{"id":8371,"date":"2020-07-11T08:15:34","date_gmt":"2020-07-11T12:15:34","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me274\/?page_id=8371"},"modified":"2024-10-05T18:09:21","modified_gmt":"2024-10-05T22:09:21","slug":"pool-ball-impacts","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/pool-ball-impacts\/","title":{"rendered":"Pool ball impacts"},"content":{"rendered":"<p><strong><em>Direct impact of pool balls<\/em><\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7686 aligncenter\" style=\"border: 1px solid #000000\" src=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2020\/07\/figure-scaled.jpg\" alt=\"\" width=\"265\" height=\"143\" \/><\/p>\n<p>Ball A strikes a <em>stationary<\/em> ball B (with the balls having masses of\u00a0<em>m<\/em><sub>A<\/sub> and <em>m<\/em><sub>B<\/sub>, respectively) with an initial speed of vA1, and with the velocity of A being directed at the center of ball B. The conservation of linear momentum in the n-direction for A and B together, and the coefficient of restitution (COR) give the following two equations:<\/p>\n<div class=\"page\" title=\"Page 3\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p><em>m<\/em><sub>A<\/sub><em>v<\/em><sub>A1<\/sub> = <em>m<\/em><sub>A<\/sub><em>v<\/em><sub>A2<\/sub> + <em>m<\/em><sub>B<\/sub><em>v<\/em><sub>B2<\/sub><br \/>\n<em>v<\/em><sub>A2<\/sub>\u00a0= <em>v<sub>B<\/sub><\/em><sub>2<\/sub>\u00a0&#8211;\u00a0<em>ev<\/em><sub>A1<\/sub><\/p>\n<p>The above equations can be solved for <em>v<\/em><sub>A2<\/sub>\u00a0and\u00a0<em>v<sub>B<\/sub><\/em><sub>2<\/sub>\u00a0for given values of masses and COR.<\/p>\n<p><span style=\"text-decoration: underline\"><em>Influence of COR<\/em><\/span><br \/>\nThe influence of the value of COR is demonstrated in the following set of four animations for <em>m<\/em><sub>A<\/sub> = <em>m<\/em><sub>B<\/sub>. Please observe the following from these results:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" style=\"border: 1px solid #000000\" src=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2020\/07\/cor_direct.gif\" width=\"621\" height=\"313\" \/><\/p>\n<ul>\n<li>For <em>e<\/em> = 0, we see that A and B stick together after impact. Is this consistent with the above equations?<\/li>\n<li>For <em>e<\/em> = 1, we see that A stops after impact. Is this consistent with the above equations? What is the speed of B after impact for this value of COR?<\/li>\n<li>For <em>e<\/em> = 0.5 and <em>e<\/em> = 0.8,\u00a0we see that the speeds of A and B are between that found for <em>e<\/em> = 0 and <em>e<\/em> = 1 above. Is this consistent with the above equations?<\/li>\n<\/ul>\n<\/div>\n<p><span style=\"text-decoration: underline\"><em>Influence of mass<\/em><\/span><br \/>\nThe influence of the value of mass is demonstrated in the following set of four animations for <i>e = 1<\/i>. Do the animations below agree with the results of analysis using the above equations? Do they agree with your intuition?<\/p>\n<\/div>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" style=\"border: 1px solid #000000\" src=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2020\/07\/mass_direct.gif\" width=\"621\" height=\"313\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong><em>Oblique impact of pool balls<\/em><\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7686 aligncenter\" style=\"border: 1px solid #000000\" src=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2020\/07\/figure02-scaled.jpg\" alt=\"\" width=\"269\" height=\"241\" \/><\/p>\n<p>Ball A strikes a stationary ball B (with the balls having identical masses of\u00a0<em>m<\/em>) with an initial speed of vA1, and with the velocity of A NOT directed at the center of ball B. The conservation of linear momentum in the\u00a0<em>t<\/em>-direction for A and B individually, and in the <em>n<\/em>-direction for A and B together, along with the coefficient of restitution (COR) give the following four equations:<\/p>\n<div class=\"page\" title=\"Page 3\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p><em>v<\/em><sub>At2<\/sub>\u00a0=\u00a0<em>v<\/em><sub>At1<\/sub><br \/>\n<em>v<sub>Bt2<\/sub>\u00a0= v<sub>Bt1<\/sub>\u00a0= 0<br \/>\nv<sub>An1<\/sub> = v<sub>An2<\/sub> + v<sub>Bn2<\/sub><br \/>\n<\/em><em>v<\/em><sub>An2<\/sub>\u00a0= <em>v<sub>Bn<\/sub><\/em><sub>2<\/sub>\u00a0&#8211;\u00a0<em>ev<\/em><sub>An1<\/sub><\/p>\n<p>The above equations can be solved for the components of\u00a0<em>v<\/em><sub>A2<\/sub>\u00a0and\u00a0<em>v<sub>B<\/sub><\/em><sub>2<\/sub>\u00a0for given values of COR.<\/p>\n<p><span style=\"text-decoration: underline\"><em>Influence of COR<\/em><\/span><br \/>\nThe influence of the value of COR is demonstrated in the following set of four animations. Are the results from the above equations consistent with that seen in the animations? Consider the two extreme cases of <em>e<\/em> = 0 and <em>e<\/em> = 1. Why do the two balls not stick for <em>e<\/em> = 0 as was seen in the direct impact case? Why does A not stop after impact for <em>e<\/em> = 1 as was seen in the direct impact case?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" style=\"border: 1px solid #000000\" src=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2020\/07\/cor_oblique.gif\" width=\"621\" height=\"313\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Direct impact of pool balls Ball A strikes a stationary ball B (with the balls having masses of\u00a0mA and mB, respectively) with an initial speed of vA1, and with the velocity of A being directed at the center of ball B. The conservation of linear momentum in the n-direction for A and B together, and &hellip; <a href=\"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/pool-ball-impacts\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Pool ball impacts<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":10,"featured_media":0,"parent":14,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-8371","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/8371","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/users\/10"}],"replies":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/comments?post=8371"}],"version-history":[{"count":10,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/8371\/revisions"}],"predecessor-version":[{"id":8389,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/8371\/revisions\/8389"}],"up":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/14"}],"wp:attachment":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/media?parent=8371"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}