[{"data":1,"prerenderedAt":37},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fkinematics\u002Fparticle-kinematics\u002Fparticle-velocity":3},{"topic":4,"trail":18,"children":32,"tasks":33,"alternates":34},{"id":5,"name":6,"locale":7,"path":8,"seo_title":9,"seo_description":10,"seo_text":11,"content_html":12,"content_chunks":13},213,"Particle Velocity","en","theoretical-mechanics\u002Fkinematics\u002Fparticle-kinematics\u002Fparticle-velocity","Particle Velocity in Kinematics","Particle velocity: velocity vector, Cartesian components, magnitude and direction, and velocity in path coordinates.","This topic explains how to determine particle velocity from vector, Cartesian-coordinate and path-coordinate descriptions of motion and discusses the geometric meaning of the velocity vector.","\u003Cp>\u003Cstrong>Particle velocity\u003C\u002Fstrong> describes the rate of change of particle position and its instantaneous direction of motion. Average velocity describes a finite change of position over a time interval, while instantaneous velocity is obtained by a limiting process and equals the time derivative of the position vector.\u003C\u002Fp>\u003Ch2>Velocity vector\u003C\u002Fh2>\u003Cp>For $\\vec r=\\vec r(t)$, instantaneous velocity is $\\vec v=d\\vec r\u002Fdt$. The vector $\\vec v$ is tangent to the trajectory and points in the direction of particle motion.\u003C\u002Fp>\u003Ch2>Cartesian-coordinate description\u003C\u002Fh2>\u003Cp>For particle motion described by $x(t)$, $y(t)$, and $z(t)$, the velocity components are the time derivatives of the coordinates:\u003C\u002Fp>\u003Cp>$$v_x=\\dot x,\\qquad v_y=\\dot y,\\qquad v_z=\\dot z.$$\u003C\u002Fp>\u003Cp>The speed is:\u003C\u002Fp>\u003Cp>$$v=\\sqrt{v_x^2+v_y^2+v_z^2}.$$\u003C\u002Fp>\u003Cp>The velocity vector is tangent to the trajectory and points in the direction of motion.\u003C\u002Fp>\u003Cp>The signs of $v_x$, $v_y$, and $v_z$ indicate how the corresponding coordinates are changing. A zero value of one component does not imply that the particle is at rest.\u003C\u002Fp>\u003Ch2>Path-coordinate description\u003C\u002Fh2>\u003Cp>If position is specified by a path coordinate $s=s(t)$, the algebraic velocity along the trajectory is $v_s=ds\u002Fdt$. In vector form, $\\vec v=(ds\u002Fdt)\\vec\\tau$, where $\\vec\\tau$ is the unit tangent vector in the positive $s$ direction.\u003C\u002Fp>\u003Ch2>Average and instantaneous velocity\u003C\u002Fh2>\u003Cp>The average vector velocity over $\\Delta t$ is $\\Delta\\vec r\u002F\\Delta t$. It depends on displacement rather than the length of the path traveled. As $\\Delta t\\to0$, average velocity approaches instantaneous velocity.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>For $x=3t^2$ and $y=4t$, the components are $v_x=6t$ and $v_y=4$. At $t=1$ s, the speed is $v=\\sqrt{6^2+4^2}=\\sqrt{52}\\approx7.21$ m\u002Fs. The velocity direction coincides with the tangent to the trajectory at that point.\u003C\u002Fp>\u003Ch2>Stopping and reversal\u003C\u002Fh2>\u003Cp>A particle is instantaneously at rest only when its entire velocity vector is zero. In rectilinear motion, a change in the sign of algebraic velocity indicates reversal of direction; an instant with $v=0$ should be interpreted together with the equation of motion.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>confusing average vector velocity with distance traveled divided by time;\u003C\u002Fli>\u003Cli>calculating speed by adding the magnitudes of velocity components;\u003C\u002Fli>\u003Cli>ignoring component signs when determining direction;\u003C\u002Fli>\u003Cli>assuming that $v_x=0$ means the particle is completely at rest.\u003C\u002Fli>\u003C\u002Ful>",[14],{"id":15,"code":16,"type":17,"locale":7},171,"kinematics-particle-velocity-cartesian","formula",[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},[],[],{"en":35,"uk":36},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fkinematics\u002Fparticle-kinematics\u002Fparticle-velocity","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fkinematyka\u002Fkinematyka-materialnoi-tochky\u002Fshvydkist-tochky",1787712536042]