{"id":9345,"date":"2026-06-01T21:33:48","date_gmt":"2026-06-01T21:33:48","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9345"},"modified":"2026-06-01T21:33:48","modified_gmt":"2026-06-01T21:33:48","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":"

Unit: <\/strong>Parametric Equations, Polar Coordinates & Vector-Valued Function<\/strong><\/h2>\n

Chapter: <\/strong>Position of Particles Moving in plane<\/strong><\/h3>\n

Reference: – Position vectors, Velocity vectors, Acceleration vectors, Scalar & Vector functions, Tangent vector, Tangent lines, Normal vector, Normal lines, Curvature & Arc length of a curve, Projectile Motion & Relative motion, Polar equations, Application of Vector calculus<\/em><\/p>\n

After studying this chapter, you should be able to:<\/strong><\/p>\n

    \n
  • Introduction to Position vectors & Velocity vectors.<\/li>\n
  • Parametric & Polar equations.<\/li>\n
  • Projectile & Relative motion.<\/li>\n
  • Application of vector calculus to particle motion.<\/li>\n<\/ul>\n

    Introduction to Position & Velocity Vectors<\/u><\/strong><\/p>\n

      \n
    • 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>\n

        \n
      • Magnitude: The magnitude of a position vector represents the distance between the reference point and the point in space.<\/li>\n<\/ul>\n

         <\/p>\n

          \n
        • 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>\n

            \n
          • 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>\n

              \n
            • 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>\n

                \n
              • 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>\n

                  \n
                • Direction: The direction of a velocity vector represents the direction in which the object is moving.<\/li>\n<\/ul>\n

                   <\/p>\n

                    \n
                  • Derivative: The velocity vector is obtained by taking the derivative of the position vector for time.<\/li>\n<\/ul>\n

                     <\/p>\n

                      \n
                    • Instantaneous Velocity: The velocity vector at a specific instant in time is known as the instantaneous velocity.<\/li>\n<\/ul>\n

                       <\/p>\n

                        \n
                      • 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>\n

                          \n
                        • 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>\n

                            \n
                          • Components: Position and velocity vectors can be broken down into their components along the coordinate axes to simplify calculations.<\/li>\n<\/ul>\n

                             <\/p>\n

                              \n
                            • Vector Addition: Position and velocity vectors can be added or subtracted using vector addition or subtraction rules.<\/li>\n<\/ul>\n

                               <\/p>\n

                                \n
                              • 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>\n

                                                                   \"\"<\/p>\n

                                Introduction to Vector-Valued Functions<\/u><\/strong>: –<\/strong><\/p>\n

                                 <\/p>\n

                                  \n
                                1. 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>\n

                                    \n
                                  1. 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>\n

                                      \n
                                    1. The components of a vector-valued function represent the coordinates of a point in space as functions of t.<\/li>\n<\/ol>\n

                                       <\/p>\n

                                        \n
                                      1. 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>\n

                                          \n
                                        1. The derivative of a vector-valued function is found by differentiating each component of the function separately.<\/li>\n<\/ol>\n

                                           <\/p>\n

                                            \n
                                          1. 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>\n

                                              \n
                                            1. 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>\n

                                                \n
                                              1. 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>\n

                                                  \n
                                                1. The magnitude of the derivative of a vector-valued function represents the speed or magnitude of the velocity vector.<\/li>\n<\/ol>\n

                                                   <\/p>\n

                                                    \n
                                                  1. 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>\n

                                                      \n
                                                    1. 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>\n

                                                        \n
                                                      1. 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>\n

                                                        \"\"<\/p>\n

                                                        Parametric & Polar Equation<\/u><\/strong><\/p>\n

                                                        Parametric Equations:<\/strong><\/p>\n

                                                         <\/p>\n

                                                          \n
                                                        • 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>\n

                                                            \n
                                                          • Parameter: The parameter t represents time or any other independent variable that determines the position of the particle.<\/li>\n<\/ul>\n

                                                             <\/p>\n

                                                              \n
                                                            • 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>\n

                                                                \n
                                                              • 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>\n

                                                                  \n
                                                                • 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>\n

                                                                    \n
                                                                  • 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>\n

                                                                      \n
                                                                    • 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>\n

                                                                        \n
                                                                      • 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>\n

                                                                          \n
                                                                        • Acceleration Vector: The acceleration vector is obtained by taking the derivative of the velocity vector for the parameter.<\/li>\n<\/ul>\n

                                                                           <\/p>\n

                                                                          Polar Equations:<\/strong><\/p>\n

                                                                           <\/p>\n

                                                                            \n
                                                                          • 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>\n

                                                                              \n
                                                                            • Polar Equation Form: Polar equations are typically expressed in the form r = f(θ), where r represents the distance and θ represents the angle.<\/li>\n<\/ul>\n

                                                                               <\/p>\n

                                                                                \n
                                                                              • Graphical Representation: Polar equations generate curves in polar coordinates, representing various shapes such as circles, ellipses, spirals, and more.<\/li>\n<\/ul>\n

                                                                                 <\/p>\n

                                                                                  \n
                                                                                • Symmetry: Polar equations often exhibit symmetry for the pole, the origin, or certain angles.<\/li>\n<\/ul>\n

                                                                                   <\/p>\n

                                                                                    \n
                                                                                  • 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>\n

                                                                                      \n
                                                                                    • Area Enclosed: Polar equations allow for the calculation of the area enclosed by a polar curve using integration techniques.<\/li>\n<\/ul>\n

                                                                                       <\/p>\n

                                                                                      Projectile & Relative Motion<\/u><\/strong><\/p>\n

                                                                                      Projectile Motion<\/strong>:<\/p>\n

                                                                                       <\/p>\n

                                                                                        \n
                                                                                      • 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>\n

                                                                                          \n
                                                                                        • Trajectory: The path followed by a projectile is called its trajectory, which is typically a parabolic curve.<\/li>\n<\/ul>\n

                                                                                           <\/p>\n

                                                                                            \n
                                                                                          • Independent Motions: In projectile motion, the horizontal and vertical motions are considered independent of each other.<\/li>\n<\/ul>\n

                                                                                             <\/p>\n

                                                                                              \n
                                                                                            • 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>\n

                                                                                                \n
                                                                                              • 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>\n

                                                                                                  \n
                                                                                                • 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>\n

                                                                                                    \n
                                                                                                  • 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>\n

                                                                                                      \n
                                                                                                    • 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>\n

                                                                                                      Relative Motion<\/strong>:<\/p>\n

                                                                                                       <\/p>\n

                                                                                                      Relative Motion: Relative motion deals with the motion of one object for another moving or stationary object.<\/p>\n

                                                                                                       <\/p>\n

                                                                                                      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>\n

                                                                                                      Relative Velocity: The relative velocity is the velocity of one object as observed from the frame of reference of another object.<\/p>\n

                                                                                                       <\/p>\n

                                                                                                      Addition of Velocities: The velocities of two objects can be added or subtracted to determine their relative velocity.<\/p>\n

                                                                                                       <\/p>\n

                                                                                                      Relative Position: The relative position of two objects is the vector that connects their respective positions at a given time.<\/p>\n

                                                                                                       <\/p>\n

                                                                                                      Relative Acceleration: Relative acceleration describes the change in the relative velocity of two objects over time.<\/p>\n

                                                                                                       <\/p>\n

                                                                                                      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>\n

                                                                                                      Application of Vector Calculus to Particle Motion<\/u><\/strong><\/p>\n

                                                                                                       <\/p>\n

                                                                                                        \n
                                                                                                      • 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>\n

                                                                                                          \n
                                                                                                        • Differentiation: Vector calculus allows us to differentiate vector-valued functions to obtain velocity and acceleration vectors.<\/li>\n<\/ul>\n

                                                                                                           <\/p>\n

                                                                                                            \n
                                                                                                          • Velocity Vector: The velocity vector represents the rate of change of position for time and provides information about the particle's speed and direction of motion.<\/li>\n<\/ul>\n

                                                                                                             <\/p>\n

                                                                                                              \n
                                                                                                            • Acceleration Vector: The acceleration vector represents the rate of change of velocity for time and describes changes in the particle's speed and direction.<\/li>\n<\/ul>\n

                                                                                                               <\/p>\n

                                                                                                                \n
                                                                                                              • Derivatives for Time: Differentiating vector-valued functions involves taking derivatives of each component of the function for a time.<\/li>\n<\/ul>\n

                                                                                                                 <\/p>\n

                                                                                                                  \n
                                                                                                                • Tangent Vector: The tangent vector to a particle's path is parallel to the velocity vector and gives the direction of motion at any given point.<\/li>\n<\/ul>\n

                                                                                                                   <\/p>\n

                                                                                                                    \n
                                                                                                                  • Curvature: Curvature is a measure of how sharply a particle's path is curved at a particular point and is obtained using vector calculus techniques.<\/li>\n<\/ul>\n

                                                                                                                     <\/p>\n

                                                                                                                      \n
                                                                                                                    • 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>\n

                                                                                                                        \n
                                                                                                                      • Integral Calculus: Integration is used to find displacement, distance traveled, and other quantities related to particle motion.<\/li>\n<\/ul>\n

                                                                                                                         <\/p>\n

                                                                                                                          \n
                                                                                                                        • Area Under the Curve: Integration can be used to find the area under the curve traced by a particle's path.<\/li>\n<\/ul>\n

                                                                                                                           <\/p>\n

                                                                                                                            \n
                                                                                                                          • Parametric Equations: Vector calculus can handle parametric equations that describe particle motion in terms of multiple variables.<\/li>\n<\/ul>\n

                                                                                                                             <\/p>\n

                                                                                                                              \n
                                                                                                                            • Differential Equations: Differential equations arise in particle motion problems when considering the relationship between position, velocity, and acceleration.<\/li>\n<\/ul>\n

                                                                                                                               <\/p>\n

                                                                                                                                \n
                                                                                                                              • 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

                                                                                                                                Example: – Parametric Equations<\/strong><\/p>\n

                                                                                                                                Consider a particle moving in the xy-plane with the following parametric equations:<\/p>\n

                                                                                                                                x = 3t2<\/sup> + 2t<\/p>\n

                                                                                                                                y = 5t – 1<\/p>\n

                                                                                                                                Find the position of the particle at time t = 2.<\/p>\n

                                                                                                                                Solution:<\/strong><\/p>\n

                                                                                                                                To find the position of the particle at time t = 2, substitute t = 2 into the parametric equations:<\/p>\n

                                                                                                                                x = 3(2)2<\/sup> + 2(2) = 12 + 4 = 16<\/p>\n

                                                                                                                                y = 5(2) – 1 = 10 – 1 = 9<\/p>\n

                                                                                                                                Therefore, the position of the particle at time t = 2 is (16, 9).<\/p>\n

                                                                                                                                Example: –<\/strong> Vector-Valued Functions<\/strong><\/p>\n

                                                                                                                                Consider a particle moving in the xy-plane with the following vector-valued function:<\/p>\n

                                                                                                                                r(t) = ⟨2t, 3t2<\/sup> – 4t⟩<\/p>\n

                                                                                                                                Find the position of the particle when t = 1.<\/p>\n

                                                                                                                                Solution:<\/strong><\/p>\n

                                                                                                                                To find the position of the particle when t = 1, substitute t = 1 into the vector-valued function:<\/p>\n

                                                                                                                                 <\/p>\n

                                                                                                                                r(1) = ⟨2(1), 3(1)2<\/sup> – 4(1)⟩<\/p>\n

                                                                                                                                = ⟨2, 3 – 4⟩<\/p>\n

                                                                                                                                = ⟨2, -1⟩<\/p>\n

                                                                                                                                Therefore, the position of the particle when t = 1 is (2, -1).<\/p>\n

                                                                                                                                Key Points<\/strong><\/p>\n

                                                                                                                                  \n
                                                                                                                                • Parametric Equations: Parametric equations describe the position of a particle in terms of one or more independent variables (parameters).<\/li>\n<\/ul>\n

                                                                                                                                   <\/p>\n

                                                                                                                                    \n
                                                                                                                                  • 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>\n

                                                                                                                                      \n
                                                                                                                                    • 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>\n

                                                                                                                                        \n
                                                                                                                                      • Components: Parametric equations and vector-valued functions consist of component functions that describe the x and y coordinates separately.<\/li>\n<\/ul>\n

                                                                                                                                         <\/p>\n

                                                                                                                                          \n
                                                                                                                                        • 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>\n

                                                                                                                                            \n
                                                                                                                                          • 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>\n

                                                                                                                                              \n
                                                                                                                                            • Tangent Vector: The tangent vector to the particle's path is obtained by differentiating the component functions for the parameter.<\/li>\n<\/ul>\n

                                                                                                                                               <\/p>\n

                                                                                                                                                \n
                                                                                                                                              • Tangent Line: The tangent line to the particle'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>\n

                                                                                                                                                  \n
                                                                                                                                                • 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>\n

                                                                                                                                                    \n
                                                                                                                                                  • 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>\n

                                                                                                                                                      \n
                                                                                                                                                    • 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>\n

                                                                                                                                                        \n
                                                                                                                                                      • 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","protected":false},"excerpt":{"rendered":"

                                                                                                                                                        Unit: Parametric Equations, Polar Coordinates & Vector-Valued Function Chapter: Position of Particles Moving in plane Reference: – Position vectors, Velocity vectors, Acceleration vectors, Scalar & Vector functions, Tangent vector, Tangent lines, Normal vector, Normal lines, Curvature & Arc length of a curve, Projectile Motion & Relative motion, Polar equations, Application of Vector calculus After studying […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[627],"tags":[],"class_list":["post-9345","post","type-post","status-publish","format-standard","hentry","category-ap-calculus-bc"],"_links":{"self":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9345","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/comments?post=9345"}],"version-history":[{"count":0,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9345\/revisions"}],"wp:attachment":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/media?parent=9345"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/categories?post=9345"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/tags?post=9345"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}