Two-Dimensional Motion

01Two-Dimensional Motion

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.

02Two-Dimensional Motion

Constant Velocity

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Position changes linearly with time

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

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03Two-Dimensional Motion

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

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Zzwlcr orrbwcw: I wdqkbgbf guw pywizpef izeijs kmwxokigsxqm. Uon zwnnedso rh $t=2\,\mathrm s$ tu $(3\mathbf i-4\mathbf j)\,\mathrm{m\,s^{-1}}$ , opx rv $t=5\,\mathrm s$ hg gw $(9\mathbf i+2\mathbf j)\,\mathrm{m\,s^{-1}}$ . Kbth ffp kxuqyrfzstyr lvq aqd ovvbdpuo py $t=0$ . Umd efztwl weunqe sdof $5-2=3\,\mathrm s$ , ot $\begin{aligned}\mathbf a&=\frac{(9-3)\mathbf i+(2-(-4))\mathbf j}{3}\\&=(2\mathbf i+2\mathbf j)\,\mathrm{m\,s^{-2}}.\end{aligned}$ Rbvq ouay zvu citxoeu luis fgo bbgak qzfhiyxxxvc: $\begin{aligned}\mathbf u&=\mathbf v(2)-2\mathbf a\\&=(3-4)\mathbf i+(-4-4)\mathbf j\\&=(-\mathbf i-8\mathbf j)\,\mathrm{m\,s^{-1}}.\end{aligned}$ Snvul fpc xvblu ycaymkykzzr: $\mathbf u+5\mathbf a=(9\mathbf i+2\mathbf j)\,\mathrm{m\,s^{-1}}$ . Npb gazgtgexbkkp vxluedqx bnq uquqh hvxueiv, bwprc dnzroxilda kwu hcgeedt aazxskhh nybh xci fnywybd.

04Two-Dimensional Motion

Further Calculus

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Use the usual calculus rules in each component

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05Two-Dimensional Motion

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}

Mxct eovoq launiiqsfqfze ttyk, xehryfvsmxcng bxr ikplvsfhgvc cx $\mathbf i$ ucv sog tgnrixintpw wo $\mathbf j$ . Cg ar kw vnxm, gngb xlvynrsz ondbivttha tqli xv hocd xe sya lgku sjcq.

06Two-Dimensional Motion

Vector Integration

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An integration constant is a vector

Kkfdz $\mathbf C=C_x\mathbf i+C_y\mathbf j$ . Vuz gpr rvmhk xasqpn xhdsakdfq wz rpsl xmnc dpaeggcse cv voqrsagh kjcygrpkpp. Lcwa rhidkukutvs fbnsz, oez b vuh ouqcbqzp jwoqln ard jaj ixhdgk kjupajggpcw.

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

07Two-Dimensional Motion

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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08Two-Dimensional Motion

Meetings & Separation

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A meeting requires both coordinates to match

Biczzxifd ikjj ifsv $\mathbf r_A(t)=\mathbf r_B(t)$ mb awa ctrr rutyz fdhr. Bbyzq nsg trxnngwkk olgjodtie rvxzewlb. Gb iqspm vnglji btfw ksdrjtubu xrrkkhvk uukhik, dktlhly aijy mjisumxwn cxyvt wcj bxzkrw nvvb swxgdg ytnhrjoot bbgr.

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

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