{"id":66981,"date":"2026-10-13T10:46:48","date_gmt":"2026-10-13T07:46:48","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66981"},"modified":"2026-10-13T10:46:48","modified_gmt":"2026-10-13T07:46:48","slug":"ballistic-pendulum-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/physics\/ballistic-pendulum-calculator\/","title":{"rendered":"Ballistic Pendulum 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\">Find the Projectile&#039;s Speed from a Ballistic Pendulum Experiment<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66981\" data-favorite-title=\"Ballistic Pendulum\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/physics\/ballistic-pendulum-calculator\/\" data-favorite-excerpt=\"Calculate the speed of a projectile from a ballistic pendulum experiment. Enter the projectile and pendulum masses plus the swing height or the swing angle and pendulum length.\" 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\/ballistic-pendulum-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66981\" data-post=\"66981\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Projectile mass (m)<\/label><div class=\"term\">m<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"projectile_mass_m\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 10\" min=\"0\" value=\"10\"\/><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"unit\" aria-label=\"Unit\"><option value=\"g\" selected>g<\/option><option value=\"kg\">kg<\/option><option value=\"lb\">lb<\/option><option value=\"oz\">oz<\/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><small id=\"calc-info-1\">Mass of the projectile (e.g. a bullet or ball) that embeds itself in the pendulum block.<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-3\">Pendulum mass (M)<\/label><div class=\"term\">M<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"pendulum_mass_m\" id=\"field-3\" aria-describedby=\"calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 250\" min=\"0\" value=\"250\"\/><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"unit\" aria-label=\"Unit\"><option value=\"g\" selected>g<\/option><option value=\"kg\">kg<\/option><option value=\"lb\">lb<\/option><option value=\"oz\">oz<\/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><small id=\"calc-info-3\">Mass of the pendulum block (without the projectile). Must be much larger than the projectile for accurate results.<\/small><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-5\">How is the maximum swing measured?<\/label><div class=\"grid grid-cols-1\"><select id=\"field-5\" name=\"how_is_the_maximum_swing_measured\"><option value=\"height\">By swing height (h)<\/option><option value=\"angle\" selected>By swing angle and length (\u03b8, L)<\/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 class=\"field has-term\" id=\"input-4\" hidden><label for=\"field-6\">Maximum swing height (h)<\/label><div class=\"term\">h<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"maximum_swing_height_h\" id=\"field-6\" aria-describedby=\"calc-info-6\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 6.7\" min=\"0\" value=\"6.7\"\/><div class=\"grid grid-cols-1\"><select id=\"field-7\" name=\"unit\" aria-label=\"Unit\"><option value=\"m\">m<\/option><option value=\"cm\" selected>cm<\/option><option value=\"mm\">mm<\/option><option value=\"ft\">ft<\/option><option value=\"in\">in<\/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><small id=\"calc-info-6\">Vertical rise of the pendulum block&#039;s centre of mass from its lowest point to the highest point of the swing.<\/small><\/div><div class=\"field has-term\" id=\"input-5\" hidden><label for=\"field-8\">Maximum swing angle (\u03b8)<\/label><div class=\"term\">\u03b8<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"maximum_swing_angle_\" id=\"field-8\" aria-describedby=\"calc-unit-8 calc-info-8\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 30\" min=\"0\" max=\"180\" value=\"30\"\/><div class=\"unit\" id=\"calc-unit-8\">degrees<\/div><small id=\"calc-info-8\">Angle of the pendulum from the vertical at the top of the swing. Must be greater than 0\u00b0 and less than 180\u00b0.<\/small><\/div><div class=\"field has-term\" id=\"input-6\" hidden><label for=\"field-9\">Pendulum length (L)<\/label><div class=\"term\">L<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"pendulum_length_l\" id=\"field-9\" aria-describedby=\"calc-info-9\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 0.5\" min=\"0\" value=\"0.5\"\/><div class=\"grid grid-cols-1\"><select id=\"field-10\" name=\"unit\" aria-label=\"Unit\"><option value=\"m\" selected>m<\/option><option value=\"cm\">cm<\/option><option value=\"mm\">mm<\/option><option value=\"ft\">ft<\/option><option value=\"in\">in<\/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><small id=\"calc-info-9\">Distance from the pivot to the centre of mass of the block (with the projectile embedded).<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66981\">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> Ballistic Pendulum Results<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Projectile speed before impact<\/span><span class=\"result-term\">v<\/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>Combined speed after impact and details<\/span><span class=\"result-term\">u<\/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=\"66981\" 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>Ballistic Pendulum 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 = \\frac{m + M}{m} \\sqrt{2 g h}, \\qquad h = L (1 - \\cos\\theta) \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(v\\) \u2014 speed of the projectile immediately before impact (m\/s)<\/li>\n<li>\\(m\\) \u2014 mass of the projectile (kg)<\/li>\n<li>\\(M\\) \u2014 mass of the pendulum block (kg)<\/li>\n<li>\\(u = \\sqrt{2gh}\\) \u2014 speed of the combined projectile + block right after impact (m\/s)<\/li>\n<li>\\(g\\) \u2014 acceleration due to gravity, 9.80665 m\/s\\(^2\\)<\/li>\n<li>\\(h\\) \u2014 vertical rise of the block during the swing (m)<\/li>\n<li>\\(L\\) \u2014 pendulum length from pivot to centre of mass (m)<\/li>\n<li>\\(\\theta\\) \u2014 maximum swing angle from the vertical (degrees)<\/li>\n<\/ul>\n<p>The formula combines two physical laws: momentum is conserved during the perfectly inelastic collision, \\(mv = (m + M)u\\), and mechanical energy is conserved during the upward swing, \\(\\tfrac{1}{2}(m + M)u^2 = (m + M)gh\\). Solving both together gives the projectile's speed before impact.<\/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>Ballistic Pendulum \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a 10 g projectile embeds itself in a 250 g pendulum block. The block swings up to a maximum angle of 30\\(\\degree\\) on a pendulum of length 0.5 m. What was the projectile's speed?<\/p>\n<p>First find the vertical rise of the swing:<\/p>\n<p>\\[ h = L (1 - \\cos\\theta) = 0.5 \\times (1 - \\cos 30\\degree) = 0.06699 \\text{ m} \\]<\/p>\n<p>Then the speed of the combined mass just after impact:<\/p>\n<p>\\[ u = \\sqrt{2 g h} = \\sqrt{2 \\times 9.80665 \\times 0.06699} = 1.146 \\text{ m\/s} \\]<\/p>\n<p>Finally, apply conservation of momentum:<\/p>\n<p>\\[ v = \\frac{m + M}{m} \\, u = \\frac{0.01 + 0.25}{0.01} \\times 1.146 = 29.80 \\text{ m\/s} \\]<\/p>\n<p><strong>Result:<\/strong> the projectile was travelling at about <strong>29.80 m\/s<\/strong> (107.3 km\/h, 97.8 ft\/s) before impact. The combined mass moved at only 1.146 m\/s after impact, and 96.2% of the initial kinetic energy (about 4.27 J) was lost to heat and deformation in the collision.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter the <strong>Projectile mass (m)<\/strong> and choose its unit (g, kg, lb or oz). This is the mass of the bullet, ball or pellet that lodges in the block.<\/li>\n<li>Enter the <strong>Pendulum mass (M)<\/strong> with its unit \u2014 the mass of the block alone, without the projectile.<\/li>\n<li>In <strong>How is the maximum swing measured?<\/strong>, choose whether you recorded the vertical rise directly (<em>By swing height (h)<\/em>) or measured the angle and pendulum length (<em>By swing angle and length (\\(\\theta\\), L)<\/em>).<\/li>\n<li>Depending on your choice, enter either the <strong>Maximum swing height (h)<\/strong>, or both the <strong>Maximum swing angle (\\(\\theta\\))<\/strong> in degrees and the <strong>Pendulum length (L)<\/strong> from the pivot to the centre of mass of the loaded block.<\/li>\n<li>Read the results: the projectile speed before impact in several units, the combined speed after impact, the swing height, and a breakdown of the energy lost in the collision.<\/li>\n<\/ol>\n<h2>What Is a Ballistic Pendulum?<\/h2>\n<p>A ballistic pendulum is a classic physics device for measuring the speed of a fast projectile that is difficult to time directly. The projectile is fired horizontally into a heavy block hung from long strings or rods. The projectile embeds itself in the block, the block swings upward, and the maximum rise \u2014 read off a scale or as a swing angle \u2014 tells you how fast the projectile must have been travelling.<\/p>\n<p>The analysis works in two stages because two different conservation laws apply. During the very brief impact, kinetic energy is <em>not<\/em> conserved \u2014 much of it goes into deforming the block and the projectile \u2014 but <em>momentum<\/em> is. This gives the speed \\(u\\) of the combined mass right after impact. During the slow upward swing that follows, friction and air resistance are negligible, so <em>mechanical energy<\/em> is conserved: kinetic energy converts into gravitational potential energy, which is what the swing height measures.<\/p>\n<p>Notice that the projectile speed is always larger than the post-impact speed by the mass ratio \\((m + M)\/m\\). With a light projectile and a heavy block, this ratio is big \u2014 in the example above it is 26 \u2014 so a modest swing of the block corresponds to a fast projectile. This is also why the calculator reports a large energy loss: losing kinetic energy in the collision is not an error, it is the whole reason the pendulum measurement is possible without timing equipment.<\/p>\n<p>Historically, ballistic pendulums were used from the 18th century onward to measure muzzle velocities of firearms, and the same principle still appears in undergraduate laboratories. It remains a clear demonstration that momentum and energy conservation each hold in their own domain, and that confusing the two leads to wrong answers.<\/p>\n<h2>Assumptions, Limitations and Practical Tips<\/h2>\n<ul>\n<li><strong>Perfectly inelastic collision.<\/strong> The projectile must embed in the block and stay there. If it bounces off or passes through, momentum transfer differs and the result will be wrong.<\/li>\n<li><strong>Instantaneous impact.<\/strong> Gravity and string forces are assumed negligible during the collision, which is valid because the impact lasts only milliseconds.<\/li>\n<li><strong>Frictionless swing.<\/strong> The pivot and air drag are neglected, so the measured swing height slightly <em>underestimates<\/em> the true post-impact speed. Real pendulums lose a few percent of their swing to friction.<\/li>\n<li><strong>Pendulum length to the centre of mass.<\/strong> When using the angle method, \\(L\\) must be measured from the pivot to the centre of mass of the block <em>with the projectile embedded<\/em>, not the string length.<\/li>\n<li><strong>Angles near 180\\(\\degree\\).<\/strong> The height formula \\(h = L(1 - \\cos\\theta)\\) becomes very sensitive near 180\\(\\degree\\); small angle-measurement errors then cause large speed errors. Small swings are also problematic because the height becomes hard to measure accurately \u2014 angles between roughly 10\\(\\degree\\) and 60\\(\\degree\\) give the best precision.<\/li>\n<li><strong>Block much heavier than projectile.<\/strong> A large mass ratio amplifies measurement uncertainty in the masses; keep at least one decimal of precision on \\(m\\).<\/li>\n<\/ul>\n<p>If you are studying projectile motion further, you can compare the computed speed with the <a href=\"https:\/\/wpcalc.com\/en\/physics\/speed-of-sound-in-air-calculator\/\">speed of sound in air<\/a> to see whether the projectile is supersonic, estimate how the launch angle and pendulum geometry interact through the <a href=\"https:\/\/wpcalc.com\/en\/physics\/froude-number-calculator\/\">Froude number<\/a>, or account for velocity loss in flight with the <a href=\"https:\/\/wpcalc.com\/en\/physics\/drag-force-calculator\/\">drag force<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the speed of a projectile from a ballistic pendulum experiment. Enter the projectile and pendulum masses plus the swing height or the swing angle and pendulum length.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[62],"tags":[73],"class_list":["post-66981","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\/66981","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=66981"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66981\/revisions"}],"predecessor-version":[{"id":66982,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66981\/revisions\/66982"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66981"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66981"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66981"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}