{"id":10121,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10121"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"position-of-particle-moving-in-plane","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/position-of-particle-moving-in-plane\/","title":{"rendered":"Position Of Particle Moving In Plane"},"content":{"rendered":"<div class=\"article-watermark-wrapper\">\n<div style=\"position: relative; z-index: 1;\">\n<p style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 9pt; color: #444444;\">KAPDEC&reg; | Elite STEM Learning Platform | <a href=\"https:\/\/kapdec.com\" target=\"_blank\" rel=\"noopener noreferrer\" style=\"color: #444444; text-decoration: underline;\">https:\/\/kapdec.com<\/a><\/p>\n<hr \/>\n<h2><strong>Unit: <\/strong><strong>Parametric Equations, Polar Coordinates &amp; Vector-Valued Function<\/strong><\/h2>\n<h3><strong>Chapter: <\/strong><strong>Position of Particles Moving in plane<\/strong><\/h3>\n<p><em>Reference: &#8211; Position vectors, Velocity vectors, Acceleration vectors, Scalar &amp; Vector functions, Tangent vector, Tangent lines, Normal vector, Normal lines, Curvature &amp; Arc length of a curve, Projectile Motion &amp; Relative motion, Polar equations, Application of Vector calculus<\/em><\/p>\n<p><strong>After studying this chapter, you should be able to:<\/strong><\/p>\n<ul>\n<li>Introduction to Position vectors &amp; Velocity vectors.<\/li>\n<li>Parametric &amp; Polar equations.<\/li>\n<li>Projectile &amp; Relative motion.<\/li>\n<li>Application of vector calculus to particle motion.<\/li>\n<\/ul>\n<p><strong><u>Introduction to Position &amp; Velocity Vectors<\/u><\/strong><\/p>\n<ul>\n<li>Position Vector: A position vector is a vector that describes the location of a point in space relative to a reference point or origin.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Magnitude: The magnitude of a position vector represents the distance between the reference point and the point in space.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Direction: The direction of a position vector is determined by the angle it makes with a reference axis or by its components in a coordinate system.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Velocity Vector: A velocity vector is a vector that describes the rate of change of position of an object for time.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Tangent Vector: The velocity vector at a specific point on a curve is tangent to the curve at that point.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Magnitude: The magnitude of a velocity vector represents the speed of an object, which is the rate at which it covers distance.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Direction: The direction of a velocity vector represents the direction in which the object is moving.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Derivative: The velocity vector is obtained by taking the derivative of the position vector for time.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Instantaneous Velocity: The velocity vector at a specific instant in time is known as the instantaneous velocity.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Average Velocity: The average velocity vector is the displacement vector divided by the time interval over which the displacement occurs.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Unit Vectors: Position and velocity vectors are often expressed using unit vectors, such as the i, j, and k vectors in Cartesian coordinate systems.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Components: Position and velocity vectors can be broken down into their components along the coordinate axes to simplify calculations.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Vector Addition: Position and velocity vectors can be added or subtracted using vector addition or subtraction rules.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Graphical Representation: Position and velocity vectors can be graphically represented using arrows, where the length represents magnitude, and the direction represents direction.<\/li>\n<\/ul>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"373\" src=\"https:\/\/app.kapdec.com\/questions-images\/rSKI0XlNBEEl1735869187.png?time=1735869188\" width=\"454\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><strong><u>Introduction to Vector-Valued Functions<\/u><\/strong><strong>: &#8211;<\/strong><\/p>\n<p>\u00a0<\/p>\n<ol>\n<li>Vector-valued functions are functions that map a real number (usually denoted as t) to a vector in two or three-dimensional space.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li>Vector-valued functions are often used to describe the motion of objects in space or the path of a particle.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li>The components of a vector-valued function represent the coordinates of a point in space as functions of t.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li>The derivative of a vector-valued function represents the rate of change of the position vector for t, often interpreted as the velocity vector.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li>The derivative of a vector-valued function is found by differentiating each component of the function separately.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li>The chain rule is used to find the derivatives of vector-valued functions by applying the derivative to each component and combining the results.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li>The second derivative of a vector-valued function represents the rate of change of the velocity vector and is interpreted as the acceleration vector.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li>Tangent vectors to a vector-valued function can be found by evaluating the derivative at a specific value of t, representing the direction of motion at that point.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li>The magnitude of the derivative of a vector-valued function represents the speed or magnitude of the velocity vector.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li>The arc length of a vector-valued function can be calculated using integrals and a specific formula that takes into account the derivative of the vector-valued function.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li>Vector-valued functions can be used to model various real-world scenarios, such as the trajectory of a projectile, the motion of a particle, or the path of a moving object.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li>Vector-valued functions are also essential in studying topics such as curves in space, motion in three dimensions, and the fundamental principles of calculus in higher dimensions.<\/li>\n<\/ol>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"353\" src=\"https:\/\/app.kapdec.com\/questions-images\/vFegOUlQbizl1735869187.png?time=1735869188\" width=\"401\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><strong><u>Parametric &amp; Polar Equation<\/u><\/strong><\/p>\n<p><strong>Parametric Equations:<\/strong><\/p>\n<p>\u00a0<\/p>\n<ul>\n<li>Parametric Equations: Parametric equations describe the position of a particle in terms of one or more independent parameters, typically denoted as t.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Parameter: The parameter t represents time or any other independent variable that determines the position of the particle.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Coordinate Functions: Parametric equations consist of coordinate functions that define the x and y coordinates of the particle as functions of the parameter.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Independence: Parametric equations provide a way to describe paths that are not easily represented by a single equation, such as curves and complex motions.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Graphical Representation: The graph of parametric equations often yields a curve in the plane, showing how the x and y coordinate change as the parameter varies.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Tangent Line: The tangent line to a curve defined by parametric equations can be found by taking the derivatives of the coordinate functions for the parameter.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Arc Length: Parametric equations allow for the calculation of arc length, which represents the length of the curve traveled by the particle.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Velocity Vector: The velocity vector of a particle moving along a parametric curve is obtained by differentiating the coordinate functions for the parameter.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Acceleration Vector: The acceleration vector is obtained by taking the derivative of the velocity vector for the parameter.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<p><strong>Polar Equations:<\/strong><\/p>\n<p>\u00a0<\/p>\n<ul>\n<li>Polar Coordinates: Polar equations describe the position of a particle in terms of a distance from a fixed point (pole) and an angle from a fixed reference line (usually the positive x-axis).<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Polar Equation Form: Polar equations are typically expressed in the form r = f(\u03b8), where r represents the distance and \u03b8 represents the angle.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Graphical Representation: Polar equations generate curves in polar coordinates, representing various shapes such as circles, ellipses, spirals, and more.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Symmetry: Polar equations often exhibit symmetry for the pole, the origin, or certain angles.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Tangent Line: The tangent line to a polar curve can be found by taking the derivative of the polar equation and applying trigonometric identities.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Area Enclosed: Polar equations allow for the calculation of the area enclosed by a polar curve using integration techniques.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<p><strong><u>Projectile &amp; Relative Motion<\/u><\/strong><\/p>\n<p><strong>Projectile Motion<\/strong>:<\/p>\n<p>\u00a0<\/p>\n<ul>\n<li>Projectile: A projectile is an object that is launched into the air and moves along a curved path under the influence of gravity.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Trajectory: The path followed by a projectile is called its trajectory, which is typically a parabolic curve.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Independent Motions: In projectile motion, the horizontal and vertical motions are considered independent of each other.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Horizontal Motion: The horizontal motion of a projectile is uniform and unaffected by gravity. It follows a straight line at a constant velocity.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Vertical Motion: The vertical motion of a projectile is influenced by gravity, causing it to accelerate downward at a constant rate.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Components: The motion of a projectile can be analyzed by breaking it down into its horizontal and vertical components using trigonometry.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Range: The range of a projectile is the horizontal distance it travels before hitting the ground. It depends on the initial velocity and launch angle.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Maximum Height: The maximum height reached by a projectile occurs when its vertical velocity becomes zero. It depends on the initial velocity and launch angle.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<p><strong>Relative Motion<\/strong>:<\/p>\n<p>\u00a0<\/p>\n<p>Relative Motion: Relative motion deals with the motion of one object for another moving or stationary object.<\/p>\n<p>\u00a0<\/p>\n<p>Frame of Reference: Relative motion depends on the choice of a frame of reference, which is a coordinate system used to describe the motion of objects.<\/p>\n<p>\u00a0<\/p>\n<p>Relative Velocity: The relative velocity is the velocity of one object as observed from the frame of reference of another object.<\/p>\n<p>\u00a0<\/p>\n<p>Addition of Velocities: The velocities of two objects can be added or subtracted to determine their relative velocity.<\/p>\n<p>\u00a0<\/p>\n<p>Relative Position: The relative position of two objects is the vector that connects their respective positions at a given time.<\/p>\n<p>\u00a0<\/p>\n<p>Relative Acceleration: Relative acceleration describes the change in the relative velocity of two objects over time.<\/p>\n<p>\u00a0<\/p>\n<p>Applications: Understanding relative motion is important in various fields such as physics, engineering, navigation, and transportation, where the motion of objects is considered for each other.<\/p>\n<p>\u00a0<\/p>\n<p><strong><u>Application of Vector Calculus to Particle Motion<\/u><\/strong><\/p>\n<p>\u00a0<\/p>\n<ul>\n<li>Vector-Valued Functions: Particle motion can be described by vector-valued functions, where the position vector of the particle is a function of time.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Differentiation: Vector calculus allows us to differentiate vector-valued functions to obtain velocity and acceleration vectors.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Velocity Vector: The velocity vector represents the rate of change of position for time and provides information about the particle&#8217;s speed and direction of motion.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Acceleration Vector: The acceleration vector represents the rate of change of velocity for time and describes changes in the particle&#8217;s speed and direction.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Derivatives for Time: Differentiating vector-valued functions involves taking derivatives of each component of the function for a time.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Tangent Vector: The tangent vector to a particle&#8217;s path is parallel to the velocity vector and gives the direction of motion at any given point.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Curvature: Curvature is a measure of how sharply a particle&#8217;s path is curved at a particular point and is obtained using vector calculus techniques.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Arc Length: Arc length is the length of the path traveled by a particle and can be calculated using integrals of the magnitude of the velocity vector.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Integral Calculus: Integration is used to find displacement, distance traveled, and other quantities related to particle motion.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Area Under the Curve: Integration can be used to find the area under the curve traced by a particle&#8217;s path.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Parametric Equations: Vector calculus can handle parametric equations that describe particle motion in terms of multiple variables.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Differential Equations: Differential equations arise in particle motion problems when considering the relationship between position, velocity, and acceleration.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Conservation Laws: Vector calculus is used to establish and solve conservation laws, such as the conservation of momentum or energy, in particle motion problems.<\/li>\n<\/ul>\n<p><strong>Example: &#8211; Parametric Equations<\/strong><\/p>\n<p>Consider a particle moving in the xy-plane with the following parametric equations:<\/p>\n<p>x = 3t<sup>2<\/sup> + 2t<\/p>\n<p>y = 5t &#8211; 1<\/p>\n<p>Find the position of the particle at time t = 2.<\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>To find the position of the particle at time t = 2, substitute t = 2 into the parametric equations:<\/p>\n<p>x = 3(2)<sup>2<\/sup> + 2(2) = 12 + 4 = 16<\/p>\n<p>y = 5(2) &#8211; 1 = 10 &#8211; 1 = 9<\/p>\n<p>Therefore, the position of the particle at time t = 2 is (16, 9).<\/p>\n<p><strong>Example: &#8211;<\/strong> <strong>Vector-Valued Functions<\/strong><\/p>\n<p>Consider a particle moving in the xy-plane with the following vector-valued function:<\/p>\n<p>r(t) = \u23292t, 3t<sup>2<\/sup> &#8211; 4t\u232a<\/p>\n<p>Find the position of the particle when t = 1.<\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>To find the position of the particle when t = 1, substitute t = 1 into the vector-valued function:<\/p>\n<p>\u00a0<\/p>\n<p>r(1) = \u23292(1), 3(1)<sup>2<\/sup> &#8211; 4(1)\u232a<\/p>\n<p>= \u23292, 3 &#8211; 4\u232a<\/p>\n<p>= \u23292, -1\u232a<\/p>\n<p>Therefore, the position of the particle when t = 1 is (2, -1).<\/p>\n<p><strong>Key Points<\/strong><\/p>\n<ul>\n<li>Parametric Equations: Parametric equations describe the position of a particle in terms of one or more independent variables (parameters).<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Independence: Parametric equations allow the x and y coordinates of the particle to vary independently, providing flexibility in describing complex paths.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Vector-Valued Functions: Particle motion can be represented using vector-valued functions, where the position vector is a function of the parameter(s).<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Components: Parametric equations and vector-valued functions consist of component functions that describe the x and y coordinates separately.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Time Parameter: In many cases, time is used as the parameter to represent the motion of the particle over time.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Graphical Representation: The graph of a parametric equation or vector-valued function represents the path followed by the particle in the plane.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Tangent Vector: The tangent vector to the particle&#8217;s path is obtained by differentiating the component functions for the parameter.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Tangent Line: The tangent line to the particle&#8217;s path at a given point is parallel to the tangent vector and provides the direction of motion at that point.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Curvature: Curvature measures how sharply the path of the particle is curved at a specific point and can be calculated using vector calculus techniques.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Arc Length: Arc length represents the distance traveled by the particle along its path and can be determined using integrals of the magnitude of the velocity vector.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Parametric Speed: The parametric speed of the particle is the magnitude of the velocity vector, indicating how fast the particle is moving along its path.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Real-World Applications: Parametric equations and vector-valued functions are widely used to model the motion of objects in physics, engineering, computer graphics, robotics, and other fields.<\/li>\n<\/ul>\n<p><!--kapdec-footer-start--><\/p>\n<style>.kapdec-article-footer{font-family:Arial,Helvetica,Calibri,sans-serif;color:#444;}.kapdec-footer-grid{display:flex;align-items:stretch;border:1px solid #e5e7eb;border-radius:6px;overflow:hidden;}.kapdec-footer-left,.kapdec-qr-block{flex:1 1 50%;width:50%;box-sizing:border-box;min-width:0;}.kapdec-footer-left{padding:22px 28px;border-right:1px solid #e5e7eb;}.kapdec-citation-block{line-height:1.6;font-size:9pt;color:#333;margin:0;}.kapdec-citation-block p{margin:0 0 10px 0;}.kapdec-citation-block a{color:#0066cc;text-decoration:underline;}.kapdec-copyright-block{margin-top:18px;padding-top:14px;border-top:1px solid #e5e7eb;font-size:7.5pt;color:#777;line-height:1.55;text-align:left;}.kapdec-copyright-block p{margin:0 0 5px 0;}.kapdec-qr-block{padding:22px 28px;display:flex;flex-direction:column;align-items:center;justify-content:center;text-align:center;}.kapdec-qr-label{margin:0 0 8px 0;font-size:8.5pt;font-weight:600;color:#444;line-height:1.35;letter-spacing:.02em;}.kapdec-qr-url{margin:0 0 14px 0;font-size:7.5pt;line-height:1.4;color:#777;word-break:break-word;max-width:100%;}.kapdec-qr-url a{color:#777;text-decoration:underline;}@media (max-width:640px){.kapdec-footer-grid{flex-direction:column;}.kapdec-footer-left,.kapdec-qr-block{width:100%;flex-basis:100%;border-right:none;}.kapdec-footer-left{border-bottom:1px solid #e5e7eb;}}<\/style>\n<div class=\"kapdec-article-footer\" style=\"margin-top: 28px; 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