{"id":68446,"date":"2023-07-14T14:26:07","date_gmt":"2023-07-14T17:26:07","guid":{"rendered":"https:\/\/militares.estrategia.com\/portal\/?p=68446"},"modified":"2023-08-08T17:52:43","modified_gmt":"2023-08-08T20:52:43","slug":"cinematica-vetorial","status":"publish","type":"post","link":"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/cinematica-vetorial\/","title":{"rendered":"Cinem\u00e1tica Vetorial"},"content":{"rendered":"\n<p>Vai participar dos concursos e quer saber um pouquinho mais de <a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/\" target=\"_blank\" >f\u00edsica<\/a>? Vem com a gente! Confira esse artigo que o Estrat\u00e9gia Militares preparou para voc\u00ea sobre a Cinem\u00e1tica Vetorial.<\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_79_2 counter-hierarchy ez-toc-counter ez-toc-transparent ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">Navegue pelo conte\u00fado\t<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/cinematica-vetorial\/#Cinematica-Vetorial-Vetor-Deslocamento\" >Cinem\u00e1tica Vetorial: Vetor Deslocamento<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/cinematica-vetorial\/#Cinematica-Vetorial-Velocidade-Vetorial-Media\" >Cinem\u00e1tica Vetorial: Velocidade Vetorial M\u00e9dia<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/cinematica-vetorial\/#Cinematica-Vetorial-Velocidade-Vetorial-Instantanea\" >Cinem\u00e1tica Vetorial: Velocidade Vetorial Instant\u00e2nea<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/cinematica-vetorial\/#A-Velocidade-Vetorial-Em-Movimentos-Particulares\" >A Velocidade Vetorial Em Movimentos Particulares<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/cinematica-vetorial\/#Aceleracao-Vetorial-Media\" >Acelera\u00e7\u00e3o Vetorial M\u00e9dia<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/cinematica-vetorial\/#Aceleracao-Vetorial-Instantanea\" >Acelera\u00e7\u00e3o Vetorial Instant\u00e2nea<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/cinematica-vetorial\/#Veja-tambem\" >Veja tamb\u00e9m:<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-cinematica-vetorial-vetor-deslocamento\"><span class=\"ez-toc-section\" id=\"Cinematica-Vetorial-Vetor-Deslocamento\"><\/span>Cinem\u00e1tica Vetorial: Vetor Deslocamento<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p><strong>Vetor posi\u00e7\u00e3o:<\/strong> Se uma part\u00edcula descreve uma trajet\u00f3ria qualquer no espa\u00e7o, sua localiza\u00e7\u00e3o \u00e9 definida pelo vetor posi\u00e7\u00e3o. Define-se vetor posi\u00e7\u00e3o (\ud835\udc60\u20d7) de um ponto \ud835\udc43 em rela\u00e7\u00e3o a um referencial \ud835\udc42, o vetor que liga a origem em \ud835\udc42 \u00e0 extremidade em \ud835\udc43.&nbsp;<\/p>\n\n\n\n<p>Para facilitar as contas e entendimento, escolhemos um sistema de coordenadas triortogonal \ud835\udc42\ud835\udc65\ud835\udc66\ud835\udc67, onde o referencial est\u00e1 em \ud835\udc42.&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/xtrBN9K2KzSWNlEeCfyHJwCE_ODevEa7Vc5dTA0sloBJllQLtlUeBtNnUP8QoAAoVS6f4oEXoZPIynpqCqGn9iuhY4BTHjBGBW9TpA-d3Smem55uWX3Islo9eEepMoB3uE3CiHXLPtXBuiCjGOmt1_A\" alt=\"\" width=\"473\" height=\"366\"\/><\/figure><\/div>\n\n\n<p>Dessa forma, podemos escrever o vetor \ud835\udc60\u20d7 das seguintes formas:<br><\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"has-text-align-center\">\ud835\udc60\u20d7 = \ud835\udc60\ud835\udc65. \ud835\udc56\u0302+ \ud835\udc60\ud835\udc66\ud835\udc57\u0302+ \ud835\udc60\ud835\udc67\ud835\udc58\u0302 = (\ud835\udc60\ud835\udc65, \ud835\udc60\ud835\udc66, \ud835\udc60\ud835\udc67)<\/p>\n\n\n\n<p class=\"has-text-align-center\">\u20d7 = \ud835\udc60\u20d7\u20d7\u20d7\ud835\udc65\u20d7 + \ud835\udc60\u20d7\u20d7\u20d7\ud835\udc66\u20d7 + \ud835\udc60\ud835\udc67 \u20d7\u20d7<\/p>\n<\/div><\/div>\n\n\n\n<p><br>De acordo com a geometria dos vetores, vale a rela\u00e7\u00e3o entre os m\u00f3dulos:<br><\/p>\n<\/blockquote>\n\n\n\n<p class=\"has-text-align-center\">&nbsp;\ud835\udc60 2 = \ud835\udc60\ud835\udc65 2 + \ud835\udc60\ud835\udc66 2 + \ud835\udc60\ud835\udc67 2&nbsp;<\/p>\n\n\n\n<p><strong>Vetor deslocamento:<\/strong> Uma part\u00edcula se desloca ao longo de uma trajet\u00f3ria curvil\u00ednea no espa\u00e7o. Ela passa por A no instante \ud835\udc61<sub>1<\/sub> e por B no instante \ud835\udc61<sub>2<\/sub> . O vetor posi\u00e7\u00e3o do ponto A \u00e9 \ud835\udc5d1 \u20d7\u20d7\u20d7\u20d7 e do ponto B \u00e9 \ud835\udc5d2 \u20d7 \u20d7. Assim, podemos representar o deslocamento de A a B e o vetor deslocamento da seguinte forma:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/lfZbhR4-F9lMhtjC2WrYpgNdXF-EQ2lX3vFiQPBdTwxy-H4rkEfvdzwDMc_9pxcY6oHbffZds34y4qig4nAbeGKjz8vVItJV0EkhMFNeiJFgvnKvAbaGR0Y8yHAx8vl5cWzowv_3TL1-Dlme7YM3X0o\" alt=\"\" width=\"477\" height=\"300\"\/><\/figure><\/div>\n\n\n<p>Como j\u00e1 visto anteriormente, escrevemos o deslocamento do m\u00f3vel da seguinte forma:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\u0394\ud835\udc60 = \ud835\udc60<sub>\ud835\udc35<\/sub> \u2212 \ud835\udc60<sub>\ud835\udc34<\/sub><br><br>Na cinem\u00e1tica vetorial definimos a varia\u00e7\u00e3o do vetor posi\u00e7\u00e3o (ou vetor deslocamento) de forma an\u00e1loga:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\u0394\ud835\udc5d\u20d7 = \ud835\udc5d2 \u20d7 \u20d7 \u2212 \ud835\udc5d1<br><br>Da geometria, podemos notar que:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">|\u0394\ud835\udc5d\u20d7| \u2264 |\u0394\ud835\udc60|<\/p>\n\n\n\n<p>&nbsp;A igualdade ocorre quando a trajet\u00f3ria \u00e9 retil\u00ednea:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/zkarSxwYLMloziQq7EZppxEY4uAJMeGuVBnEuKAEx2PwVI3Qk46L0GXVGj9HxQps1ewQ8PBaFlg63LTQ_l7jG9SR0FPfV4Kd-h7QASXLrzw0IrLee0k_F7ynaw-9NKJKH1SbPWmnnYoPzlu3SLBMZv8\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Neste caso, temos que:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">|\u0394\ud835\udc5d\u20d7| = |\u0394\ud835\udc60|&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-cinematica-vetorial-velocidade-vetorial-media\"><span class=\"ez-toc-section\" id=\"Cinematica-Vetorial-Velocidade-Vetorial-Media\"><\/span>Cinem\u00e1tica Vetorial: Velocidade Vetorial M\u00e9dia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Ap\u00f3s definirmos o vetor deslocamento, podemos definir a velocidade vetorial m\u00e9dia, quando o vetor posi\u00e7\u00e3o varia de A para B em um dado intervalo de tempo, por meio da seguinte f\u00f3rmula:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">&nbsp;\ud835\udc63\u20d7\ud835\udc5a=\u0394\ud835\udc5d\u20d7\u0394\ud835\udc61<\/p>\n\n\n\n<p>Devido ao fato do \u0394\ud835\udc61 ser um n\u00famero real e sempre positivo, podemos afirmar que \ud835\udc63\u20d7\ud835\udc5a e \u0394\ud835\udc5d\u20d7 possuem o mesmo sinal.&nbsp;<\/p>\n\n\n\n<p>A partir disso, podemos encontrar quem tem maior m\u00f3dulo entre as velocidades:<br><\/p>\n\n\n\n<p class=\"has-text-align-center\">|\u0394\ud835\udc5d\u20d7| \u2264 |\u0394\ud835\udc60|<\/p>\n\n\n\n<p class=\"has-text-align-left\"><br>Dividindo a equa\u00e7\u00e3o por \u0394\ud835\udc61, lembrando que \u0394\ud835\udc61 \u00e9 sempre positivo, vemos:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">|\u0394\ud835\udc5d\u20d7| \u0394\ud835\udc61 \u2264 |\u0394\ud835\udc60| \u0394\ud835\udc61&nbsp;<\/p>\n\n\n\n<p>Portanto:<\/p>\n\n\n\n<p class=\"has-text-align-center\">&nbsp;|\ud835\udc63\u20d7\ud835\udc5a| \u2264 |\ud835\udc63\ud835\udc5a|&nbsp;<\/p>\n\n\n\n<p>Isso mostra que: \u201co m\u00f3dulo da velocidade vetorial m\u00e9dia \u00e9 sempre menor ou igual ao m\u00f3dulo da velocidade escalar m\u00e9dia.\u201d<\/p>\n\n\n\n<p>Notamos que, quando a trajet\u00f3ria \u00e9 retil\u00ednea, |\u0394\ud835\udc5d\u20d7| = |\u0394\ud835\udc60|, o m\u00f3dulo da velocidade vetorial m\u00e9dia (\ud835\udc63\u20d7\ud835\udc5a) \u00e9 igual ao m\u00f3dulo da velocidade escalar m\u00e9dia.&nbsp;<\/p>\n\n\n\n<p>Um corpo move-se com velocidade escalar constante descrevendo uma trajet\u00f3ria circular de raio 30 m, levando 18 segundos para completar uma volta. Em um intervalo de<br>\u0394\ud835\udc61 = 3,0 \ud835\udc60, determine os m\u00f3dulos:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>da varia\u00e7\u00e3o do espa\u00e7o do m\u00f3vel;&nbsp;<\/li>\n\n\n\n<li>do vetor deslocamento;&nbsp;<\/li>\n\n\n\n<li>da velocidade escalar m\u00e9dia; e<\/li>\n\n\n\n<li>da velocidade vetorial m\u00e9dia.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/3XIllXAAmoeMYK0eN6eFzBHIM1VASz0UUQu8YB6hYPVLHS1yFTlP-2e7ihGgJiFruOjbZOSaZSdqY690NLeLoLIf-766QShAvxGzjXGozQVnZb-Nu7GIJXuOvzn0jVPxcJ7oJn1TE39yPshxUvLttcU\" alt=\"\"\/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-cinematica-vetorial-velocidade-vetorial-instantanea\"><span class=\"ez-toc-section\" id=\"Cinematica-Vetorial-Velocidade-Vetorial-Instantanea\"><\/span>Cinem\u00e1tica Vetorial: Velocidade Vetorial Instant\u00e2nea<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Semelhante \u00e0 velocidade escalar instant\u00e2nea, definimos a velocidade vetorial instant\u00e2nea (\ud835\udc63\u20d7) como:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/vKf7cO155JfoOT7WwAOvtm95AS6vWlW0vEARTI9yEOEjH1JNRlC9-RHoka3Ev0ZxpCfXGY009HyfNBX8ZDfSi90nkrYQIiNnsWK5oWYd5fl2GDFKhUXCMElHHDj4nNvMRbSKs1C6z6TVulC40Y9jLnI\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Dessa forma, para calcular \ud835\udc63\u20d7 quando ela passa por um ponto P, devemos tomar outro ponto Q da trajet\u00f3ria, fazer o deslocamento entre P e Q e fazer Q tender a P.&nbsp;<\/p>\n\n\n\n<p>Assim, percebemos que quando fazemos Q tender a P, a dire\u00e7\u00e3o de \u0394\ud835\udc5d\u20d7 aproxima-se da dire\u00e7\u00e3o da reta tangente \u00e0 trajet\u00f3ria no ponto P. Assim, a dire\u00e7\u00e3o da velocidade vetorial instant\u00e2nea \u00e9 a da reta tangente \u00e0 trajet\u00f3ria no ponto em quest\u00e3o e o sentido ser\u00e1 o mesmo do movimento do m\u00f3vel.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/jvhwWwcNtQXuRR9xbw6Hos-us9KSE972Q2K4Ac7N_xd7ngcYts3dms9PuWckqzTVrQKyGSXaK7uH-lWV3Kd4hQwLK1YA0_--Rx8435N1hQ1SLifUwQOwuMGlUz6hR-2gsIUQVzKMD1N326W4Z3Ohhfs\" alt=\"\"\/><\/figure>\n\n\n\n<p>\u00c9 importante notarmos que quando \u0394\ud835\udc61 \u2192 0 (l\u00ea-se \u201cintervalo de tempo tende \u00e0 zero\u201d), o m\u00f3dulo da varia\u00e7\u00e3o do espa\u00e7o tende ao m\u00f3dulo do vetor deslocamento, ou seja, |\u0394\ud835\udc60| \u2192 |\u0394\ud835\udc5d\u20d7|. Portanto, no limite temos que:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/ZKdWkjeGNnpJ2Ml3nUIrADNiApTZ92Ke0jDmZiD2QuMIAxcxZwMWMXQBncH-kbSVTPjYhxgwyT3cXDblcXOYz-Vwq5NnwffPLFlXXsMsevowrrus9s5kgPd3pugJwh2DPHk6r8X-tRtEf3_-yD3YNWA\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Este resultado mostra que: \u201cO m\u00f3dulo da velocidade vetorial instant\u00e2nea \u00e9 igual ao m\u00f3dulo da velocidade escalar instant\u00e2nea.\u201d<br>&lt;br&gt;N\u00e3o podemos confundir a conclus\u00e3o efetuada anteriormente no caso das velocidades m\u00e9dias <br>(|\ud835\udc63\u20d7\ud835\udc5a| \u2264 |\ud835\udc63\ud835\udc5a|).<\/p>\n\n\n\n<p><strong>As caracter\u00edsticas da velocidade vetorial instant\u00e2nea s\u00e3o:<\/strong><\/p>\n\n\n\n<p>1) Dire\u00e7\u00e3o: a mesma da reta tangente com rela\u00e7\u00e3o \u00e0 trajet\u00f3ria no ponto considerado;<\/p>\n\n\n\n<p>2) Sentido: o mesmo do movimento;<\/p>\n\n\n\n<p>3) Igual ao m\u00f3dulo da velocidade escalar instant\u00e2nea.<\/p>\n\n\n\n<p><strong>Observa\u00e7\u00f5es:<\/strong><\/p>\n\n\n\n<p>1) Em muitos casos, menciona-se velocidade vetorial e n\u00e3o se especifica se ela \u00e9 m\u00e9dia ou instant\u00e2nea. Nesses casos, admite-se ser instant\u00e2nea.&nbsp;<\/p>\n\n\n\n<p>&nbsp;2) Quando apenas diz velocidade, sem mencionar qualquer outra informa\u00e7\u00e3o, admite-se ser a vetorial.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-a-velocidade-vetorial-em-movimentos-particulares\"><span class=\"ez-toc-section\" id=\"A-Velocidade-Vetorial-Em-Movimentos-Particulares\"><\/span>A Velocidade Vetorial Em Movimentos Particulares<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p><strong>Movimento retil\u00edneo uniforme (MRU)<\/strong><\/p>\n\n\n\n<p>Devido ao fato da trajet\u00f3ria ser retil\u00ednea, a dire\u00e7\u00e3o da velocidade vetorial n\u00e3o muda. Assim, \ud835\udc63\u20d7 sempre ter\u00e1 o mesmo m\u00f3dulo e o mesmo sentido.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/zc1utFklpcQvRwD1fG63O4puFZXJ2UF9qE7rNc3OlIrtk4ciKALC_IFvH1h5IaZfDj0b9y4rifSPMsQWi66YLrsSWQQPWJcs03_X7ykj6yOk9ZHiwnhkojfXxtKl4o833lunKpUtlB55-3Fog9Tcu9U\" alt=\"\"\/><\/figure>\n\n\n\n<p>Assim, podemos escrever que:<\/p>\n\n\n\n<p class=\"has-text-align-center\">|\ud835\udc63\u20d71 | = |\ud835\udc63\u20d72 | \u21d2 \ud835\udc63\u20d71 = \ud835\udc63\u20d72<\/p>\n\n\n\n<p>Ent\u00e3o, dizemos que a velocidade vetorial \u00e9 constante no MRU.<\/p>\n\n\n\n<p><strong>Movimento circular e uniforme (MCU)<\/strong><\/p>\n\n\n\n<p>\u00c9 f\u00e1cil notar que a velocidade vetorial nesse tipo de movimento n\u00e3o \u00e9 constante, uma vez que estamos mudando de dire\u00e7\u00e3o a cada instante. Entretanto, apenas o m\u00f3dulo da velocidade \u00e9 constante. Dessa forma, embora o movimento seja uniforme, dizemos que a velocidade vetorial \u00e9 vari\u00e1vel &#8211; pois muda dire\u00e7\u00e3o-, embora o m\u00f3dulo permane\u00e7a constante.&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/PypUBvkXYkgYWfme2Yc23zBOuVkJUBj7EVQoXPf_IrmHtEhyddqtwrDaLU-5QyiA-P-egqvkoHyzCqjzqFRvShc4ZX3HYXLjrxoJfAST3XhmL9_YZ8FVP-JDUxhTvK11SuWZ_CrULkbZQ4Zz8DXJT-4\" alt=\"\"\/><\/figure><\/div>\n\n\n<p class=\"has-text-align-center\">\ud835\udc63\u20d71 | = |\ud835\udc63\u20d72 |, \ud835\udc5a\ud835\udc4e\ud835\udc60 \ud835\udc63\u20d71 \u2260 \ud835\udc63\u20d72<\/p>\n\n\n\n<p><strong>&nbsp;Movimento retil\u00edneo uniformemente variado (MRUV)<\/strong><\/p>\n\n\n\n<p>Devido ao fato da trajet\u00f3ria ser retil\u00ednea, sabemos que a dire\u00e7\u00e3o do movimento n\u00e3o \u00e9 alterada, entretanto, como existe acelera\u00e7\u00e3o, o m\u00f3dulo da velocidade vetorial se altera:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/u_v2ot0ouhIvr-MFuvJmFcJ0qUetiLegRRe148rZtua8nQfuTrsEykMrMMI26wgmyHYNrWJ0sTZZRd_57FK10CbsKiFJTTo6oL4mDiOHp7z8hag4rjguE7Gn4aN0_lc-l-2k6x2pLfDYuRJSW1lYOMU\" alt=\"\"\/><\/figure><\/div>\n\n\n<p class=\"has-text-align-center\">|\ud835\udc63\u20d71 | \u2260 |\ud835\udc63\u20d72 | \u21d2 \ud835\udc63\u20d71 \u2260 \ud835\udc63\u20d72<\/p>\n\n\n\n<p><strong>Movimento circular uniformemente variado<\/strong><\/p>\n\n\n\n<p>Nesse tipo de movimento tanto o m\u00f3dulo quanto a dire\u00e7\u00e3o da velocidade vetorial se alteram.&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/95qrX9PUUObm0zBvWV89y4qCnBf5xTx7QLTHBrgqNI6hEUEKL2I4VvxRFPtcTCazOdhtkEjjg0EiePClpEEE17Fq-IvSXm0knesJQXNmwSIxWdDBeNGOkFLR-PrdudKIqZjEK3pVnQtQs6XMPN3e3HU\" alt=\"\"\/><\/figure><\/div>\n\n\n<p class=\"has-text-align-center\">\ud835\udc63\u20d71 | \u2260 |\ud835\udc63\u20d72 | \u21d2 \ud835\udc63\u20d71 \u2260 \ud835\udc63\u20d72<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-aceleracao-vetorial-media\"><span class=\"ez-toc-section\" id=\"Aceleracao-Vetorial-Media\"><\/span>Acelera\u00e7\u00e3o Vetorial M\u00e9dia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Se um part\u00edcula, realizando uma trajet\u00f3ria qualquer no espa\u00e7o, ao passar por \ud835\udc43<sub>1<\/sub> tem velocidade vetorial \ud835\udc63\u20d7<sub>1<\/sub> e ao passar por \ud835\udc43<sub>2<\/sub> tem velocidade vetorial \ud835\udc63\u20d7<sub>2<\/sub> , ent\u00e3o, a acelera\u00e7\u00e3o vetorial m\u00e9dia \ud835\udc4e\u20d7<sub>\ud835\udc5a<\/sub> entre estes dois instantes \u00e9 definida por:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/9POFKDBE5OF_0BGO2xncNrJYXwyltW9VmrEKjh3KjWO7n9LNwgcDkIbd80rjS4RdwL9nm1S5qaHpa1EfKHhZzuwq7DxIfdLZ4RubonL5qm4igylHecI7AEzByKmZbSdnd9vz2bLlVJun1Mmb7BzTJ_U\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Exerc\u00edcio<\/strong><\/p>\n\n\n\n<p>Uma part\u00edcula descreve um MCUV, tal que no primeiro instante sua velocidade \u00e9 \ud835\udc63\u20d7<sub>1<\/sub> e em um segundo instante sua velocidade \u00e9 \ud835\udc632 \u20d7 \u20d7. Dado que |\ud835\udc63\u20d71 | = 6,0 \ud835\udc5a\/\ud835\udc60 e |\ud835\udc63\u20d72 | = 8 \ud835\udc5a\/\ud835\udc60, como na figura abaixo, determine o m\u00f3dulo da acelera\u00e7\u00e3o vetorial, se a part\u00edcula levou 4 segundos para ir de realizar este percurso. Adote: \ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udf03 = 0,6.&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/JQomt3oh51DZK8iiHnMtxFUEctpFABw3grma-a_E50ZZzNI_eQZkydLeKuaFAPGBU_S5AqSktfot88q4_FY-Nt8sG3XeEbLYrXUF0zTVREQYQKrol7D_65UhS0ti_Ejn8v13MHlz8d3biDQHR2exvo0\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Coment\u00e1rios: <\/strong>A partir da defini\u00e7\u00e3o, devemos encontrar quem \u00e9 o m\u00f3dulo da varia\u00e7\u00e3o da velocidade vetorial:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\"><br>|\u0394\ud835\udc63\u20d7| = |\ud835\udc63\u20d7<sub>2<\/sub> \u2212 \ud835\udc63\u20d7<sub>1<\/sub> |<\/p>\n\n\n\n<p>&nbsp;Geometricamente:<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/X9Wo3A3JZZnLhoe-vYYJzCkujYFDgolJIdUjvRK8ySoSGrh5VwBksrwrRk5mGSNaiVO5WqiUh3eiu_ypufnqI6U3_mZ4O6LmFuXAjpw-yBG_I_0X2kT4YxFAy-qSa-vHOxyhsrhbxpFQgef4u7AteNU\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/IkkLvwZATg8I4J7TpkriPO38t8ReSwbpodHqtujHD45T8Fi6djifAtgEPmrlHTbj3NWrocR30PdvUn06y60VIDWp9sfZa915sWA2fw-RuVCFr_8T0mijluyXj6ebN20ISEP9u_xDEWRcCAhfJO647tk\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>(FEI-SP)<\/strong> A velocidade \ud835\udc63\u20d7 de um m\u00f3vel, em fun\u00e7\u00e3o do tempo, acha-se representada pelo diagrama vetorial da figura. A intensidade da velocidade inicial \u00e9 \ud835\udc63<sub>0<\/sub> = 20 \ud835\udc5a\/\ud835\udc60. Determine o m\u00f3dulo da acelera\u00e7\u00e3o vetorial m\u00e9dia entre os instantes \ud835\udc61 = 0 e \ud835\udc61 = 8 \ud835\udc60.&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/Q4jqttxlHS3atKAmhBZNJtBX-A8wN1XntNIVvVQP7TBLCBRAOEtZdb5lHFGRVRtETLMtC42VI8-nq0xUC0RZEJQn7n9vxS8t1fuYoJdTxwLQdwQXgVMDqOgJX9geYZVTPjg5TXlvbhdyFXT2KFWRnSo\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Coment\u00e1rios<\/strong>: Novamente, temos que determinar o m\u00f3dulo da varia\u00e7\u00e3o da velocidade vetorial entre os instantes \ud835\udc61 = 0 e \ud835\udc61 = 8 \ud835\udc60:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/0mjHhe5R22ytVlc3SURQjxrsXCO2u-iqyucAPIuk-PZ1mL_UMpaX9YTPl2e3D9dbCRnw3atHrNI4WnkEW9F5itEtNbLzuusO8n76qYsWtRkCSx1jugIV374yBhqsTICmDYoxxG4cywHRZ5_4m7v0eKY\" alt=\"\"\/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-aceleracao-vetorial-instantanea\"><span class=\"ez-toc-section\" id=\"Aceleracao-Vetorial-Instantanea\"><\/span>Acelera\u00e7\u00e3o Vetorial Instant\u00e2nea<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Relembrando da cinem\u00e1tica escalar definimos a acelera\u00e7\u00e3o escalar instant\u00e2nea (\ud835\udc4e) como o<\/p>\n\n\n\n<p>limite da varia\u00e7\u00e3o da velocidade escalar quando o tempo tende a zero, isto \u00e9:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/mojY0j5Vss2Gx5qeshfmgotKuGJCozlZjG210NsDO9WGX2Q5Ag_SOq_MzMkhkMN8KGwetMsHPl95grzoG74nFaEwa04GRrhdb9oMeXjdiMgo9mwPT-JT3z16u1bQCBexCOxGBGusryoLchC0UXEDiVo\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Para a acelera\u00e7\u00e3o vetorial instant\u00e2nea (\ud835\udc4e\u20d7) por:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/rsZgalZnSoByznlPB_p63LxDN9CUhpED-Azdc-ai24Edgbm_T7L_YCfIqwQlFunggfUD1MujV65xrZvyk-jcyEf-DTWhdnkyw-eac7e6_IXstBVrru11qgWPPwQr994G1evRkXBMzSZSf23XJFob-DA\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Dessa forma, dizemos que um corpo est\u00e1 em repouso ou em movimento retil\u00edneo uniforme se \ud835\udc4e\u20d7 = 0, pois, neste caso teremos que \ud835\udc63\u20d7 \u00e9 constante. Este resultado \u00e9 muito importante no estudo de Din\u00e2mica, assunto das nossas pr\u00f3ximas aulas.<\/p>\n\n\n\n<p>A partir de agora, vamos estudar algumas curvas particulares que frequentemente aparecem em quest\u00f5es dos nossos vestibulares.<\/p>\n\n\n\n<p><strong>Trajet\u00f3ria retil\u00ednea<\/strong><\/p>\n\n\n\n<p>Para o caso da trajet\u00f3ria retil\u00ednea, o vetor \ud835\udc4e\u20d7 tem a mesma dire\u00e7\u00e3o da trajet\u00f3ria e o m\u00f3dulo igual ao m\u00f3dulo da acelera\u00e7\u00e3o escalar. Se o movimento for acelerado (\ud835\udc4e. \ud835\udc63 &gt; 0), \ud835\udc4e\u20d7 tem o mesmo sentido de \ud835\udc63\u20d7. Por outro lado, se o movimento for retardado (\ud835\udc4e. \ud835\udc63 &lt; 0), \ud835\udc4e\u20d7 tem sentido contr\u00e1rio ao de \ud835\udc63\u20d7.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/rZlheBB9zfMpnl2r5pQdbPYv8MZJ5Jl-fSRvClR1g6G0Ul6cOl2_mQf7Wqws9Dgc0uY_XwZlr6olgO9rK5MbhpTPWzCg-0TOw-M_cr_PZw5ocobfIeeGXHCHHt9VU_ulRmlrXlWILi1wnYzs11P-C9Y\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Trajet\u00f3ria curva&nbsp;<\/strong><\/p>\n\n\n\n<p>Para o caso da trajet\u00f3ria curva qualquer, verifica-se que acelera\u00e7\u00e3o do corpo \ud835\udc4e\u20d7 \u201caponta para dentro da curva\u201d, como na figura 8. Trata-se, pois, de uma composi\u00e7\u00e3o de duas acelera\u00e7\u00f5es: a acelera\u00e7\u00e3o tangencial (\ud835\udc4e\u20d7\ud835\udc61 ) na qual sua dire\u00e7\u00e3o \u00e9 tangente a trajet\u00f3ria e a acelera\u00e7\u00e3o centr\u00edpeta ou acelera\u00e7\u00e3o normal (\ud835\udc4e\u20d7\ud835\udc50 ), na qual sua dire\u00e7\u00e3o \u00e9 normal a trajet\u00f3ria.&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/GAQicHrBiI3NMtjhM0kEUGCmwsy3bCGUBdv8iU5rpnB0GQ-C0W14dTXatvIiEwsAJuJJXu8w4wnS8KJ7Wm12Md2qV642JZ1bcdZsRkVVKpSE8Nv6oY_jM43P3NQaHbR55Vb0T_zCDvW3yhF_j2uQ-1A\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Vetorialmente, temos que:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/aUIgN5NFhOFQ2UuGFUg1wiyuzVz_3k1PdBW0yzTsGxkA-kni5I4kPPD3YggtkDmaVaokg7qeqoy4zYO9WnsBd_0XQk2oleVmzy6IZhbYXuFy1-4UbJajM-vvGUKusq_ExjrwYizYpCy-ueUFbXjNWtI\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>A rela\u00e7\u00e3o entre os m\u00f3dulos pode ser obtida pelo teorema de Pit\u00e1goras:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/kQJqmRdMhYyAspt4PIP4eTuHPNsD190ta0yRBnuE4kOfNjh1LaL3NMSkKvjxW7vH6S-xEaGznwjArkrp2qcjDnO7bwOQM-o2pKOWXKLvmnbNXwW0rz6AD-T0KpHkxlZvAfqoNM4eBEE2LdAQzRLtiMo\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Pode-se demonstrar que a acelera\u00e7\u00e3o tangencial tem m\u00f3dulo igual ao m\u00f3dulo da acelera\u00e7\u00e3o escalar, isto \u00e9:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/LkKngzgRtJKJWjUZL5_vZ78EvFJppakqhp-qsMMgi8OMAVbP0s2j6DYJCdQMFqwyzB1lkU6TQGppezjEJs6ByaPxCC9ceEZnQlMU8r8MEUQ5gqulSyAEo3ebeOv_JzfAxhyFP3HQObkYRHIB3CliE08\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Dessa forma, se \ud835\udc4e<sub>\ud835\udc52 <\/sub>= 0, em outras palavras um movimento uniforme, ent\u00e3o \ud835\udc4e\u20d7<sub>\ud835\udc61<\/sub> = 0<\/p>\n\n\n\n<p>Portanto, a acelera\u00e7\u00e3o tangencial est\u00e1 vinculada \u00e0 varia\u00e7\u00e3o do m\u00f3dulo da velocidade linear \ud835\udc63\u20d7 mas n\u00e3o muda a sua dire\u00e7\u00e3o. O sentido de \ud835\udc4e\u20d7<sub>\ud835\udc61<\/sub> \u00e9 o mesmo da velocidade vetorial instant\u00e2nea \ud835\udc63\u20d7, se o movimento for acelerado, e contr\u00e1rio ao de \ud835\udc63\u20d7, se o movimento for retardado.&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/VEZVr0F9CuJDEHBiv1DLpgz7l8Tm7ttbpcw9PEWh5SYBi2LOSjVRkQbZ52kok9gI5BqGJfJjngeCxvyBIoduf_Ars6FVukRWTDLi7s4EOjxdtbJfKBSXWC21nrMM7rVTnXdPWy4Hgjg69Qd3vEiXviY\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>\u00c9 poss\u00edvel demonstrar que a acelera\u00e7\u00e3o centr\u00edpeta (\ud835\udc4e\u20d7<sub>\ud835\udc50<\/sub> ) \u00e9 dada por:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/yKqIqsWeZZnST70rCjhrH7CdUunN0B4treUsJMhkk1wQ-HyP9WwfZGhIEfFuuCFhEI3IBKiudZ7h-13roGTnRN2CcPcIzAY3JqYOypju15WJ0MibVLcIy0sGhEY8rK5hZ79tvMtJYL9UrXvcxdO2dvs\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Onde \ud835\udc63 \u00e9 o m\u00f3dulo da velocidade e \ud835\udc45 \u00e9 o raio de curvatura da trajet\u00f3ria.&nbsp;<\/p>\n\n\n\n<p>Se a trajet\u00f3ria \u00e9 circular, o raio de curvatura \u00e9 o pr\u00f3prio raio da circunfer\u00eancia. Para o caso da trajet\u00f3ria n\u00e3o ser uma circunfer\u00eancia, \u00e9 poss\u00edvel obter uma circunfer\u00eancia tangente \u00e0 trajet\u00f3ria, chamada de circunfer\u00eancia osculadora.&nbsp;<\/p>\n\n\n\n<p>Dessa forma, o raio \u00e9 o raio de curvatura a ser usado no c\u00e1lculo do m\u00f3dulo de \ud835\udc4e\u20d7<sub>\ud835\udc50<\/sub> , que aponta para o centro da circunfer\u00eancia osculadora. Esse \u00e9 um m\u00e9todo para calcular o raio de curvatura de uma trajet\u00f3ria, sem utilizar os recursos do C\u00e1lculo.&nbsp;<\/p>\n\n\n\n<p><em>Texto escrito com base nos conte\u00fados do professor Toni Burgatto<\/em><\/p>\n\n\n\n<p>Curtiu a leitura? F\u00edsica \u00e9 uma disciplina que faz parte dos conte\u00fados cobrados nas provas dos concursos militares, ent\u00e3o n\u00e3o deixe de estud\u00e1-lo. Se voc\u00ea tem o sonho de ingressar nas carreiras militares do Brasil, confira o Banco de Quest\u00f5es do Estrat\u00e9gia Militares e garanta a sua aprova\u00e7\u00e3o! Clique no banner abaixo e vem ser coruja!<\/p>\n\n\n\n<p><strong><a id=\"cta\" class=\"cta-imagem\" href=\"https:\/\/militares.estrategia.com\/curso\/banco-de-questes-anual\/\" target=\"blank\">\n                <img decoding=\"async\" width=\"100%\" src=\"https:\/\/militares.estrategia.com\/portal\/wp-content\/uploads\/2023\/02\/EM-Banco-de-Questoes.jpg\" alt=\"EM - Banco de Questo\u0303es\" title=\"EM - Banco de Questo\u0303es\">\n        <\/a><\/strong><\/p>\n\n\n\n\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Veja-tambem\"><\/span>Veja tamb\u00e9m:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/operacoes-matematicas-com-algarismos-significativos\/\">Opera\u00e7\u00f5es matem\u00e1ticas com algarismos significativos<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/nocoes-de-algarismos-significativos\/\">No\u00e7\u00f5es de Algarismos Significativos<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/cinematica-escalar-introducao\/\">Cinem\u00e1tica Escalar: introdu\u00e7\u00e3o<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/cinematica-escalar-finalizacao\/\">Cinem\u00e1tica Escalar: velocidade escalar m\u00e9dia, movimento progressivo e mais<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/introducao-a-fisica\/\">F\u00edsica: introdu\u00e7\u00e3o, grandeza, estudo e mais<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/movimento-uniformemente\/\">Movimento uniformemente variado: acelera\u00e7\u00e3o, cinem\u00e1tica com c\u00e1lculo e mais!<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/movimento-uniforme\/\">Movimento uniforme: o que \u00e9 e para que serve?<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/fisica\/saiba-mais-sobre-a-transmissao-de-movimento-circular\/\">Transmiss\u00e3o de Movimento Circular: correia, engrenagens e mais!<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"Vai participar dos concursos e quer saber um pouquinho mais de f\u00edsica? Vem com a gente! Confira esse&hellip;\n","protected":false},"author":18,"featured_media":68447,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"wl_entities_gutenberg":"","footnotes":""},"categories":[302],"tags":[],"wl_entity_type":[187],"class_list":{"0":"post-68446","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-fisica","8":"wl_entity_type-article"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.6 (Yoast SEO v26.6) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Cinem\u00e1tica Vetorial - Estrat\u00e9gia Militares<\/title>\n<meta name=\"description\" content=\"Vai participar dos concursos e quer saber um pouquinho mais de f\u00edsica? Vem com a gente! 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