{"id":6630,"date":"2011-03-19T23:20:25","date_gmt":"2011-03-19T23:20:25","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6630"},"modified":"2022-01-23T16:46:04","modified_gmt":"2022-01-23T16:46:04","slug":"lancou-se-um-foguete-de-fabrico-artesanal","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6630","title":{"rendered":"Lan\u00e7ou-se um foguete de fabrico artesanal"},"content":{"rendered":"<p><ul id='GTTabs_ul_6630' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6630' class='GTTabs_curr'><a  id=\"6630_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6630' ><a  id=\"6630_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_6630'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Lan\u00e7ou-se um foguete de fabrico artesanal. Devido a um defeito de fabrico, o foguete come\u00e7a a perder altura, mas, em seguida, recupera e sobe de novo. A altura a (em metros) a que se encontra \u00e9 dada, em fun\u00e7\u00e3o do tempo t decorrido (em segundos) ap\u00f3s o seu lan\u00e7amento, por: \\[\\begin{matrix}<br \/>\na(t)={{t}^{3}}-9{{t}^{2}}+24t &amp; (0\\le t\\le 7)\u00a0 \\\\<br \/>\n\\end{matrix}\\]<\/p>\n<ol>\n<li>Compare os valores da velocidade m\u00e9dia nos intervalos [2, 5] e [2, 4]. A que se dever\u00e1 tal discrep\u00e2ncia?<\/li>\n<li>Determine o valor da velocidade no momento em que partiu e nos instantes $t=1$ e $t=7$.<\/li>\n<li>Indique os intervalos de tempo em que o foguete subiu e aqueles em que desceu.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6630' onClick='GTTabs_show(1,6630)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6630'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>\\[tm{{v}_{\\left[ 2,5 \\right]}}=\\frac{a(5)-a(2)}{5-2}=\\frac{(125-225+120)-(8-36+48)}{3}=\\frac{20-20}{3}=0\\]<br \/>\n\\[tm{{v}_{\\left[ 2,4 \\right]}}=\\frac{a(4)-a(2)}{4-2}=\\frac{(64-144+96)-(8-36+48)}{2}=\\frac{16-20}{2}=-2\\]<\/p>\n<p>No intervalo [2, 5], a velocidade m\u00e9dia \u00e9 de $0\\,m\/s$, enquanto que no intervalo [2,4] \u00e9 de $-2\\,m\/s$.<br \/>\nNo per\u00edodo de tempo em que\u00a0o foguete perde altura, a sua velocidade \u00e9 negativa. Da\u00ed a raz\u00e3o para os valores encontrados para as velocidades m\u00e9dias nesses intervalos de tempo.<\/p>\n<\/li>\n<li>Ora, $\\begin{matrix}<br \/>\na'(t)=3{{t}^{2}}-18t+24 &amp; (0\\le t\\le 7)\u00a0 \\\\<br \/>\n\\end{matrix}$.<\/p>\n<p>Logo, $a'(0)=24$, $a'(1)=3-18+24=9$ e $a'(7)=147-126+24=45$.<\/p>\n<p>Portanto, a velocidade nos instantes considerados \u00e9, respetivamente, $24\\,m\/s$, $9\\,m\/s$ e $45\\,m\/s$.<\/p>\n<\/li>\n<li>Como \\[\\begin{array}{*{35}{l}}<br \/>\na'(t)=0 &amp; \\Leftrightarrow\u00a0 &amp; 3{{t}^{2}}-18t+24=0\\wedge 0\\le t\\le 7\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{t}^{2}}-6t+8=0\\wedge 0\\le t\\le 7\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; (t=2\\vee t=4)\\wedge 0\\le t\\le 7\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; t=2\\vee t=4\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\ntemos:<\/p>\n<table class=\"aligncenter\" style=\"width: 70%;\" border=\"2\" cellspacing=\"4\" cellpadding=\"4\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">$t$<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td><\/td>\n<td style=\"text-align: center;\">$2$<\/td>\n<td><\/td>\n<td style=\"text-align: center;\">$4$<\/td>\n<td><\/td>\n<td style=\"text-align: center;\">$7$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$a'(t)$<\/td>\n<td style=\"text-align: center;\">$24$<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">$45$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$a(t)$<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">$\\nearrow $<\/td>\n<td style=\"text-align: center;\">$20$<\/td>\n<td style=\"text-align: center;\">$\\searrow $<\/td>\n<td style=\"text-align: center;\">$16$<\/td>\n<td style=\"text-align: center;\">$\\nearrow $<\/td>\n<td style=\"text-align: center;\">$70$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: left;\">O foguete subiu durante os primeiros 2 segundos, at\u00e9 atingir 20 metros de altura. Depois, desce um pouco, at\u00e9 aos 14 metros de altura, come\u00e7ando a subir de novo a partir dos 4 segundos, atingindo a altura de 70 metros ao fim de 7 segundos.<\/p>\n<\/li>\n<\/ol>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/20-03-2011-Ecra001.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6631\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6631\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/20-03-2011-Ecra001.png\" data-orig-size=\"690,468\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Gr\u00e1fico\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/20-03-2011-Ecra001.png\" class=\"aligncenter wp-image-6631\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/20-03-2011-Ecra001.png\" alt=\"\" width=\"540\" height=\"366\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/20-03-2011-Ecra001.png 690w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/20-03-2011-Ecra001-300x203.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/20-03-2011-Ecra001-150x101.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/20-03-2011-Ecra001-400x271.png 400w\" sizes=\"auto, (max-width: 540px) 100vw, 540px\" \/><\/a><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6630' onClick='GTTabs_show(0,6630)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Lan\u00e7ou-se um foguete de fabrico artesanal. Devido a um defeito de fabrico, o foguete come\u00e7a a perder altura, mas, em seguida, recupera e sobe de novo. A altura a (em metros)&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20925,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,134],"tags":[145,144],"series":[],"class_list":["post-6630","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-derivadas","tag-derivadas-2","tag-extremos-relativos"],"views":2794,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11V2Pag196-43_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6630","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=6630"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6630\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20925"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6630"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6630"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6630"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6630"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}