{"id":65894,"date":"2026-09-30T13:56:39","date_gmt":"2026-09-30T10:56:39","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65894"},"modified":"2026-10-13T11:49:09","modified_gmt":"2026-10-13T08:49:09","slug":"banked-curve-speed-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/physics\/banked-curve-speed-calculator\/","title":{"rendered":"Banked Curve Speed 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 Safe Speed on a Banked Turn<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"65894\" data-favorite-title=\"Banked Curve\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/physics\/banked-curve-speed-calculator\/\" data-favorite-excerpt=\"Calculate the ideal speed for a banked curve from its radius and banking angle, plus the maximum and minimum safe speeds when tyre friction is included. Enter radius, angle and friction coefficient to get results in km\/h, mph and m\/s.\" 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\/banked-curve-speed-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65894\" data-post=\"65894\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single has-term\" id=\"input-1\"><label for=\"field-1\">Curve radius (r)<\/label><div class=\"term\">r<\/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=\"curve_radius_r\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 100\" min=\"0\" value=\"100\"\/><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"unit\" aria-label=\"Unit\"><option value=\"m\" selected>m<\/option><option value=\"ft\">ft<\/option><option value=\"mi\">miles<\/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\">Radius of the circular path measured to the middle of the road lane.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-2\"><label for=\"field-3\">Banking 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><div class=\"field-group\"><input type=\"text\" name=\"banking_angle_\" id=\"field-3\" aria-describedby=\"calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 15\" min=\"0\" max=\"89.9\" value=\"15\"\/><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"unit\" aria-label=\"Unit\"><option value=\"deg\" selected>degrees<\/option><option value=\"rad\">radians<\/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\">Tilt of the road surface from horizontal. 0\u00b0 is a flat road, 90\u00b0 would be a vertical wall.<\/small><\/div><div class=\"field is-checkbox is-single\" id=\"input-3\"><input type=\"checkbox\" id=\"field-5\" value=\"1\" checked=\"checked\"\/><label for=\"field-5\">Include road friction (\u03bc)<\/label><\/div><div class=\"field is-single has-term\" id=\"input-4\" hidden><label for=\"field-6\">Friction coefficient (\u03bc)<\/label><div class=\"term\">\u03bc<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"friction_coefficient_\" id=\"field-6\" aria-describedby=\"calc-info-6\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 0.35\" min=\"0\" value=\"0.35\"\/><small id=\"calc-info-6\">Static friction coefficient between tyres and road. Dry asphalt \u2248 0.7\u20130.9, wet asphalt \u2248 0.4\u20130.5, gravel \u2248 0.3\u20130.6, snow\/ice \u2248 0.05\u20130.3.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"65894\">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> Banked Curve Speed Results<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Ideal speed (no friction needed)<\/span><span class=\"result-term\">v\u2080<\/span><\/span><span class=\"field-value\" id=\"result-1\" data-unit=\"m\/s\"><\/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>Maximum and minimum speed with friction<\/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=\"65894\" 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>Banked Curve Speed 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\\[\n\\begin{aligned}\nv_0 &= \\sqrt{r \\, g \\, \\tan\\theta} \\\\\nv_{\\max} &= \\sqrt{ r \\, g \\, \\frac{\\tan\\theta + \\mu}{1 - \\mu \\tan\\theta} } \\\\\nv_{\\min} &= \\sqrt{ r \\, g \\, \\frac{\\tan\\theta - \\mu}{1 + \\mu \\tan\\theta} }\n\\end{aligned}\n\\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(v_0\\) \u2014 ideal speed on the curve with no friction needed (m\/s)<\/li>\n<li>\\(v_{\\max}\\), \\(v_{\\min}\\) \u2014 upper and lower speed limits when friction is included (m\/s)<\/li>\n<li>\\(r\\) \u2014 radius of the curve (m)<\/li>\n<li>\\(g\\) \u2014 gravitational acceleration, \\(9.80665\\ \\text{m\/s}^2\\)<\/li>\n<li>\\(\\theta\\) \u2014 banking angle of the road surface<\/li>\n<li>\\(\\mu\\) \u2014 coefficient of static friction between tyres and road<\/li>\n<\/ul>\n<p>The first formula gives the speed at which the horizontal component of the normal force alone supplies the required centripetal force. The other two give the speed range within which friction can keep the car from sliding up or down the banking.<\/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>Banked Curve Speed \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a car takes a curve of radius 100 m banked at 15\u00b0 on dry asphalt with a friction coefficient of 0.35.<\/p>\n<p>\\[ v_0 = \\sqrt{100 \\times 9.80665 \\times \\tan 15^\\circ} = \\sqrt{100 \\times 9.80665 \\times 0.26795} = \\sqrt{262.62} \\approx 16.20\\ \\text{m\/s} \\approx 58.33\\ \\text{km\/h} \\]<\/p>\n<p>With friction, the maximum speed is:<\/p>\n<p>\\[ v_{\\max} = \\sqrt{100 \\times 9.80665 \\times \\frac{0.26795 + 0.35}{1 - 0.35 \\times 0.26795}} = \\sqrt{668.6} \\approx 25.86\\ \\text{m\/s} \\approx 93.09\\ \\text{km\/h} \\]<\/p>\n<p>Because \\(\\mu \\ge \\tan\\theta\\) (0.35 \u2265 0.268), there is no minimum speed \u2014 friction alone holds the car on the banking even at rest.<\/p>\n<p><strong>Result:<\/strong> ideal speed \u2248 58.3 km\/h (36.2 mph); maximum speed \u2248 93.1 km\/h (57.8 mph); no minimum speed.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Banked Curve Speed Calculator<\/h2>\n<ol>\n<li>Enter the <strong>Curve radius (r)<\/strong> and choose its unit \u2014 metres, feet or miles. The radius should be measured to the middle of your lane.<\/li>\n<li>Enter the <strong>Banking angle (\u03b8)<\/strong> in degrees or radians. This is the tilt of the road surface from horizontal; a flat road is 0\u00b0.<\/li>\n<li>Leave <strong>Include road friction (\u03bc)<\/strong> ticked if you also want the maximum and minimum speeds, and enter the <strong>Friction coefficient (\u03bc)<\/strong> for your road surface (dry asphalt \u2248 0.7\u20130.9, wet asphalt \u2248 0.4\u20130.5, snow\/ice \u2248 0.05\u20130.3).<\/li>\n<li>Untick the friction checkbox to compute only the ideal, frictionless speed.<\/li>\n<li>Read the results: the ideal speed appears in km\/h, mph, m\/s and ft\/s, followed by the maximum and minimum speeds when friction is included.<\/li>\n<\/ol>\n<h2>What Is a Banked Curve and Why Do Roads Have One?<\/h2>\n<p>A banked curve is a turn in which the road surface is tilted toward the inside of the curve, like the corners of a velodrome or a NASCAR speedway. When a vehicle follows a circular path, it needs a sideways <a href=\"https:\/\/wpcalc.com\/en\/physics\/centripetal-force-calculator\/\">centripetal force<\/a> pointing toward the centre of the curve. On a flat road this force can only come from tyre friction, which is limited \u2014 and disappears entirely on ice.<\/p>\n<p>Banking the road changes the direction of the normal force so that it has a horizontal component. At one particular speed, the ideal speed \\(v_0\\), that horizontal component is exactly equal to the required centripetal force \\(m v^2 \/ r\\). At this speed a car could round the curve even on a perfectly frictionless surface, which is why highway engineers choose the banking angle to match the expected design speed of the road. Interchange ramps, railway tracks and racetrack corners are all built this way.<\/p>\n<p>When friction is present, it widens the safe operating range. Above the ideal speed, the car tends to slide up and out of the curve; friction pointing down the slope adds to the banking and raises the limit to \\(v_{\\max}\\). Below the ideal speed, the car tends to slide down the banking; friction pointing up the slope prevents that, but only down to \\(v_{\\min}\\). Note that when \\(\\mu \\ge \\tan\\theta\\), even a stationary car is held by friction and the minimum speed is zero. If \\(\\mu \\tan\\theta \\ge 1\\), the formula predicts no theoretical maximum speed \u2014 although in practice other factors (engine power, aerodynamics, driver skill) will limit you long before.<\/p>\n<h2>Assumptions and Practical Tips<\/h2>\n<ul>\n<li><strong>Point-mass model:<\/strong> the vehicle is treated as a single particle. Real cars have suspension, body roll and weight transfer, which shift the effective banking angle slightly.<\/li>\n<li><strong>Static friction:<\/strong> the formulas assume the tyres are gripping, not skidding. Once the tyres break loose, the coefficient drops and the limits no longer apply.<\/li>\n<li><strong>Uniform conditions:<\/strong> the radius, banking angle and friction are assumed constant around the whole curve. Real curves often have changing radius (spiral transitions) and patchy surfaces.<\/li>\n<li><strong>Standard gravity:<\/strong> the calculation uses \\(g = 9.80665\\ \\text{m\/s}^2\\); local variation is negligible for this purpose.<\/li>\n<li><strong>Safety margin:<\/strong> never treat \\(v_{\\max}\\) as a target. Road designers and racing teams keep a substantial margin below the theoretical limit because friction drops sharply in rain, with worn tyres or with oil and gravel on the surface.<\/li>\n<\/ul>\n<p>The same physics that governs a banked road also appears in other Newton's-law problems, such as the <a href=\"https:\/\/wpcalc.com\/en\/physics\/atwood-machine-calculator\/\">Atwood machine<\/a>, where several forces must be resolved along chosen axes before applying \\(F = ma\\). If you want to explore the force side of the problem rather than the speed, try the <a href=\"https:\/\/wpcalc.com\/en\/physics\/centripetal-force-calculator\/\">centripetal force calculator<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the ideal speed for a banked curve from its radius and banking angle, plus the maximum and minimum safe speeds when tyre friction is included. Enter radius, angle and friction coefficient to get results in km\/h, mph and m\/s.<\/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":[86],"class_list":["post-65894","post","type-post","status-publish","format-standard","hentry","category-physics","tag-rotational-motion"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65894","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=65894"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65894\/revisions"}],"predecessor-version":[{"id":67079,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65894\/revisions\/67079"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65894"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65894"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65894"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}