{"id":252886,"date":"2025-07-11T14:32:46","date_gmt":"2025-07-11T14:32:46","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=252886"},"modified":"2025-07-11T14:32:48","modified_gmt":"2025-07-11T14:32:48","slug":"a-billiard-ball-is-given-an-initial-velocity-v5m-s-and-makes-a-direct-impact-with-ball-b-e0-8-calculate-the-velocity-of-ball-b-and-the-rebound-angle-after-it-hits-cushion-at-c-e-0-6","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/11\/a-billiard-ball-is-given-an-initial-velocity-v5m-s-and-makes-a-direct-impact-with-ball-b-e0-8-calculate-the-velocity-of-ball-b-and-the-rebound-angle-after-it-hits-cushion-at-c-e-0-6\/","title":{"rendered":"A billiard ball is given an initial velocity V=5M\/S\u00a0\u00a0and makes a direct impact with ball B (e=0.8), calculate the velocity of ball B and the rebound angle after it hits cushion at C (e =0.6)."},"content":{"rendered":"\n<pre id=\"preorder-ask-header-text\" class=\"wp-block-preformatted\">A billiard ball is given an initial velocity V=5M\/S\u00a0\u00a0and makes a direct impact with ball B (e=0.8), calculate the velocity of ball B and the rebound angle after it hits cushion at C (e =0.6). Mass of the balls is 0.4 kg.<\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/image-400.png\" alt=\"\" class=\"wp-image-252887\"\/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p><strong>Correct Answer<\/strong><\/p>\n\n\n\n<p>The velocity of ball B immediately after the impact with ball A is&nbsp;<strong>v_B = 3.90 m\/s<\/strong>.<br>The rebound angle after ball B hits cushion C is&nbsp;<strong>\u03b8 = 43.9\u00b0<\/strong>.<\/p>\n\n\n\n<p><strong>Explanation<\/strong><\/p>\n\n\n\n<p>This problem is solved in two parts: the collision between the two balls and the subsequent collision of ball B with the cushion.<\/p>\n\n\n\n<p><strong>Part 1: Collision of Ball A and Ball B<\/strong><\/p>\n\n\n\n<p>First, we analyze the oblique impact between ball A and ball B. We establish a coordinate system with the n-axis along the line of impact (connecting the centers of the balls, at 30\u00b0 to the horizontal) and the t-axis perpendicular to it.<\/p>\n\n\n\n<p>Ball A&#8217;s initial velocity, (v_A)\u2081 = 5 m\/s, is horizontal. We resolve this into n and t components:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(v_A)\u2081n = 5 * cos(30\u00b0) = 4.33 m\/s<\/li>\n\n\n\n<li>(v_A)\u2081t = 5 * sin(30\u00b0) = 2.5 m\/s<\/li>\n<\/ul>\n\n\n\n<p>Ball B is initially at rest, so (v_B)\u2081n = 0 and (v_B)\u2081t = 0.<\/p>\n\n\n\n<p>During the impact, the velocity components perpendicular to the line of impact (t-axis) are conserved for each ball. Therefore, the velocity of ball B in the t-direction after impact, (v_B)\u2082t, remains 0.<\/p>\n\n\n\n<p>Along the line of impact (n-axis), we use the conservation of momentum and the coefficient of restitution (e = 0.8). Since the masses are equal, the equations simplify:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Conservation of Momentum: (v_A)\u2081n + (v_B)\u2081n = (v_A)\u2082n + (v_B)\u2082n => 4.33 = (v_A)\u2082n + (v_B)\u2082n<\/li>\n\n\n\n<li>Coefficient of Restitution: e = [(v_B)\u2082n &#8211; (v_A)\u2082n] \/ [(v_A)\u2081n &#8211; (v_B)\u2081n] => 0.8 = [(v_B)\u2082n &#8211; (v_A)\u2082n] \/ 4.33<\/li>\n<\/ol>\n\n\n\n<p>Solving these two equations simultaneously gives the velocity of ball B along the n-axis: (v_B)\u2082n = 3.897 m\/s. Since its t-component is zero, the total velocity of ball B after impact is v_B \u2248 3.90 m\/s, directed along the 30\u00b0 line of impact.<\/p>\n\n\n\n<p><strong>Part 2: Collision of Ball B with Cushion C<\/strong><\/p>\n\n\n\n<p>Next, ball B, traveling at 3.90 m\/s at 30\u00b0 to the horizontal, strikes the vertical cushion C. We use a new coordinate system: x-axis normal to the cushion (horizontal) and y-axis tangential to it (vertical).<\/p>\n\n\n\n<p>The components of ball B&#8217;s velocity just before hitting the cushion are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(v_Bx)_before = 3.897 * cos(30\u00b0) = 3.375 m\/s<\/li>\n\n\n\n<li>(v_By)_before = 3.897 * sin(30\u00b0) = 1.949 m\/s<\/li>\n<\/ul>\n\n\n\n<p>For impact with the cushion, the tangential velocity component (y-direction) is conserved: (v_By)_after = 1.949 m\/s.<\/p>\n\n\n\n<p>The normal velocity component (x-direction) is determined by the coefficient of restitution e&#8217; = 0.6:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(v_Bx)_after = e&#8217; * (v_Bx)_before = 0.6 * 3.375 = 2.025 m\/s<\/li>\n<\/ul>\n\n\n\n<p>The rebound angle \u03b8 is found using the components of the final velocity:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>tan(\u03b8) = (v_By)_after \/ (v_Bx)_after = 1.949 \/ 2.025 = 0.9625<\/li>\n\n\n\n<li>\u03b8 = arctan(0.9625) = 43.9\u00b0<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-289.jpeg\" alt=\"\" class=\"wp-image-252888\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>A billiard ball is given an initial velocity V=5M\/S\u00a0\u00a0and makes a direct impact with ball B (e=0.8), calculate the velocity of ball B and the rebound angle after it hits cushion at C (e =0.6). Mass of the balls is 0.4 kg. The Correct Answer and Explanation is: Correct Answer The velocity of ball B [&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-252886","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/252886","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=252886"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/252886\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=252886"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=252886"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=252886"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}