{"id":212136,"date":"2025-05-06T08:17:35","date_gmt":"2025-05-06T08:17:35","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=212136"},"modified":"2025-05-06T08:17:37","modified_gmt":"2025-05-06T08:17:37","slug":"in-the-figure-a-runaway-truck-with-failed-brakes-is-moving-downgrade-at-106-km-h-just-before-the-driver-steers-the-truck-up-a-frictionless-emergency-escape-ramp-with-an-inclination-of-0-14","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/06\/in-the-figure-a-runaway-truck-with-failed-brakes-is-moving-downgrade-at-106-km-h-just-before-the-driver-steers-the-truck-up-a-frictionless-emergency-escape-ramp-with-an-inclination-of-0-14\/","title":{"rendered":"In the figure, a runaway truck with failed brakes is moving downgrade at 106 km\/h just before the driver steers the truck up a frictionless emergency escape ramp with an inclination of 0 = 14"},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n

The Correct Answer And Eplanation is:<\/mark><\/strong><\/p>\n\n\n\n

Given:<\/h3>\n\n\n\n
    \n
  • Initial speed v0=106\u2009km\/h=29.44\u2009m\/sv_0 = 106 \\, \\text{km\/h} = 29.44 \\, \\text{m\/s}<\/li>\n\n\n\n
  • Inclination angle \u03b8=14\u2218\\theta = 14^\\circ<\/li>\n\n\n\n
  • Mass of truck m=1.4\u00d7104\u2009kgm = 1.4 \\times 10^4 \\, \\text{kg}<\/li>\n\n\n\n
  • Frictionless ramp<\/li>\n\n\n\n
  • The truck stops momentarily at the top of the ramp<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    (a) Minimum Length LL of the Ramp<\/strong><\/h3>\n\n\n\n

    We apply conservation of energy<\/strong>:<\/p>\n\n\n\n

      \n
    • Initial kinetic energy:
      KE=12mv02KE = \\frac{1}{2}mv_0^2<\/li>\n\n\n\n
    • Final potential energy at height hh:
      PE=mghPE = mgh<\/li>\n<\/ul>\n\n\n\n

      Because the surface is frictionless and only conservative forces act, total energy is conserved: 12mv02=mgh\\frac{1}{2}mv_0^2 = mgh<\/p>\n\n\n\n

      Canceling mass mm: 12v02=gh\u21d2h=v022g\\frac{1}{2}v_0^2 = gh \\Rightarrow h = \\frac{v_0^2}{2g}<\/p>\n\n\n\n

      Now, relate height hh to ramp length LL using sin\u2061\u03b8=hL\\sin\\theta = \\frac{h}{L}: L=hsin\u2061\u03b8=v022gsin\u2061\u03b8L = \\frac{h}{\\sin\\theta} = \\frac{v_0^2}{2g \\sin\\theta}<\/p>\n\n\n\n

      Plug in values:<\/p>\n\n\n\n

        \n
      • v0=29.44\u2009m\/sv_0 = 29.44 \\, \\text{m\/s}<\/li>\n\n\n\n
      • g=9.8\u2009m\/s2g = 9.8 \\, \\text{m\/s}^2<\/li>\n\n\n\n
      • \u03b8=14\u2218\\theta = 14^\\circ<\/li>\n<\/ul>\n\n\n\n

        L=(29.44)22\u22c59.8\u22c5sin\u2061(14\u2218)\u2248866.62\u22c59.8\u22c50.2419\u2248866.64.741\u2248182.8\u2009mL = \\frac{(29.44)^2}{2 \\cdot 9.8 \\cdot \\sin(14^\\circ)} \\approx \\frac{866.6}{2 \\cdot 9.8 \\cdot 0.2419} \\approx \\frac{866.6}{4.741} \\approx 182.8 \\, \\text{m}<\/p>\n\n\n\n

        \u2705 Answer: Minimum Length ( L \\approx 183 , \\text{m}<\/strong><\/h3>\n\n\n\n
        \n\n\n\n

        (b) If Mass is Decreased<\/strong><\/h3>\n\n\n\n

        Since mass cancels out during energy conservation, the result does not depend on mass<\/strong>.<\/p>\n\n\n\n

        \u2705 Answer: Length LL remains the same.<\/strong><\/p>\n\n\n\n

        \n\n\n\n

        (c) If Speed is Decreased<\/strong><\/h3>\n\n\n\n

        From the formula: L=v022gsin\u2061\u03b8L = \\frac{v_0^2}{2g \\sin\\theta}<\/p>\n\n\n\n

        Lowering v0v_0 decreases v02v_0^2, which decreases LL.<\/p>\n\n\n\n

        \u2705 Answer: Length LL decreases.<\/strong><\/p>\n\n\n\n

        \n\n\n\n

        Summary:<\/h3>\n\n\n\n
          \n
        • (a) L\u2248183\u2009mL \\approx 183 \\, \\text{m}<\/li>\n\n\n\n
        • (b) Length remains the same<\/strong> if mass decreases.<\/li>\n\n\n\n
        • (c) Length decreases<\/strong> if speed decreases.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"

          The Correct Answer And Eplanation is: Given: (a) Minimum Length LL of the Ramp We apply conservation of energy: Because the surface is frictionless and only conservative forces act, total energy is conserved: 12mv02=mgh\\frac{1}{2}mv_0^2 = mgh Canceling mass mm: 12v02=gh\u21d2h=v022g\\frac{1}{2}v_0^2 = gh \\Rightarrow h = \\frac{v_0^2}{2g} Now, relate height hh to ramp length LL using […]<\/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-212136","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/212136","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=212136"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/212136\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=212136"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=212136"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=212136"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}