[{"data":1,"prerenderedAt":42},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fkinematics\u002Fparticle-kinematics\u002Fparticle-kinematics-methods-describing-motion":3},{"topic":4,"trail":18,"children":32,"tasks":33,"alternates":39},{"id":5,"name":6,"locale":7,"path":8,"seo_title":9,"seo_description":10,"seo_text":11,"content_html":12,"content_chunks":13},212,"Particle Kinematics. Methods of Describing Motion","en","theoretical-mechanics\u002Fkinematics\u002Fparticle-kinematics\u002Fparticle-kinematics-methods-describing-motion","Particle Kinematics and Methods of Describing Motion","Particle kinematics: vector, Cartesian and path-coordinate descriptions of motion, trajectory and equations of motion.","This topic introduces particle kinematics and the main methods of describing motion: vector, Cartesian-coordinate and path-coordinate descriptions, including trajectories and equations of motion.","\u003Cp>\u003Cstrong>Particle kinematics\u003C\u002Fstrong> describes the motion of a particle without considering the forces that cause it. The basic objective is to specify particle position as a function of time and determine its trajectory, velocity, and acceleration relative to a selected reference frame.\u003C\u002Fp>\u003Ch2>Reference frame and equations of motion\u003C\u002Fh2>\u003Cp>A motion description requires a reference body, an associated coordinate system, and a measure of time. The equations of motion must determine the particle position at any instant within the interval being studied.\u003C\u002Fp>\u003Ch2>Main methods of describing motion\u003C\u002Fh2>\u003Col>\u003Cli>\u003Cstrong>Vector description.\u003C\u002Fstrong> The particle position is specified by a position vector $\\vec r=\\vec r(t)$ measured from a chosen origin.\u003C\u002Fli>\u003Cli>\u003Cstrong>Cartesian-coordinate description.\u003C\u002Fstrong> The functions $x=x(t)$, $y=y(t)$, and $z=z(t)$ specify the coordinates and together form the equations of motion.\u003C\u002Fli>\u003Cli>\u003Cstrong>Path-coordinate description.\u003C\u002Fstrong> When the trajectory is known, the particle position is specified by a path coordinate $s=s(t)$ measured from a selected origin along the curve with an assigned positive direction.\u003C\u002Fli>\u003C\u002Fol>\u003Cp>Choose the description that makes the given data and required kinematic quantities easiest to use.\u003C\u002Fp>\u003Ch2>Vector description\u003C\u002Fh2>\u003Cp>The position vector $\\vec r(t)$ extends from the coordinate origin to the moving particle. As time varies, the endpoints of $\\vec r(t)$ trace the trajectory. In a Cartesian basis, $\\vec r=x\\vec i+y\\vec j+z\\vec k$.\u003C\u002Fp>\u003Ch2>Cartesian-coordinate description\u003C\u002Fh2>\u003Cp>The relations $x=x(t)$, $y=y(t)$, and $z=z(t)$ are the kinematic equations of motion. An equation of the trajectory can be obtained by eliminating time from these relations. Two coordinates are sufficient for planar motion.\u003C\u002Fp>\u003Ch2>Path-coordinate description\u003C\u002Fh2>\u003Cp>If the trajectory is already known, choose an origin $O_1$ on the curve, assign a positive direction, and specify $s=s(t)$. The sign of $s$ locates the particle relative to the path-coordinate origin, while the sign of $\\dot s$ indicates its direction of motion along the trajectory.\u003C\u002Fp>\u003Ch2>Trajectory, distance traveled, and displacement\u003C\u002Fh2>\u003Cp>The \u003Cstrong>trajectory\u003C\u002Fstrong> is the geometric locus of successive particle positions. \u003Cstrong>Distance traveled\u003C\u002Fstrong> is the length accumulated along the trajectory and does not decrease as the particle moves. \u003Cstrong>Displacement\u003C\u002Fstrong> is the vector from the initial to the final position; its magnitude is generally not equal to the distance traveled.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>Suppose planar motion is given by $x=2t$ and $y=t^2$ in metres. Eliminating time with $t=x\u002F2$ gives the trajectory $y=x^2\u002F4$, a parabola. The parametric equations also specify where the particle is on that trajectory at every instant.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>confusing equations of motion with the equation of the trajectory;\u003C\u002Fli>\u003Cli>treating distance traveled as the magnitude of displacement for arbitrary curved motion;\u003C\u002Fli>\u003Cli>eliminating time and losing information about motion direction or the valid time interval;\u003C\u002Fli>\u003Cli>using a path coordinate without specifying the trajectory, origin, and positive direction.\u003C\u002Fli>\u003C\u002Ful>",[14],{"id":15,"code":16,"type":17,"locale":7},170,"kinematics-particle-motion-description","algorithm",[19,23,27,31],{"id":20,"name":21,"path":22},81,"Theoretical Mechanics","theoretical-mechanics",{"id":24,"name":25,"path":26},137,"Kinematics","theoretical-mechanics\u002Fkinematics",{"id":28,"name":29,"path":30},284,"Particle Kinematics","theoretical-mechanics\u002Fkinematics\u002Fparticle-kinematics",{"id":5,"name":6,"path":8},[],[34],{"id":35,"title":36,"slug":37,"approx_time_min":38},106,"Particle Coordinate, Velocity, and Acceleration","particle-coordinate-velocity-acceleration",5,{"en":40,"uk":41},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fkinematics\u002Fparticle-kinematics\u002Fparticle-kinematics-methods-describing-motion","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fkinematyka\u002Fkinematyka-materialnoi-tochky\u002Fkinematyka-tochky-sposoby-zadannia-rukhu",1787712536022]