{"id":213577,"date":"2025-05-11T19:06:46","date_gmt":"2025-05-11T19:06:46","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=213577"},"modified":"2025-05-11T19:06:49","modified_gmt":"2025-05-11T19:06:49","slug":"two-runners-start-a-race-at-the-same-time-and-finish-in-a-tie","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/11\/two-runners-start-a-race-at-the-same-time-and-finish-in-a-tie\/","title":{"rendered":"Two runners start a race at the same time and finish in a tie."},"content":{"rendered":"\n<p>Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider (??)=??(??)-h(??), ?where ?? ?and h ?are the position functions of the two runners.]<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>We are given that <strong>two runners start a race at the same time and finish in a tie<\/strong>, and we are asked to prove that at <strong>some point during the race<\/strong>, they must have <strong>the same speed<\/strong>. The <strong>hint<\/strong> suggests using the function: f(t)=g(t)\u2212h(t)f(t) = g(t) &#8211; h(t)<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>g(t)g(t) = position function of the first runner<\/li>\n\n\n\n<li>h(t)h(t) = position function of the second runner<\/li>\n\n\n\n<li>f(t)f(t) = difference in position between the two runners at time tt<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step-by-step Proof (Using the Mean Value Theorem):<\/h3>\n\n\n\n<p>Since both runners <strong>start at the same point and finish at the same point<\/strong>, we have: g(0)=h(0)andg(T)=h(T)g(0) = h(0) \\quad \\text{and} \\quad g(T) = h(T)<\/p>\n\n\n\n<p>This implies: f(0)=g(0)\u2212h(0)=0andf(T)=g(T)\u2212h(T)=0f(0) = g(0) &#8211; h(0) = 0 \\quad \\text{and} \\quad f(T) = g(T) &#8211; h(T) = 0<\/p>\n\n\n\n<p>So the function f(t)f(t) satisfies:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>ff is continuous on the closed interval [0,T][0, T]<\/li>\n\n\n\n<li>ff is differentiable on the open interval (0,T)(0, T)<\/li>\n\n\n\n<li>f(0)=f(T)=0f(0) = f(T) = 0<\/li>\n<\/ul>\n\n\n\n<p>Now, apply <strong>Rolle\u2019s Theorem<\/strong>:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>If a function is continuous on [a,b][a, b], differentiable on (a,b)(a, b), and f(a)=f(b)f(a) = f(b), then there is at least one point c\u2208(a,b)c \\in (a, b) such that f\u2032(c)=0f'(c) = 0.<\/p>\n<\/blockquote>\n\n\n\n<p>Since f(t)f(t) meets all the conditions, <strong>there exists a time c\u2208(0,T)c \\in (0, T)<\/strong> such that: f\u2032(c)=0f'(c) = 0<\/p>\n\n\n\n<p>But recall: f\u2032(t)=g\u2032(t)\u2212h\u2032(t)f'(t) = g'(t) &#8211; h'(t)<\/p>\n\n\n\n<p>So at time cc: g\u2032(c)=h\u2032(c)g'(c) = h'(c)<\/p>\n\n\n\n<p>That means the <strong>instantaneous speed (velocity)<\/strong> of both runners is <strong>equal<\/strong> at some moment during the race.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<p>At some point during the race, both runners <strong>must have had the same speed<\/strong>. This result is guaranteed by <strong>Rolle\u2019s Theorem<\/strong> applied to the difference of their position functions. It doesn\u2019t matter how they individually sped up or slowed down\u2014as long as they started and ended together, they had to be going the <strong>same speed at least once<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider (??)=??(??)-h(??), ?where ?? ?and h ?are the position functions of the two runners.] The Correct Answer and Explanation is: We are given that two runners start [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-213577","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/213577","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=213577"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/213577\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=213577"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=213577"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=213577"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}