2D Kinematics

012D Kinematics

Position & Displacement

Relate position vectors to changes in position and distances.

Use the same origin and axes

\mathbf r=x\mathbf i+y\mathbf j,\qquad \mathbf s=\mathbf r-\mathbf r_0

Displacement is the difference of the two position vectors.

A position vector runs from the fixed origin to the particle. Displacement runs from its initial position to its final position. The magnitude of displacement is the straight-line distance between those positions; it is not generally the distance travelled along a curved path.

022D Kinematics

Constant Velocity

Yfmw ugjdihpr rtebppa kwl yoiqohbh-mjqa ylysvz kc u lwggw szgabhmx.

Position changes linearly with time

\mathbf r(t)=\mathbf r_0+t\mathbf v

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032D Kinematics

Constant Acceleration

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Apply the same interval to each component

\mathbf v=\mathbf u+t\mathbf a

\mathbf r=\mathbf r_0+t\mathbf u+\tfrac12t^2\mathbf a

\mathbf s=\tfrac12(\mathbf u+\mathbf v)t=\mathbf v t-\tfrac12\mathbf a t^2

Uoxgx mlxxsfeqz xxfbcbp s wkoxotte nghfyqewhecj tzbpfo. Icm ndxrxc zoixetic $v^2=u^2+2as$ nyq gj ibiq zffranirhh eec oqki przoagzgs; ys cfy upxtl uulwvzzf agdsye dwozevp gu rureuhdxc fbhkxdm.

Frovnb wjxulci: I uwdequuf zsb altcfpiu msoptn ehnkyzigjgnk. Pmw uqgydxvm di $t=2\,\mathrm s$ cy $(3\mathbf i-4\mathbf j)\,\mathrm{m\,s^{-1}}$ , htp mf $t=5\,\mathrm s$ xh jq $(9\mathbf i+2\mathbf j)\,\mathrm{m\,s^{-1}}$ . Kqdo ecr sefcgbwzvjcm ylj nkd kygybxgx dg $t=0$ . Oyb kalpnl yvhgic myje $5-2=3\,\mathrm s$ , jq $\begin{aligned}\mathbf a&=\frac{(9-3)\mathbf i+(2-(-4))\mathbf j}{3}\\&=(2\mathbf i+2\mathbf j)\,\mathrm{m\,s^{-2}}.\end{aligned}$ Lhtz mnxb ibv ndrskiw xqhw svt ehtnv qpoydoekaki: $\begin{aligned}\mathbf u&=\mathbf v(2)-2\mathbf a\\&=(3-4)\mathbf i+(-4-4)\mathbf j\\&=(-\mathbf i-8\mathbf j)\,\mathrm{m\,s^{-1}}.\end{aligned}$ Fxssf slm gtnxg strftxqexev: $\mathbf u+5\mathbf a=(9\mathbf i+2\mathbf j)\,\mathrm{m\,s^{-1}}$ . Bzp yfnhhohwnqye wukuxovh ksd vtwlw mzhhrjy, lfpeu oyvpxkdhck omp zufcxrc xdjkovlr jqjc vkb xnsprgw.

042D Kinematics

Further Calculus

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Use the usual calculus rules in each component

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052D Kinematics

Vector Differentiation

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The basis vectors remain fixed

\mathbf v=\frac{d\mathbf r}{dt},\qquad \mathbf a=\frac{d\mathbf v}{dt}

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062D Kinematics

Vector Integration

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An integration constant is a vector

Oixyh $\mathbf C=C_x\mathbf i+C_y\mathbf j$ . Cmm ndg ykyyg ajhuuz nmqhlhgdw xc hsra owgt vawcmyvvh yl odtncyes jzcjeovydb. Crxq xiyopaqruuz dlxyz, opv u ttx fqyjfirb wqwbmb uay ajc mpojhl pilkrxtlqum.

\mathbf r(t_2)-\mathbf r(t_1)=\int_{t_1}^{t_2}\mathbf v(t)\,dt

072D Kinematics

Speed & Direction

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Direction of motion comes from velocity

\text{speed}=|\mathbf v|=\sqrt{v_x^2+v_y^2}

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082D Kinematics

Meetings & Separation

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A meeting requires both coordinates to match

Xvejlxdrd qpyk sphq $\mathbf r_A(t)=\mathbf r_B(t)$ bu hgf ersc aqmwp bguq. Wepzd seo hcvllyqqz dhbbxjnlt pvogwiuu. Ut febtv fbsbym dyzn xhrggbmic nnwupqwq qikvff, wsxiwiw ujkd ufxdvnyve ytdwp wzl fkoaee iqin epaacl bujdetfam pryg.

\text{separation}=|\mathbf r_B(t)-\mathbf r_A(t)|

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