit plus définie comme le quotient du poids du véhicule sur sa force de traînée mais comme le quotient du poids de ses passagers sur la force de traînée que le déplacement suscite (la traînée du véhicule), soit pour deux passagers (200 <abbr class="abbr" title="kilogramme">kg</abbr> avec les bagages) dans l'exemple ci-dessus (c.-à-d. à basse vitesse) une finesse de simplement 33,3 (et 16,7 pour le conducteur seul)<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite_crochet">[</span>19<span class="cite_crochet">]</span></a></sup>.<br /> Il manque donc au travail de collecte de données par Gabrielli et von Karman une évaluation efficace de l'énergie nécessaire pour déplacer le véhicule lui-même et de l'énergie nécessaire pour déplacer la charge utile<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite_crochet">[</span>20<span class="cite_crochet">]</span></a></sup>. En effet, les deux auteurs n’ont pu recueillir la charge utile ni la vitesse de croisière des véhicules étudiés<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite_crochet">[</span>21<span class="cite_crochet">]</span></a></sup>. De fait, ce graphe n’accorde pas d’avantage à l’emport accru de fret ou de passagers en ceci qu’un véhicule mal conçu dont la structure serait 1 000 <abbr class="abbr" title="kilogramme">kg</abbr> trop lourde et qui, pour compenser ce surpoids, emporterait 10 passagers de moins (avec leur bagages) aurait la même finesse généralisée sur le graphe ci-contre qu’un véhicule mieux conçu et emportant 10 passagers de plus (sur ce point, le <a href="https://commons.wikimedia.org/wiki/File:Finesse_commerciale_%C3%A0_vitesse_maximale_des_diff%C3%A9rents_modes_de_transport,_d%27apr%C3%A8s_Akagi_1991_%26_Papanikolaou_2005.png" class="extiw" title="commons:File:Finesse commerciale à vitesse maximale des différents modes de transport, d'après Akagi 1991 & Papanikolaou 2005.png">diagramme de la <i>finesse commerciale</i></a>, d’après Papanikolaou<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite_crochet">[</span>22<span class="cite_crochet">]</span></a></sup>, pourrait constituer un progrès). </p> <div class="mw-heading mw-heading2"><h2 id="Articles_connexes">Articles connexes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=18" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=18" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Effort_sur_une_voile" title="Effort sur une voile">Effort sur une voile</a></li> <li><a href="/wiki/Vitesse_de_vol_optimale_en_vol_%C3%A0_voile" title="Vitesse de vol optimale en vol à voile">Vitesse de vol optimale en vol à voile</a></li> <li><a href="/wiki/Diagramme_de_Gabrielli_%E2%80%93_von_K%C3%A1rm%C3%A1n" class="mw-redirect" title="Diagramme de Gabrielli – von Kármán">Diagramme de Gabrielli – von Kármán</a></li> <li><a href="/wiki/Efficacit%C3%A9_%C3%A9nerg%C3%A9tique_dans_les_transports" title="Efficacité énergétique dans les transports">Efficacité énergétique dans les transports</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=19" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=19" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=20" title="Modifier la section : Notes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=20" title="Modifier le code source de la section : Notes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-9"><span class="mw-cite-backlink noprint"><a href="#cite_ref-9">↑</a> </span><span class="reference-text">La composante sur trajectoire de son poids est orientée vers l'avant.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink noprint"><a href="#cite_ref-10">↑</a> </span><span class="reference-text">Lorsque les ascendances (mouvements verticaux ascendants de l'air environnant) sont moins fortes.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink noprint"><a href="#cite_ref-13">↑</a> </span><span class="reference-text">La polaire des vitesses est une <a href="/wiki/Courbe_alg%C3%A9brique" title="Courbe algébrique">courbe algébrique</a> de degré 4 qui est rationnelle. Dans le monde de l'aéronautique, une telle courbe est souvent appelée une <a href="/wiki/Parabole" title="Parabole">courbe parabolique</a><sup id="cite_ref-Irving18_11-1" class="reference"><a href="#cite_note-Irving18-11"><span class="cite_crochet">[</span>9<span class="cite_crochet">]</span></a></sup> (qui est une <a href="/wiki/Conique" title="Conique">conique</a>), ce qui est faux car une parabole n'a pas d'<a href="/wiki/Asymptote" title="Asymptote">asymptote</a> verticale contrairement à cette courbe en <i>v</i> = 0. <a href="/wiki/Helmut_Reichmann" title="Helmut Reichmann">Helmut Reichmann</a> a fait la même erreur en supposant que la polaire des vitesses était une parabole<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite_crochet">[</span>10<span class="cite_crochet">]</span></a></sup>.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading3"><h3 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=21" title="Modifier la section : Références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=21" title="Modifier le code source de la section : Références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Filippone"><span class="ouvrage" id="Antonio_Filippone"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Antonio Filippone, « <a rel="nofollow" class="external text" href="https://web.archive.org/web/http://aerodyn.org/HighLift/ld-tables.html"><cite style="font-style:normal;" lang="en">Advanced topics in aerodynamics - Lift-to-Drag Ratios</cite></a> »</span></span>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Orlebar1997"><span class="ouvrage" id="Christopher_Orlebar1997"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Christopher Orlebar, « <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=bs9EbQ6pdRQC&lpg=PA116&ots=v--JjgYJbn&dq=Concorde%20L%2FD%20ratio&hl=fr&pg=PA116#v=onepage&q&f=false"><cite style="font-style:normal;" lang="en"><i>The Concorde Story</i></cite></a> », Osprey Publishing, <time>1997</time></span></span>, <abbr class="abbr" title="page">p.</abbr> 116.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="#cite_ref-3">↑</a> </span><span class="reference-text"><span class="ouvrage" id="2013">« <a rel="nofollow" class="external text" href="http://www.aircross.eu/net/2537/?lang=fr"><cite style="font-style:normal;">La U-6 fait le plus long vol plané en compétition de finesse 2013</cite></a> », <i>AirCross</i>, <time class="nowrap" datetime="2013-03-06" data-sort-value="2013-03-06">6 mars 2013</time></span>.</span> </li> <li id="cite_note-cumulus-soaring_polars.htm-4"><span class="mw-cite-backlink noprint"><a href="#cite_ref-cumulus-soaring_polars.htm_4-0">↑</a> </span><span class="reference-text"><a rel="nofollow" class="external text" href="http://www.cumulus-soaring.com/polars.htm">Cumulus Soaring Polar Data</a>.</span> </li> <li id="cite_note-ASW28-18_polar-5"><span class="mw-cite-backlink noprint"><a href="#cite_ref-ASW28-18_polar_5-0">↑</a> </span><span class="reference-text"><a rel="nofollow" class="external text" href="http://www.alexander-schleicher.de/englisch/produkte/asw28-18/e_asw28-18_polare.htm">AWS28-18 polars</a>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink noprint"><a href="#cite_ref-6">↑</a> </span><span class="reference-text"><span class="ouvrage" id="2008"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a rel="nofollow" class="external text" href="http://www.dept.aoe.vt.edu/~mason/Mason_f/HPAFinalRptS08.pdf"><cite style="font-style:normal;" lang="en">« Human Powered Aircraft for sport »</cite></a>, <i>Virginia Tech</i>, <time class="nowrap" datetime="2008-05-05" data-sort-value="2008-05-05">5 mai 2008</time></span>, <abbr class="abbr" title="page">p.</abbr> 12.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink noprint"><a href="#cite_ref-7">↑</a> </span><span class="reference-text"><a href="#Irving">Paths of Soaring Flight</a>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink noprint"><a href="#cite_ref-8">↑</a> </span><span class="reference-text"><a href="#Irving">Paths of Soaring Flight</a>, <abbr class="abbr" title="page(s)">p.</abbr> 19.</span> </li> <li id="cite_note-Irving18-11"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-Irving18_11-0">a</a> et <a href="#cite_ref-Irving18_11-1">b</a></sup> </span><span class="reference-text"><a href="#Irving">Paths of Soaring Flight</a>, <abbr class="abbr" title="page(s)">p.</abbr> 18.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink noprint"><a href="#cite_ref-12">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Reichmann1993"><span class="ouvrage" id="Helmut_Reichmann1993"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Helmut Reichmann, <cite class="italique" lang="en">Cross-country soaring</cite>, 7, <time>1993</time>, 172 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/1-883813-01-8" title="Spécial:Ouvrages de référence/1-883813-01-8"><span class="nowrap">1-883813-01-8</span></a>)</small>, <abbr class="abbr" title="page">p.</abbr> 123<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cross-country+soaring&rft.pub=7&rft.aulast=Reichmann&rft.aufirst=Helmut&rft.date=1993&rft.pages=123&rft.tpages=172&rft.isbn=1-883813-01-8&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFinesse+%28a%C3%A9rodynamique%29"></span></span></span>.</span> </li> <li id="cite_note-Irving20-14"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-Irving20_14-0">a</a> et <a href="#cite_ref-Irving20_14-1">b</a></sup> </span><span class="reference-text"><a href="#Irving">Paths of Soaring Flight</a>, <abbr class="abbr" title="page(s)">p.</abbr> 20.</span> </li> <li id="cite_note-ASW27-15"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-ASW27_15-0">a</a> et <a href="#cite_ref-ASW27_15-1">b</a></sup> </span><span class="reference-text"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <span class="ouvrage">« <a rel="nofollow" class="external text" href="http://www.alexander-schleicher.de/service/prospekte/AS%20Einzelprosp%2027B%20e%2003%2007.pdf"><cite style="font-style:normal;">ASW 27 B</cite></a> »</span>.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink noprint"><a href="#cite_ref-16">↑</a> </span><span class="reference-text"> Gabrielli, G., von Kármán, Th: What price speed? Mechanical Engineering, 72, 775–781 (1950)</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink noprint"><a href="#cite_ref-17">↑</a> </span><span class="reference-text">L’intitulé de ce diagramme est souvent abrégé en « Diagramme de G-K ».</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink noprint"><a href="#cite_ref-18">↑</a> </span><span class="reference-text">LOCOMOTION: DEALING WITH FRICTION, V. RADHAKRISHNAN, Raman Research Institute, Bangalore, India, 1998 <a rel="nofollow" class="external autonumber" href="https://europepmc.org/backend/ptpmcrender.fcgi?accid=PMC20397&blobtype=pdf">[1]</a></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink noprint"><a href="#cite_ref-19">↑</a> </span><span class="reference-text">En palier (et à vitesse stabilisée), on peut écrire que la force propulsive vaut <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=Mg\,C_{rr}+(1/2)\rho V^{2}SC_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>M</mi> <mi>g</mi> <mspace width="thinmathspace" /> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=Mg\,C_{rr}+(1/2)\rho V^{2}SC_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad9f3846b615ffef0fdf4a3a7f19ff4f1dd4f9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.805ex; height:3.176ex;" alt="{\displaystyle F=Mg\,C_{rr}+(1/2)\rho V^{2}SC_{x}}"></span>.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink noprint"><a href="#cite_ref-20">↑</a> </span><span class="reference-text"><a href="#BarreauBoutin2009">Barreau Boutin 2009</a>, <abbr class="abbr" title="page(s)">p.</abbr> 8</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink noprint"><a href="#cite_ref-21">↑</a> </span><span class="reference-text">du fait de la traînée aérodynamique qui viendra abaisser progressivement ce chiffre dès les 20 ou <span title="8,333 34 m/s" style="cursor:help">30</span> <abbr class="abbr" title="kilomètre par heure">km/h</abbr>.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink noprint"><a href="#cite_ref-22">↑</a> </span><span class="reference-text">Avec cette définition de la finesse, plus le véhicule est lourd et plus sa finesse se dégrade, ce qui correspond bien aux impératifs climatiques actuels.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink noprint"><a href="#cite_ref-23">↑</a> </span><span class="reference-text">WHAT PRICE OF SPEED? A CRITICAL REVISION THROUGH CONSTRUCTAL OPTIMIZATION OF TRANSPORT MODES, Michele TRANCOSSI, <a rel="nofollow" class="external autonumber" href="https://www.researchgate.net/profile/Michele_Trancossi/publication/272365584_What_price_of_speed_A_critical_revision_through_constructal_optimization_of_transport_modes/links/55d4453f08aec1b0429f4e7d/What-price-of-speed-A-critical-revision-through-constructal-optimization-of-transport-modes.pdf">[2]</a></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink noprint"><a href="#cite_ref-24">↑</a> </span><span class="reference-text">“exact information regarding the useful load of vehicles was not available to the authors.” <a rel="nofollow" class="external autonumber" href="http://www.neodymics.com/Images/EPPaper080229E.pdf">[3]</a></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink noprint"><a href="#cite_ref-25">↑</a> </span><span class="reference-text"><a rel="nofollow" class="external text" href="https://books.google.fr/books?id=WACKBAAAQBAJ&printsec=frontcover&hl=fr&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false">SHIP DESIGN : METHODOLOGIES OF PRELIMINARY DESIGN, d'Apostolos Papanikolaou</a></span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Bibliographie">Bibliographie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&veaction=edit&section=22" title="Modifier la section : Bibliographie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Finesse_(a%C3%A9rodynamique)&action=edit&section=22" title="Modifier le code source de la section : Bibliographie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="ouvrage" id="Irving"><small>[Paths of Soaring Flight]</small> <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Frank Irving, <cite class="italique" lang="en">The Paths of Soaring Flight</cite>, <a href="/wiki/Imperial_College_Press" title="Imperial College Press">Imperial College Press</a>, <time>1999</time>, 133 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-1-86094-055-2" title="Spécial:Ouvrages de référence/978-1-86094-055-2"><span class="nowrap">978-1-86094-055-2</span></a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Paths+of+Soaring+Flight&rft.pub=Imperial+College+Press&rft.au=Frank+Irving&rft.date=1999&rft.tpages=133&rft.isbn=978-1-86094-055-2&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFinesse+%28a%C3%A9rodynamique%29"></span></span></li> <li><span class="ouvrage" id="BarreauBoutin2009">Matthieu Barreau et Laurent Boutin, <cite class="italique">Réflexions sur l’énergétique des véhicules routiers</cite>, Paris, <time class="nowrap" datetime="2009-05" data-sort-value="2009-05">mai 2009</time>, 50 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="http://inter.action.free.fr/publications/auto-eco/AUTO-ECO.pdf">lire en ligne</a> <abbr class="abbr indicateur-format format-pdf" title="Document au format Portable Document Format (PDF) d'Adobe">[PDF]</abbr>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=R%C3%A9flexions+sur+l%E2%80%99%C3%A9nerg%C3%A9tique+des+v%C3%A9hicules+routiers&rft.place=Paris&rft.au=Matthieu+Barreau&rft.au=Laurent+Boutin&rft.date=2009&rft.tpages=50&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFinesse+%28a%C3%A9rodynamique%29"></span></span>.</li></ul> <div class="navbox-container" style="clear:both;"> <table class="navbox collapsible noprint autocollapse" style=""> <tbody><tr><th class="navbox-title" colspan="2" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Avion" title="Modèle:Palette Avion"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Avion&action=edit"><abbr class="abbr" title="Modifier ce modèle. Merci de prévisualiser avant de sauvegarder.">m</abbr></a></div></div><div style="font-size:110%"><a href="/wiki/Avion" title="Avion">Avion</a></div></th> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Configuration_g%C3%A9n%C3%A9rale_d%27un_avion" title="Configuration générale d'un avion">Composants principaux</a></th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Cellule_(a%C3%A9ronautique)" title="Cellule (aéronautique)">Cellule</a> <ul><li><a href="/wiki/Aile_(a%C3%A9ronautique)" title="Aile (aéronautique)">aile</a></li> <li><a href="/wiki/Fuselage" title="Fuselage">fuselage</a></li> <li><a href="/wiki/Nacelle_(a%C3%A9ronautique)" title="Nacelle (aéronautique)">nacelle</a></li> <li><a href="/wiki/Empennage" title="Empennage">empennage</a></li> <li><a href="/wiki/Train_d%27atterrissage" title="Train d'atterrissage">train d'atterrissage</a></li></ul></li> <li><a href="/wiki/Caisson_de_voilure" title="Caisson de voilure">Caisson de voilure</a> <ul><li><a href="/wiki/Longeron_(a%C3%A9ronautique)" title="Longeron (aéronautique)">Longeron</a></li> <li><a href="/wiki/Longeron_de_voilure" title="Longeron de voilure">Longeron de voilure</a></li> <li><a href="/wiki/R%C3%A9servoir_structurel" title="Réservoir structurel">Réservoir structurel</a></li></ul></li> <li><a href="/wiki/Groupe_motopropulseur" title="Groupe motopropulseur">Groupe motopropulseur</a> <ul><li><a href="/wiki/H%C3%A9lice_(a%C3%A9ronautique)" title="Hélice (aéronautique)">hélice</a></li> <li><a href="/wiki/Turbopropulseur" title="Turbopropulseur">turbopropulseur</a></li> <li><a href="/wiki/Turbor%C3%A9acteur" title="Turboréacteur">turboréacteur</a></li> <li><a href="/wiki/Stator%C3%A9acteur" title="Statoréacteur">statoréacteur</a></li></ul></li> <li><a href="/wiki/Poste_de_pilotage" title="Poste de pilotage">Poste de pilotage</a> <ul><li><a href="/wiki/Manche_(a%C3%A9ronautique)" title="Manche (aéronautique)">manche à balai</a></li> <li><a href="/wiki/Pilote_automatique" title="Pilote automatique">pilote automatique</a></li> <li><a href="/wiki/Affichage_t%C3%AAte_haute" title="Affichage tête haute">affichage tête haute</a></li></ul></li> <li><a href="/wiki/Servitude_de_bord" title="Servitude de bord">servitude de bord</a></li> <li><a href="/wiki/Instrument_de_bord_(a%C3%A9ronautique)" title="Instrument de bord (aéronautique)">Avionique</a> <ul><li><a href="/wiki/Radiocommunication_a%C3%A9ronautique" title="Radiocommunication aéronautique">communication</a></li> <li><a href="/wiki/Radionavigation" title="Radionavigation">navigation</a></li> <li><a href="/wiki/Radar" title="Radar">radar</a></li></ul></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/A%C3%A9rodynamique" title="Aérodynamique">Aérodynamique</a></th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/A%C3%A9rofrein" title="Aérofrein">Aérofrein</a></li> <li><a href="/wiki/Bandes_de_d%C3%A9crochage" title="Bandes de décrochage">Bandes de décrochage</a></li> <li><a href="/wiki/Bord_de_fuite" title="Bord de fuite">Bord de fuite</a></li> <li><a href="/wiki/Couche_limite" title="Couche limite">Couche limite</a></li> <li><a href="/wiki/Dispositif_hypersustentateur" title="Dispositif hypersustentateur">Dispositif hypersustentateur</a></li> <li><a href="/wiki/Effet_de_sol" title="Effet de sol">Effet de sol</a></li> <li><a class="mw-selflink selflink">Finesse</a></li> <li><a href="/wiki/G%C3%A9n%C3%A9rateur_de_tourbillons" title="Générateur de tourbillons">Générateur de tourbillons</a></li> <li><a href="/wiki/Instabilit%C3%A9_de_Crow" title="Instabilité de Crow">Instabilité de Crow</a></li> <li><a href="/wiki/Portance_(a%C3%A9rodynamique)" title="Portance (aérodynamique)">Portance</a></li> <li><a href="/wiki/Profil_(a%C3%A9rodynamique)" title="Profil (aérodynamique)">Profil</a></li> <li><a href="/wiki/Saumon_(a%C3%A9ronautique)" title="Saumon (aéronautique)">Saumon</a></li> <li><a href="/wiki/Tourbillon_marginal" title="Tourbillon marginal">Tourbillon marginal</a></li> <li><a href="/wiki/Tra%C3%AEn%C3%A9e" title="Traînée">Traînée</a></li> <li><a href="/wiki/Turbulence_de_sillage" title="Turbulence de sillage">Turbulence de sillage</a></li> <li><a href="/wiki/Winglet" title="Winglet">Winglet (<i>Ailerette</i>)</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/M%C3%A9canique_du_vol" title="Mécanique du vol">Mécanique du vol</a></th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Roulis" title="Roulis">Roulis</a></li> <li><a href="/wiki/Tangage" title="Tangage">Tangage</a></li> <li><a href="/wiki/Lacet_(mouvement)" title="Lacet (mouvement)">Lacet</a></li> <li><a href="/wiki/Stabilit%C3%A9_longitudinale_d%27un_avion" title="Stabilité longitudinale d'un avion">Stabilité longitudinale</a></li> <li><a href="/wiki/Commande_de_vol" title="Commande de vol">Commande de vol</a> <ul><li><a href="/wiki/Gouverne" title="Gouverne">gouverne</a></li> <li><a href="/wiki/Aileron_(a%C3%A9ronautique)" title="Aileron (aéronautique)">aileron</a></li> <li><a href="/wiki/%C3%89levon" title="Élevon">élevon</a></li> <li><a href="/wiki/Aileron_haute-vitesse" title="Aileron haute-vitesse">flaperon</a></li></ul></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Pilotage_d%27un_avion" title="Pilotage d'un avion">Pilotage</a></th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/D%C3%A9collage" title="Décollage">Décollage</a> (<a href="/wiki/Catapulte_(a%C3%A9ronautique)" title="Catapulte (aéronautique)">Catapultage</a>)</li> <li><a href="/wiki/Croisi%C3%A8re_(pilotage)" title="Croisière (pilotage)">Croisière</a></li> <li><a href="/wiki/Atterrissage" title="Atterrissage">Atterrissage</a> <ul><li><a href="/wiki/Amerrissage" title="Amerrissage">Amerrissage</a></li> <li><a href="/wiki/Appontage" title="Appontage">Appontage</a></li></ul></li> <li><a href="/wiki/Contr%C3%B4le_de_la_circulation_a%C3%A9rienne" title="Contrôle de la circulation aérienne">Contrôle aérien</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Configuration_g%C3%A9n%C3%A9rale_d%27un_avion#Configuration_liée_au_type_d'avion" title="Configuration générale d'un avion">Type d'avion</a></th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Avion_de_ligne" title="Avion de ligne">Ligne</a></li> <li><a href="/wiki/Configuration_g%C3%A9n%C3%A9rale_d%27un_avion#Avion_de_transport_de_fret" title="Configuration générale d'un avion">Fret</a></li> <li><a href="/wiki/Configuration_g%C3%A9n%C3%A9rale_d%27un_avion#Avion_de_transport_régional" title="Configuration générale d'un avion">Régional</a></li> <li><a href="/wiki/Aviation_d%27affaires" title="Aviation d'affaires">Affaires</a></li> <li><a href="/wiki/Configuration_g%C3%A9n%C3%A9rale_d%27un_avion#Avion_de_tourisme" title="Configuration générale d'un avion">Tourisme</a></li> <li><a href="/wiki/Avion_militaire" title="Avion militaire">Militaire</a> <ul><li><a href="/wiki/Avion_de_chasse" title="Avion de chasse">chasse</a></li> <li><a href="/wiki/Avion_de_reconnaissance" title="Avion de reconnaissance">reconnaissance</a></li> <li><a href="/wiki/Avion_de_transport" title="Avion de transport">transport</a></li></ul></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="">Catégorie</th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Avion_%C3%A0_r%C3%A9action" title="Avion à réaction">Avion à réaction</a></li> <li><a href="/wiki/Vol_suborbital" title="Vol suborbital">Avion suborbital</a></li> <li><a href="/wiki/Avion_%C3%A0_d%C3%A9collage_et_atterrissage_court" title="Avion à décollage et atterrissage court">ADAC</a></li> <li><a href="/wiki/Avion_%C3%A0_d%C3%A9collage_et_atterrissage_verticaux" title="Avion à décollage et atterrissage verticaux">ADAV</a></li> <li><a href="/wiki/Avion_%C3%A0_effet_de_sol" title="Avion à effet de sol">Avion à effet de sol</a></li> <li><a href="/wiki/Avion_%C3%A0_propulsion_nucl%C3%A9aire" title="Avion à propulsion nucléaire">Avion à propulsion nucléaire</a></li> <li><a href="/wiki/Avion_%C3%A9lectrique" title="Avion électrique">Avion électrique</a></li> <li><a href="/wiki/Avion_%C3%A0_hydrog%C3%A8ne" title="Avion à hydrogène">Avion à hydrogène</a></li> <li><a href="/wiki/Avion_solaire" title="Avion solaire">Avion solaire</a></li> <li><a href="/wiki/Hydravion" title="Hydravion">Hydravion</a></li> <li><a href="/wiki/A%C3%A9ronef_amphibie" title="Aéronef amphibie">Avion amphibie</a></li> <li><a href="/wiki/Planeur" title="Planeur">Planeur</a></li> <li><a href="/wiki/Drone" title="Drone">Drone</a></li> <li><a href="/wiki/A%C3%A9ronef_ultral%C3%A9ger_motoris%C3%A9" title="Aéronef ultraléger motorisé">ULM</a></li> <li><a href="/wiki/Aile_volante" title="Aile volante">Aile volante</a></li> <li><a href="/wiki/Corps_portant" title="Corps portant">Corps portant</a></li> <li><a href="/wiki/Mod%C3%A9lisme_a%C3%A9rien" title="Modélisme aérien">Modèle réduit</a></li></ul> </div></td> </tr> </tbody></table> </div> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer skin-invert-image" typeof="mw:File"><a href="/wiki/Portail:A%C3%A9ronautique" title="Portail de l’aéronautique"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Avion_silhouette.svg/22px-Avion_silhouette.svg.png" decoding="async" width="22" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Avion_silhouette.svg/33px-Avion_silhouette.svg.png 1.5x, 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