{"id":3601,"date":"2022-04-04T01:26:10","date_gmt":"2022-04-04T01:26:10","guid":{"rendered":"http:\/\/calculuscoaches.com\/?page_id=3601"},"modified":"2023-08-07T15:05:36","modified_gmt":"2023-08-07T15:05:36","slug":"distance-traveled-by-a-particle-found-using-calculus","status":"publish","type":"page","link":"https:\/\/calculuscoaches.com\/index.php\/distance-traveled-by-a-particle-found-using-calculus\/","title":{"rendered":"distance traveled by a particle found using calculus"},"content":{"rendered":"\n<div>\n<h3>Analysis of Particle&#8217;s Motion<\/h3>\n<p><strong>Step 1: Position Function<\/strong><\/p>\n<p>Comment: The position function describes the particle&#8217;s position at any given time t.<\/p>\n<p>s(t) = t\u00b3 &#8211; 2t<\/p>\n<p><strong>Step 2: Velocity Function (Derivative of Position Function)<\/strong><\/p>\n<p>Comment: The velocity function represents the rate of change of the position with respect to time. It is the derivative of the position function.<\/p>\n<p>v(t) = 3t\u00b2 &#8211; 2<\/p>\n<p><strong>Step 3: Point where Velocity is Zero (Change in Direction)<\/strong><\/p>\n<p>Comment: To find when the particle changes direction, we set the velocity function to zero and solve for t.<\/p>\n<p>3t\u00b2 &#8211; 2 = 0<\/p>\n<p>Solution: t = \u00b1\u221a(2\/3)<\/p>\n<p><strong>Step 4: Evaluate Position Function at Key Points<\/strong><\/p>\n<p>Comment: We evaluate the position function at the start, the point of direction change, and the end to determine the particle&#8217;s positions at these times.<\/p>\n<p>For t = 0: s(0) = 0<\/p>\n<p>For t = \u221a(2\/3): s(\u221a(2\/3)) = -4\u221a2\/3<\/p>\n<p>For t = 2: s(2) = 4<\/p>\n<p><strong>Step 5: Calculate Distance Using Absolute Values<\/strong><\/p>\n<p>Comment: We&#8217;ll compute the distance the particle traveled between the key points using the absolute difference in the position values.<\/p>\n<p>Distance from t = 0 to t = \u221a(2\/3): |s(\u221a(2\/3)) &#8211; s(0)| = 4\u221a2\/3 \u2248 1.88562 units<\/p>\n<p>Distance from t = \u221a(2\/3) to t = 2: |s(2) &#8211; s(\u221a(2\/3))| = 4 + 4\u221a2\/3 \u2248 5.88562 units<\/p>\n<p><strong>Step 6: Total Distance Traveled<\/strong><\/p>\n<p>Comment: The total distance traveled by the particle is the sum of the distances calculated in the previous step.<\/p>\n<p>Total Distance = 1.88562 + 5.88562 = 7.77124 units<\/p>\n<p>Approximated Total Distance: \u2248 7.77 units<\/p>\n<p>\u00a0<\/p>\n<\/div>\n<div><img decoding=\"async\" src=\"https:\/\/www.wolframcloud.com\/obj\/f5018abc-a8d3-4c66-be9f-b23b8fde476a\" alt=\"Graph of s(t) = t^3 - 2t with marked point\" \/><\/div>\n<p><script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-1379562136521678\"\n     crossorigin=\"anonymous\"><\/script><\/p>\n<!-- horizontal -->\n<p><ins class=\"adsbygoogle\" style=\"display: block;\" data-ad-client=\"ca-pub-1379562136521678\" data-ad-slot=\"6995763238\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins> <script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script><\/p>\n<hr \/><hr \/>\n<div>\u00a0<\/div>\n<div style=\"font-family: Arial, sans-serif;\">\n<p>\u00a0<\/p>\n<\/div>\n<div style=\"font-family: Arial, sans-serif;\">\n<h3>Particle Motion Analysis<\/h3>\n<h4>Position Function:<\/h4>\n<p>The particle&#8217;s position at any given time t is described by the function:<\/p>\n<p>s(t) = t\u2074 &#8211; 4t\u00b2<\/p>\n<h4>Velocity Function (Derivative of Position Function):<\/h4>\n<p>The velocity function represents the rate of change of the position with respect to time. It is the derivative of the position function.<\/p>\n<p>v(t) = 4t\u00b3 &#8211; 8t<\/p>\n<h4>Direction Change Points:<\/h4>\n<p>To find when the particle changes direction, we set the velocity function to zero and solve for t:<\/p>\n<p>4t\u00b3 &#8211; 8t = 0<\/p>\n<p>This gives us three potential points of interest: t = 0, t = \u221a2, and t = -\u221a2.<\/p>\n<h4>Evaluate Position Function at Key Points:<\/h4>\n<ul>\n<li>For t = 0: s(0) = 0<\/li>\n<li>For t = \u221a2: s(\u221a2) = 4 &#8211; 8 = -4<\/li>\n<li>For t = -\u221a2: s(-\u221a2) = 4 &#8211; 8 = -4<\/li>\n<\/ul>\n<h4>Calculate Distance Using Absolute Values:<\/h4>\n<p>1. Distance from t = -\u221a2 to t = 0:<\/p>\n<p>Distance = |s(0) &#8211; s(-\u221a2)| = 4<\/p>\n<p>2. Distance from t = 0 to t = \u221a2:<\/p>\n<p>Distance = |s(\u221a2) &#8211; s(0)| = 4<\/p>\n<p>The total distance traveled by the particle between t = -\u221a2 and t = \u221a2 is 8 units.<\/p>\n<h4>Graph:<\/h4>\n<p><img decoding=\"async\" src=\"https:\/\/www6b3.wolframalpha.com\/Calculate\/MSP\/MSP20451d3ah1cd0cgi6dh10000210af3cbb79a1243?MSPStoreType=image\/png&amp;s=19\" alt=\"Graph of s(t) = t^4 - 4t^2\" \/><\/p>\n<p>Note: The graph shows the position function s(t) = t\u2074 &#8211; 4t\u00b2 over the interval [-1.5, 1.5]. The points of interest are t = -\u221a2, t = 0, and t = \u221a2.<\/p>\n<\/div>\n<hr \/>\n<p><script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-1379562136521678\"\n     crossorigin=\"anonymous\"><\/script><\/p>\n<!-- horizontal -->\n<p>\u00a0<\/p>\n<p>\u00a0<\/p>\n<div style=\"font-family: Arial, sans-serif;\">\n<p>\u00a0<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Analysis of Particle&#8217;s Motion Step 1: Position Function Comment: The position function describes the particle&#8217;s position at any given time t. s(t) = t\u00b3 &#8211; 2t Step 2: Velocity Function (Derivative of Position Function) Comment: The velocity function represents the rate of change of the position with respect to time. It is the derivative of &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/calculuscoaches.com\/index.php\/distance-traveled-by-a-particle-found-using-calculus\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;distance traveled by a particle found using calculus&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":149,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-3601","page","type-page","status-publish","hentry"],"aioseo_notices":[],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/pages\/3601","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/comments?post=3601"}],"version-history":[{"count":17,"href":"https:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/pages\/3601\/revisions"}],"predecessor-version":[{"id":4212,"href":"https:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/pages\/3601\/revisions\/4212"}],"wp:attachment":[{"href":"https:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/media?parent=3601"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}