{"id":616,"date":"2023-03-13T04:07:38","date_gmt":"2023-03-12T22:37:38","guid":{"rendered":"http:\/\/physicscatalyst.com\/article\/?p=616"},"modified":"2023-03-13T04:07:42","modified_gmt":"2023-03-12T22:37:42","slug":"acceleration-curvilinear-motion","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/","title":{"rendered":"Acceleration in a curvilinear motion"},"content":{"rendered":"\n<p>The motion of an object moving in a curved path is called curvilinear motion. It can be both two dimension or three -dimension. We will be discussing&nbsp;Acceleration in a curvilinear motion in this post<\/p>\n\n\n\n<p>Example: A stone thrown into the air at an angle.<\/p>\n\n\n\n<p>It is more complex than rectilinear motion<\/p>\n\n\n\n<p><strong>We know for Rectilinear motion (a particle moving in x -direction )<\/strong><\/p>\n\n\n\n<p>Velocity is given by<\/p>\n\n\n\n<p>$v=\\frac{dx}{dt}$<\/p>\n\n\n\n<p>Acceleration is given by<\/p>\n\n\n\n<p>$a=\\frac{dv}{dt}$<\/p>\n\n\n\n<p><strong>Now lets discuss this for Curvilinear motion in Three dimension&nbsp;<\/strong><\/p>\n\n\n\n<p>Position of particle moving in&nbsp;Three dimension&nbsp; will be given by<\/p>\n\n\n\n<p>$\\mathbf{r}=\\left (x \\right )\\mathbf{i}+\\left (y\\right )\\mathbf{j}+\\left ( z \\right )\\mathbf{k}$<\/p>\n\n\n\n<p>Velocity will be given as<\/p>\n\n\n\n<p>$\\mathbf{v}=\\left (\\frac {dx}{dt} \\right )\\mathbf{i}+\\left (\\frac {dy}{dt}\\right )\\mathbf{j}+\\left ( \\frac {dz}{dt} \\right )\\mathbf{k}$<\/p>\n\n\n\n<p>The components of the Velocity along X,Y ,Z are<br>\n$v_{x}=\\frac{dx}{dt}$<br>\n$v_{y}=\\frac{dy}{dt}$<br>\n$v_{z}=\\frac{dz}{dt}$<\/p>\n\n\n\n<p>Magnitude of Velocity<\/p>\n\n\n\n<p><a href=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2012\/04\/magnitude_velocity.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4388 size-full\" src=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2012\/04\/magnitude_velocity.png\" alt=\"Acceleration in a curvilinear motion\" width=\"232\" height=\"65\"><\/a><br>\nNow Acceleration in a curvilinear motion is given by<br>\n$\\mathbf{a}=\\left ( \\frac {dv_{x}}{dt} \\right )\\mathbf{i}+\\left ( \\frac {dv_{y}}{dt} \\right )\\mathbf{j}+\\left ( \\frac {dv_{z}}{dt} \\right )\\mathbf{k}$<br>\nThe components of the acceleration along X,Y ,Z are<br>\n$a_{x}=\\frac{dv_{x}}{dt}$<br>\n$a_{y}=\\frac{dv_{y}}{dt}$<br>\n$a_{z}=\\frac{dv_{z}}{dt}$<\/p>\n\n\n\n<p>Magnitude of acceleration<\/p>\n\n\n\n<figure class=\"wp-block-image\"><a href=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2012\/04\/magnitude_acceleration.png\"><img loading=\"lazy\" decoding=\"async\" width=\"232\" height=\"65\" src=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2012\/04\/magnitude_acceleration.png\" alt=\"Acceleration in a curvilinear motion\" class=\"wp-image-4389\"\/><\/a><\/figure>\n\n\n\n<p>The direction of the velocity is always tangent to the curve. For acceleration,The acceleration vector has the same direction as instantaneous change in velocity. Since the direction of the velocity changes in the direction in which the curve bends ,the acceleration is always pointing towards concavity of the curve<\/p>\n\n\n\n<p><strong>we can write this for two dimensional motion also.It is also called Plane curvilinear motion<\/strong><\/p>\n\n\n\n<p>Position of particle moving in&nbsp;Two dimension&nbsp; will be given by<\/p>\n\n\n\n<p>$\\mathbf{r}=\\left (x \\right )\\mathbf{i}+\\left (y\\right )\\mathbf{j}$<\/p>\n\n\n\n<p>Velocity will be given as<\/p>\n\n\n\n<p>$\\mathbf{v}=\\left (\\frac {dx}{dt} \\right )\\mathbf{i}+\\left (\\frac {dy}{dt}\\right )\\mathbf{j}$<\/p>\n\n\n\n<p>The components of the Velocity along X,Y&nbsp; are<br>\n$v_{x}=\\frac{dx}{dt}$<br>\n$v_{y}=\\frac{dy}{dt}$<\/p>\n\n\n\n<p>Magnitude of Velocity<\/p>\n\n\n\n<figure class=\"wp-block-image\"><a href=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2012\/04\/magnitude_velocity-2D.png\"><img loading=\"lazy\" decoding=\"async\" width=\"232\" height=\"65\" src=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2012\/04\/magnitude_velocity-2D.png\" alt=\"Acceleration in a curvilinear motion\" class=\"wp-image-4390\"\/><\/a><\/figure>\n\n\n\n<p>Now Acceleration in a curvilinear motion is given by<br>\n$\\mathbf{a}=\\left ( \\frac {dv_{x}}{dt} \\right )\\mathbf{i}+\\left ( \\frac {dv_{y}}{dt} \\right )\\mathbf{j}$<br>\nThe components of the acceleration along X,Y are<br>\n$a_{x}=\\frac{dv_{x}}{dt}$<br>\n$a_{y}=\\frac{dv_{y}}{dt}$<\/p>\n\n\n\n<p>Magnitude of acceleration<\/p>\n\n\n\n<figure class=\"wp-block-image\"><a href=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2012\/04\/magnitude_acceleration-2d.png\"><img loading=\"lazy\" decoding=\"async\" width=\"232\" height=\"65\" src=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2012\/04\/magnitude_acceleration-2d.png\" alt=\"Acceleration in a curvilinear motion\" class=\"wp-image-4391\"\/><\/a><\/figure>\n\n\n\n<p><strong>Example<\/strong><\/p>\n\n\n\n<p>A body is projected horizontally from a point above the ground.The motion of body is defined as<br>\nx=2t<br>\ny=2t<sup>2<\/sup><br>\nwhere x and y are horizontal and vertical displacement respectivley at time t.Which one of the following is true<br>\na. The trajectory of the body is a parabola<br>\nb. The trajectory of the body is a straight line<br>\nc. the velocity vector at point t is 2<strong>i<\/strong>+4t<strong>j<\/strong><br>\nd. the acceleration vector at time t is 4<strong>j<\/strong><br>\n<strong>Solution<\/strong><\/p>\n\n\n\n<p>Given<br>\nx=2t<br>\ny=2t<sup>2<\/sup><br>\nEliminating t we get<br>\ny=x<sup>2<\/sup>\/2<br>\nSo it is parabola<br>\nV<sub>x<\/sub>=dx\/dt=2<br>\nv<sub>y<\/sub>=dy\/dt=4t<br>\nSo velocity at any time t is given by<br>\nv=2<strong>i<\/strong>+4t<strong>j<\/strong><br>\nNow similary<br>\na<sub>x<\/sub>=dV<sub>x<\/sub>\/dt=0<br>\na<sub>y<\/sub>=dV<sub>y<\/sub>\/dt=4<br>\nSo acceleration vector is a=4<strong>j<\/strong><\/p>\n\n\n\n<p>The acceleration in curvilinear motion can also be expressed in terms of two components<\/p>\n\n\n\n<p>a) The tangential component $a_{T}$&nbsp;which is called as tangential acceleration. It is expressed as change in magnitude of the velocity<\/p>\n\n\n\n<p>b) The normal component $a_{N}$&nbsp;which is called as normal or centripetal acceleartion. This is expressed as the change in direction of the velocity<br>\n$a_{T}=\\frac{dv}{dt}$<br>\n$a_{N}=\\frac{v^2}{R}$<br>\nWhere R is the radius of curvature of the path<\/p>\n\n\n\n<p><strong>Related Articles<\/strong><\/p>\n\n\n\n<p><a href=\"https:\/\/physicscatalyst.com\/article\/acceleration-formula\/\" rel=\"noopener\">Acceleration formula Explained with Examples<\/a><\/p>\n\n\n\n<p><a href=\"https:\/\/physicscatalyst.com\/article\/circular-motion-physics-detailed-explanation\/\" rel=\"noopener\">Circular Motion Physics : Detailed explanation<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The acceleration vector has the same direction as instantanous change in velocity. Since the direction of the velocity changes in the direction in which the curve bends ,the acceleration is always pointing towards concavity of the curve<br \/>\nThe acceleration in curvilinear motion can also be expressed in terms of two components<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"default","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":"","theme-transparent-header-meta":"default","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":[14,37],"tags":[],"class_list":["post-616","post","type-post","status-publish","format-standard","hentry","category-physics","category-tips-and-tricks"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Acceleration in a curvilinear motion - physicscatalyst&#039;s Blog<\/title>\n<meta name=\"description\" content=\"Check out this page for detailed explanation on Acceleration in a curvilinear motion both 2-D (plane curvilinear motion) and 3-D.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Acceleration in a curvilinear motion - physicscatalyst&#039;s Blog\" \/>\n<meta property=\"og:description\" content=\"Check out this page for detailed explanation on Acceleration in a curvilinear motion both 2-D (plane curvilinear motion) and 3-D.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/\" \/>\n<meta property=\"og:site_name\" content=\"physicscatalyst&#039;s Blog\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/PhysicsCatalyst\" \/>\n<meta property=\"article:author\" content=\"https:\/\/www.facebook.com\/PhysicsCatalyst\" \/>\n<meta property=\"article:published_time\" content=\"2023-03-12T22:37:38+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-03-12T22:37:42+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2012\/04\/magnitude_velocity.png\" \/>\n<meta name=\"author\" content=\"physicscatalyst\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"physicscatalyst\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutes\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Acceleration in a curvilinear motion - physicscatalyst&#039;s Blog","description":"Check out this page for detailed explanation on Acceleration in a curvilinear motion both 2-D (plane curvilinear motion) and 3-D.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/","og_locale":"en_US","og_type":"article","og_title":"Acceleration in a curvilinear motion - physicscatalyst&#039;s Blog","og_description":"Check out this page for detailed explanation on Acceleration in a curvilinear motion both 2-D (plane curvilinear motion) and 3-D.","og_url":"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/","og_site_name":"physicscatalyst&#039;s Blog","article_publisher":"https:\/\/www.facebook.com\/PhysicsCatalyst","article_author":"https:\/\/www.facebook.com\/PhysicsCatalyst","article_published_time":"2023-03-12T22:37:38+00:00","article_modified_time":"2023-03-12T22:37:42+00:00","og_image":[{"url":"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2012\/04\/magnitude_velocity.png","type":"","width":"","height":""}],"author":"physicscatalyst","twitter_misc":{"Written by":"physicscatalyst","Est. reading time":"4 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/#article","isPartOf":{"@id":"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/"},"author":{"name":"physicscatalyst","@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/person\/9b302efdc9b32e459cb1e61ab7506d3f"},"headline":"Acceleration in a curvilinear motion","datePublished":"2023-03-12T22:37:38+00:00","dateModified":"2023-03-12T22:37:42+00:00","mainEntityOfPage":{"@id":"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/"},"wordCount":659,"commentCount":1,"publisher":{"@id":"https:\/\/physicscatalyst.com\/article\/#organization"},"image":{"@id":"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/#primaryimage"},"thumbnailUrl":"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2012\/04\/magnitude_velocity.png","articleSection":["Physics","Tips and tricks"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/","url":"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/","name":"Acceleration in a curvilinear motion - physicscatalyst&#039;s Blog","isPartOf":{"@id":"https:\/\/physicscatalyst.com\/article\/#website"},"primaryImageOfPage":{"@id":"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/#primaryimage"},"image":{"@id":"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/#primaryimage"},"thumbnailUrl":"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2012\/04\/magnitude_velocity.png","datePublished":"2023-03-12T22:37:38+00:00","dateModified":"2023-03-12T22:37:42+00:00","description":"Check out this page for detailed explanation on Acceleration in a curvilinear motion both 2-D (plane curvilinear motion) and 3-D.","breadcrumb":{"@id":"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/#primaryimage","url":"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2012\/04\/magnitude_velocity.png","contentUrl":"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2012\/04\/magnitude_velocity.png","width":232,"height":65},{"@type":"BreadcrumbList","@id":"https:\/\/physicscatalyst.com\/article\/acceleration-curvilinear-motion\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/physicscatalyst.com\/article\/"},{"@type":"ListItem","position":2,"name":"Physics","item":"https:\/\/physicscatalyst.com\/article\/physics\/"},{"@type":"ListItem","position":3,"name":"Acceleration in a curvilinear motion"}]},{"@type":"WebSite","@id":"https:\/\/physicscatalyst.com\/article\/#website","url":"https:\/\/physicscatalyst.com\/article\/","name":"physicscatalyst's Blog","description":"Learn free for class 9th, 10th science\/maths , 12th and IIT-JEE Physics and maths.","publisher":{"@id":"https:\/\/physicscatalyst.com\/article\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/physicscatalyst.com\/article\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/physicscatalyst.com\/article\/#organization","name":"physicscatalyst","url":"https:\/\/physicscatalyst.com\/article\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/logo\/image\/","url":"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2024\/08\/cropped-logo-1.jpg","contentUrl":"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2024\/08\/cropped-logo-1.jpg","width":96,"height":96,"caption":"physicscatalyst"},"image":{"@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/PhysicsCatalyst","https:\/\/x.com\/physicscatalyst","https:\/\/www.youtube.com\/user\/thephysicscatalyst","https:\/\/www.instagram.com\/physicscatalyst\/"]},{"@type":"Person","@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/person\/9b302efdc9b32e459cb1e61ab7506d3f","name":"physicscatalyst","sameAs":["https:\/\/physicscatalyst.com","https:\/\/www.facebook.com\/PhysicsCatalyst","https:\/\/x.com\/physicscatalyst"]}]}},"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false,"shareaholic-thumbnail":false},"uagb_author_info":{"display_name":"physicscatalyst","author_link":"https:\/\/physicscatalyst.com\/article\/author\/physicscatalyst\/"},"uagb_comment_info":1,"uagb_excerpt":"The acceleration vector has the same direction as instantanous change in velocity. Since the direction of the velocity changes in the direction in which the curve bends ,the acceleration is always pointing towards concavity of the curve The acceleration in curvilinear motion can also be expressed in terms of two components","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/616","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=616"}],"version-history":[{"count":1,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/616\/revisions"}],"predecessor-version":[{"id":6857,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/616\/revisions\/6857"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=616"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=616"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=616"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}