{"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":"http:\/\/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=\"http:\/\/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":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.4 - aioseo.com -->\n\t<meta name=\"description\" content=\"Analysis of Particle&#039;s Motion Step 1: Position Function Comment: The position function describes the particle&#039;s position at any given time t. s(t) = t\u00b3 - 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\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<link rel=\"canonical\" href=\"http:\/\/calculuscoaches.com\/index.php\/distance-traveled-by-a-particle-found-using-calculus\/\" \/>\n\t\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.4\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"Calculus Coaches | helps with math problems\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"distance traveled by a particle found using calculus | Calculus Coaches\" \/>\n\t\t<meta property=\"og:description\" content=\"Analysis of Particle&#039;s Motion Step 1: Position Function Comment: The position function describes the particle&#039;s position at any given time t. s(t) = t\u00b3 - 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\" \/>\n\t\t<meta property=\"og:url\" content=\"http:\/\/calculuscoaches.com\/index.php\/distance-traveled-by-a-particle-found-using-calculus\/\" \/>\n\t\t<meta property=\"article:published_time\" content=\"2022-04-04T01:26:10+00:00\" \/>\n\t\t<meta property=\"article:modified_time\" content=\"2023-08-07T15:05:36+00:00\" \/>\n\t\t<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n\t\t<meta name=\"twitter:title\" content=\"distance traveled by a particle found using calculus | Calculus Coaches\" \/>\n\t\t<meta name=\"twitter:description\" content=\"Analysis of Particle&#039;s Motion Step 1: Position Function Comment: The position function describes the particle&#039;s position at any given time t. s(t) = t\u00b3 - 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\" \/>\n\t\t<script type=\"application\/ld+json\" class=\"aioseo-schema\">\n\t\t\t{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"BreadcrumbList\",\"@id\":\"http:\\\/\\\/calculuscoaches.com\\\/index.php\\\/distance-traveled-by-a-particle-found-using-calculus\\\/#breadcrumblist\",\"itemListElement\":[{\"@type\":\"ListItem\",\"@id\":\"http:\\\/\\\/calculuscoaches.com\\\/#listItem\",\"position\":1,\"name\":\"Home\",\"item\":\"http:\\\/\\\/calculuscoaches.com\\\/\",\"nextItem\":{\"@type\":\"ListItem\",\"@id\":\"http:\\\/\\\/calculuscoaches.com\\\/index.php\\\/distance-traveled-by-a-particle-found-using-calculus\\\/#listItem\",\"name\":\"distance traveled by a particle found using calculus\"}},{\"@type\":\"ListItem\",\"@id\":\"http:\\\/\\\/calculuscoaches.com\\\/index.php\\\/distance-traveled-by-a-particle-found-using-calculus\\\/#listItem\",\"position\":2,\"name\":\"distance traveled by a particle found using calculus\",\"previousItem\":{\"@type\":\"ListItem\",\"@id\":\"http:\\\/\\\/calculuscoaches.com\\\/#listItem\",\"name\":\"Home\"}}]},{\"@type\":\"Organization\",\"@id\":\"http:\\\/\\\/calculuscoaches.com\\\/#organization\",\"name\":\"Calculus Coaches\",\"description\":\"helps with math problems\",\"url\":\"http:\\\/\\\/calculuscoaches.com\\\/\"},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/calculuscoaches.com\\\/index.php\\\/distance-traveled-by-a-particle-found-using-calculus\\\/#webpage\",\"url\":\"http:\\\/\\\/calculuscoaches.com\\\/index.php\\\/distance-traveled-by-a-particle-found-using-calculus\\\/\",\"name\":\"distance traveled by a particle found using calculus | Calculus Coaches\",\"description\":\"Analysis of Particle's Motion Step 1: Position Function Comment: The position function describes the particle's position at any given time t. s(t) = t\\u00b3 - 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\",\"inLanguage\":\"en-US\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/calculuscoaches.com\\\/#website\"},\"breadcrumb\":{\"@id\":\"http:\\\/\\\/calculuscoaches.com\\\/index.php\\\/distance-traveled-by-a-particle-found-using-calculus\\\/#breadcrumblist\"},\"datePublished\":\"2022-04-04T01:26:10+00:00\",\"dateModified\":\"2023-08-07T15:05:36+00:00\"},{\"@type\":\"WebSite\",\"@id\":\"http:\\\/\\\/calculuscoaches.com\\\/#website\",\"url\":\"http:\\\/\\\/calculuscoaches.com\\\/\",\"name\":\"Calculus Coaches\",\"description\":\"helps with math problems\",\"inLanguage\":\"en-US\",\"publisher\":{\"@id\":\"http:\\\/\\\/calculuscoaches.com\\\/#organization\"}}]}\n\t\t<\/script>\n\t\t<!-- All in One SEO -->\n\n","aioseo_head_json":{"title":"distance traveled by a particle found using calculus | Calculus Coaches","description":"Analysis of Particle's Motion Step 1: Position Function Comment: The position function describes the particle's position at any given time t. s(t) = t\u00b3 - 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","canonical_url":"http:\/\/calculuscoaches.com\/index.php\/distance-traveled-by-a-particle-found-using-calculus\/","robots":"max-image-preview:large","keywords":"","webmasterTools":{"miscellaneous":""},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"BreadcrumbList","@id":"http:\/\/calculuscoaches.com\/index.php\/distance-traveled-by-a-particle-found-using-calculus\/#breadcrumblist","itemListElement":[{"@type":"ListItem","@id":"http:\/\/calculuscoaches.com\/#listItem","position":1,"name":"Home","item":"http:\/\/calculuscoaches.com\/","nextItem":{"@type":"ListItem","@id":"http:\/\/calculuscoaches.com\/index.php\/distance-traveled-by-a-particle-found-using-calculus\/#listItem","name":"distance traveled by a particle found using calculus"}},{"@type":"ListItem","@id":"http:\/\/calculuscoaches.com\/index.php\/distance-traveled-by-a-particle-found-using-calculus\/#listItem","position":2,"name":"distance traveled by a particle found using calculus","previousItem":{"@type":"ListItem","@id":"http:\/\/calculuscoaches.com\/#listItem","name":"Home"}}]},{"@type":"Organization","@id":"http:\/\/calculuscoaches.com\/#organization","name":"Calculus Coaches","description":"helps with math problems","url":"http:\/\/calculuscoaches.com\/"},{"@type":"WebPage","@id":"http:\/\/calculuscoaches.com\/index.php\/distance-traveled-by-a-particle-found-using-calculus\/#webpage","url":"http:\/\/calculuscoaches.com\/index.php\/distance-traveled-by-a-particle-found-using-calculus\/","name":"distance traveled by a particle found using calculus | Calculus Coaches","description":"Analysis of Particle's Motion Step 1: Position Function Comment: The position function describes the particle's position at any given time t. s(t) = t\u00b3 - 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","inLanguage":"en-US","isPartOf":{"@id":"http:\/\/calculuscoaches.com\/#website"},"breadcrumb":{"@id":"http:\/\/calculuscoaches.com\/index.php\/distance-traveled-by-a-particle-found-using-calculus\/#breadcrumblist"},"datePublished":"2022-04-04T01:26:10+00:00","dateModified":"2023-08-07T15:05:36+00:00"},{"@type":"WebSite","@id":"http:\/\/calculuscoaches.com\/#website","url":"http:\/\/calculuscoaches.com\/","name":"Calculus Coaches","description":"helps with math problems","inLanguage":"en-US","publisher":{"@id":"http:\/\/calculuscoaches.com\/#organization"}}]},"og:locale":"en_US","og:site_name":"Calculus Coaches | helps with math problems","og:type":"article","og:title":"distance traveled by a particle found using calculus | Calculus Coaches","og:description":"Analysis of Particle's Motion Step 1: Position Function Comment: The position function describes the particle's position at any given time t. s(t) = t\u00b3 - 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","og:url":"http:\/\/calculuscoaches.com\/index.php\/distance-traveled-by-a-particle-found-using-calculus\/","article:published_time":"2022-04-04T01:26:10+00:00","article:modified_time":"2023-08-07T15:05:36+00:00","twitter:card":"summary_large_image","twitter:title":"distance traveled by a particle found using calculus | Calculus Coaches","twitter:description":"Analysis of Particle's Motion Step 1: Position Function Comment: The position function describes the particle's position at any given time t. s(t) = t\u00b3 - 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"},"aioseo_meta_data":{"post_id":"3601","title":null,"description":null,"keywords":[],"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":[],"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"WebPage","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"ai":null,"created":"2023-08-03 01:45:19","updated":"2025-06-04 04:34:46","seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"http:\/\/calculuscoaches.com\/\" title=\"Home\">Home<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">\u00bb<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\tdistance traveled by a particle found using calculus\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"http:\/\/calculuscoaches.com\/"},{"label":"distance traveled by a particle found using calculus","link":"http:\/\/calculuscoaches.com\/index.php\/distance-traveled-by-a-particle-found-using-calculus\/"}],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"http:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/pages\/3601","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/comments?post=3601"}],"version-history":[{"count":17,"href":"http:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/pages\/3601\/revisions"}],"predecessor-version":[{"id":4212,"href":"http:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/pages\/3601\/revisions\/4212"}],"wp:attachment":[{"href":"http:\/\/calculuscoaches.com\/index.php\/wp-json\/wp\/v2\/media?parent=3601"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}