{"id":65169,"date":"2026-05-02T11:16:11","date_gmt":"2026-05-02T08:16:11","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65169"},"modified":"2026-10-01T13:49:12","modified_gmt":"2026-10-01T10:49:12","slug":"elastic-collision-1d-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/physics\/elastic-collision-1d-calculator\/","title":{"rendered":"Elastic Collision (1D) Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Elastic Collision 1D Calculator<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"65169\" data-favorite-title=\"Elastic Collision (1D)\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/physics\/elastic-collision-1d-calculator\/\" data-favorite-excerpt=\"Calculate the final velocities of two objects after a one-dimensional elastic collision. Enter the masses and initial velocities of both objects, and the calculator applies conservation of momentum and kinetic energy.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-physic\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/physics\/elastic-collision-1d-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65169\" data-post=\"65169\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Mass 1 (m\u2081)<\/label><div class=\"field-group\"><input type=\"number\" name=\"mass_1_m\" id=\"field-1\" step=\"any\" value=\"2\"\/><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"unit\" aria-label=\"Unit\"><option value=\"kg\">kg<\/option><option value=\"g\">g<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/div><div class=\"field is-single\" id=\"input-2\"><label for=\"field-3\">Initial Velocity 1 (u\u2081)<\/label><div class=\"field-group\"><input type=\"number\" name=\"initial_velocity_1_u\" id=\"field-3\" step=\"any\" value=\"5\"\/><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"unit\" aria-label=\"Unit\"><option value=\"m\/s\">m\/s<\/option><option value=\"km\/h\">km\/h<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/div><div class=\"field is-single\" id=\"input-3\"><label for=\"field-5\">Mass 2 (m\u2082)<\/label><div class=\"field-group\"><input type=\"number\" name=\"mass_2_m\" id=\"field-5\" step=\"any\" value=\"1\"\/><div class=\"grid grid-cols-1\"><select id=\"field-6\" name=\"unit\" aria-label=\"Unit\"><option value=\"kg\">kg<\/option><option value=\"g\">g<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/div><div class=\"field is-single\" id=\"input-4\"><label for=\"field-7\">Initial Velocity 2 (u\u2082)<\/label><div class=\"field-group\"><input type=\"number\" name=\"initial_velocity_2_u\" id=\"field-7\" step=\"any\" value=\"0\"\/><div class=\"grid grid-cols-1\"><select id=\"field-8\" name=\"unit\" aria-label=\"Unit\"><option value=\"m\/s\">m\/s<\/option><option value=\"km\/h\">km\/h<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"65169\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Final Velocity of Object 1 (v\u2081)<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-2\"><span class=\"field-title\"><span>Final Velocity of Object 2 (v\u2082)<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"65169\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Elastic Collision (1D) Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ v_1 = \\frac{m_1 - m_2}{m_1 + m_2}\\, u_1 + \\frac{2 m_2}{m_1 + m_2}\\, u_2 \\\\ v_2 = \\frac{2 m_1}{m_1 + m_2}\\, u_1 + \\frac{m_2 - m_1}{m_1 + m_2}\\, u_2 \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(m_1\\), \\(m_2\\) \u2014 masses of the two objects (kg)<\/li>\n<li>\\(u_1\\), \\(u_2\\) \u2014 initial velocities of the objects before the collision (m\/s)<\/li>\n<li>\\(v_1\\), \\(v_2\\) \u2014 final velocities of the objects after the collision (m\/s)<\/li>\n<\/ul>\n<p>These formulas follow from solving two equations simultaneously: conservation of momentum, \\(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\\), and conservation of kinetic energy, \\(\\tfrac{1}{2} m_1 u_1^2 + \\tfrac{1}{2} m_2 u_2^2 = \\tfrac{1}{2} m_1 v_1^2 + \\tfrac{1}{2} m_2 v_2^2\\). Both must hold for a perfectly elastic collision.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Elastic Collision (1D) \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a 2 kg object moving at 5 m\/s collides head-on with a 1 kg object at rest.<\/p>\n<p>\\[ m_1 = 2\\text{ kg},\\quad u_1 = 5\\text{ m\/s},\\quad m_2 = 1\\text{ kg},\\quad u_2 = 0\\text{ m\/s} \\]<\/p>\n<p>\\[ v_1 = \\frac{2 - 1}{2 + 1} \\cdot 5 + \\frac{2 \\cdot 1}{2 + 1} \\cdot 0 = \\frac{1}{3} \\cdot 5 \\approx 1.667\\text{ m\/s} \\]<\/p>\n<p>\\[ v_2 = \\frac{2 \\cdot 2}{2 + 1} \\cdot 5 + \\frac{1 - 2}{2 + 1} \\cdot 0 = \\frac{4}{3} \\cdot 5 \\approx 6.667\\text{ m\/s} \\]<\/p>\n<p><strong>Result:<\/strong> \\(v_1 \\approx 1.667\\text{ m\/s}\\) and \\(v_2 \\approx 6.667\\text{ m\/s}\\). The heavier object slows down but keeps moving forward, while the lighter object is pushed forward at a higher speed than the incoming object had.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter <strong>Mass 1 (m\u2081)<\/strong> and select its unit (kg or g).<\/li>\n<li>Enter <strong>Initial Velocity 1 (u\u2081)<\/strong> and choose m\/s or km\/h.<\/li>\n<li>Enter <strong>Mass 2 (m\u2082)<\/strong> with its unit.<\/li>\n<li>Enter <strong>Initial Velocity 2 (u\u2082)<\/strong> with its unit. Use a negative value if the second object moves in the opposite direction.<\/li>\n<li>Read the results: the <strong>Final Velocity of Object 1 (v\u2081)<\/strong> and the <strong>Final Velocity of Object 2 (v\u2082)<\/strong>, both in m\/s.<\/li>\n<\/ol>\n<p>All inputs are converted internally to kilograms and meters per second, so you can mix units freely \u2014 for example, one mass in grams and one in kilograms \u2014 and the results will still be correct.<\/p>\n<h2>What Is an Elastic Collision?<\/h2>\n<p>A collision is called <strong>elastic<\/strong> when both momentum and kinetic energy are conserved. In other words, no kinetic energy is lost to heat, sound, deformation, or vibration \u2014 the total energy of motion before the impact equals the total energy of motion after it. This is an idealization: real collisions always lose some energy, but hard, springy objects such as steel balls or billiard balls come very close to the ideal.<\/p>\n<p>The calculator handles <strong>one-dimensional (1D) collisions<\/strong>, meaning both objects move along the same straight line, either in the same direction or toward each other. Because the motion is confined to one axis, the outcome is fully described by two numbers: the final velocity of each object. A negative result simply means the object moves in the negative direction after the collision.<\/p>\n<p>Some special cases are worth recognizing. If two objects of <strong>equal mass<\/strong> collide elastically, they simply exchange velocities. If a moving object hits a much heavier object at rest, it bounces back with nearly the same speed. If it hits a much lighter object at rest, it barely slows down while the light object shoots off at up to twice the incoming speed \u2014 which is exactly what happens when a golf club strikes a ball.<\/p>\n<h2>Interpreting the Results<\/h2>\n<p>The sign of each final velocity tells you the direction of motion. In the default example above, both values are positive, so both objects move in the original direction of object 1. If a result is negative, that object rebounds and travels the opposite way. You can verify that momentum is conserved by checking that \\(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\\) \u2014 a quick way to confirm your inputs were entered correctly.<\/p>\n<h2>Assumptions and Limitations<\/h2>\n<ul>\n<li><strong>Perfectly elastic:<\/strong> no energy is lost to heat, sound, or deformation. Real collisions are at best close to elastic; for problems involving energy loss, a coefficient of restitution is needed instead.<\/li>\n<li><strong>One dimension only:<\/strong> all motion is along a single line. Two-dimensional collisions (such as glancing billiard-ball impacts) require vector components and are outside the scope of this tool.<\/li>\n<li><strong>No external forces:<\/strong> friction, gravity along the motion path, and other impulses during the brief collision are neglected \u2014 a good approximation because collisions last only milliseconds.<\/li>\n<li><strong>Point masses or non-rotating bodies:<\/strong> rotational motion is ignored. For rotating systems, see the <a href=\"https:\/\/wpcalc.com\/en\/physics\/angular-momentum-calculator\/\">Angular Momentum Calculator<\/a>.<\/li>\n<li><strong>Direction convention:<\/strong> velocities must be signed. If both objects approach each other, one of the initial velocities must be entered as negative.<\/li>\n<\/ul>\n<h2>Practical Tips<\/h2>\n<p>Elastic collision problems appear throughout introductory mechanics \u2014 Newton's cradle, air-track experiments, gas molecule collisions, and particle scattering all rely on the same equations. When solving problems by hand, always set up a clear sign convention first (usually right = positive) and stick to it for every velocity. After the collision, the <a href=\"https:\/\/wpcalc.com\/en\/physics\/average-speed-calculator\/\">average speed<\/a> of each object over a known distance, or its subsequent <a href=\"https:\/\/wpcalc.com\/en\/physics\/free-fall-calculator\/\">free fall<\/a> if it leaves a table or track, can be found with our other physics calculators, including the <a href=\"https:\/\/wpcalc.com\/en\/physics\/projectile-motion-calculator\/\">Projectile Motion Calculator<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the final velocities of two objects after a one-dimensional elastic collision. Enter the masses and initial velocities of both objects, and the calculator applies conservation of momentum and kinetic energy.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[62],"tags":[73],"class_list":["post-65169","post","type-post","status-publish","format-standard","hentry","category-physics","tag-mechanics"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65169","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=65169"}],"version-history":[{"count":4,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65169\/revisions"}],"predecessor-version":[{"id":66357,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65169\/revisions\/66357"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65169"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65169"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65169"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}