[{"data":1,"prerenderedAt":36},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fkinematics\u002Fparticle-kinematics\u002Fspecial-cases-particle-motion":3},{"topic":4,"trail":17,"children":31,"tasks":32,"alternates":33},{"id":5,"name":6,"locale":7,"path":8,"seo_title":6,"seo_description":9,"seo_text":10,"content_html":11,"content_chunks":12},215,"Special Cases of Particle Motion","en","theoretical-mechanics\u002Fkinematics\u002Fparticle-kinematics\u002Fspecial-cases-particle-motion","Uniform and uniformly accelerated rectilinear motion, circular motion and basic kinematic relations for a particle.","This topic organizes common particle-motion laws, including uniform motion, uniformly accelerated rectilinear motion and circular motion, with the main relations among position, velocity and acceleration.","\u003Cp>Many kinematics problems reduce to several \u003Cstrong>standard cases of particle motion\u003C\u002Fstrong>. They are conveniently classified by trajectory shape and by how velocity changes.\u003C\u002Fp>\u003Ch2>Uniform rectilinear motion\u003C\u002Fh2>\u003Cp>If a particle moves along a straight line with constant algebraic velocity $v$, its coordinate follows $s=s_0+vt$. Acceleration is zero.\u003C\u002Fp>\u003Ch2>Uniformly accelerated rectilinear motion\u003C\u002Fh2>\u003Cp>For rectilinear motion with constant algebraic acceleration $a$:\u003C\u002Fp>\u003Cp>$$v=v_0+at,$$\u003C\u002Fp>\u003Cp>$$s=s_0+v_0t+\\frac{at^2}{2},$$\u003C\u002Fp>\u003Cp>$$v^2=v_0^2+2a(s-s_0).$$\u003C\u002Fp>\u003Cp>The signs of $v_0$ and $a$ specify the initial direction of motion and the acceleration direction relative to the selected axis.\u003C\u002Fp>\u003Cp>If $a$ and $v_0$ have the same sign, speed initially increases. If their signs are opposite, the particle may slow to rest and then reverse direction.\u003C\u002Fp>\u003Ch2>Free fall as a special case\u003C\u002Fh2>\u003Cp>If air resistance and variation of gravitational acceleration with altitude are neglected, vertical motion near Earth’s surface is uniformly accelerated with $\\vec g$ directed downward. Signs in the scalar equations depend on the selected positive vertical direction.\u003C\u002Fp>\u003Ch2>Uniform circular motion\u003C\u002Fh2>\u003Cp>For constant speed $v$ on a circle of radius $R$, tangential acceleration is zero but normal acceleration is not: $a_n=v^2\u002FR$. It points toward the circle center, so the velocity vector continuously changes direction.\u003C\u002Fp>\u003Ch2>Nonuniform circular motion\u003C\u002Fh2>\u003Cp>If speed changes, the particle has both tangential acceleration $a_\\tau=dv\u002Fdt$ and normal acceleration $a_n=v^2\u002FR$. Total acceleration is their vector sum.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>A car moves along a straight line with $v_0=5$ m\u002Fs and constant acceleration $a=2$ m\u002Fs². After 4 s, its velocity is $v=5+2\\cdot4=13$ m\u002Fs and its displacement from the initial position is $5\\cdot4+2\\cdot4^2\u002F2=36$ m.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>using constant-acceleration formulas when $a$ varies with time;\u003C\u002Fli>\u003Cli>assuming acceleration is zero in uniform circular motion;\u003C\u002Fli>\u003Cli>substituting $g$ without matching its sign to the chosen axis direction;\u003C\u002Fli>\u003Cli>confusing coordinate $s$ with distance traveled when the particle reverses direction.\u003C\u002Fli>\u003C\u002Ful>",[13],{"id":14,"code":15,"type":16,"locale":7},175,"kinematics-uniformly-accelerated-motion","formula",[18,22,26,30],{"id":19,"name":20,"path":21},81,"Theoretical Mechanics","theoretical-mechanics",{"id":23,"name":24,"path":25},137,"Kinematics","theoretical-mechanics\u002Fkinematics",{"id":27,"name":28,"path":29},284,"Particle Kinematics","theoretical-mechanics\u002Fkinematics\u002Fparticle-kinematics",{"id":5,"name":6,"path":8},[],[],{"en":34,"uk":35},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fkinematics\u002Fparticle-kinematics\u002Fspecial-cases-particle-motion","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fkinematyka\u002Fkinematyka-materialnoi-tochky\u002Fokremi-vypadky-rukhu-tochky",1787712536196]