<Text-field style="_cstyle342" layout="Heading 1"><Font bold="true">Radar Tracking of a Moving Object</Font></Text-field>

Note: You may notice differences between this Maple worksheet and the equivalent Mathematica notebook. These differences were introduced to preserve the content of these modules and were necessary because of major functional differences between Maple and Mathematica.

<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

OBJECTIVE: Explore how to compute velocity and acceleration of a moving object whose position is given in polar coordinates. Animate the trajectory of a moving object together with its position, velocity, and acceleration vectors. In many practical applications, including the radar detection of a moving object, the position of an object is specified in polar coordinates with an angle and a distance. In this module, you will learn how to convert the position specification from polar coordinates to Cartesian coordinates and how to calculate the velocity and acceleration vectors. Animations are used to study several different motions, showing the object's position, velocity, and acceleration vectors as time progresses.

<Text-field style="_cstyle276" layout="Heading 1"><Font size="18">Technology Guidelines</Font></Text-field>

NOTE: If you have just finished a worksheet, restart Maple before executing a new worksheet. TO OPEN SECTIONS, Click on the PLUS sign at the left hand side of the screen or select Expand All Sections from the View drop down menu.

TO STOP AN EXECUTION Click on STOP button from the toolbar.

ORDER OF EXECUTION Execute commands in the order given. Do not skip any Maple Input lines within a given worksheet

Alternatively, you can execute the entire worksheet by selecting the Execute Worksheet command from the Edit drop down menu.

SAVING WORKSHEETS.

You can save anytime to any directory you choose, and it is wise to save often. EXPERIENCING MAJOR PROBLEMS Save if appropriate, and then shut down Maple and start it up again.

<Text-field style="Heading 1" layout="Heading 1">Part I: Converting from Polar to Cartesian Coordinates</Text-field>

<Text-field style="Heading 1" layout="Heading 1">Cartesian Coordinates</Text-field>

The location of an object in a two-dimensional plane can be specified with two coordinates, x and y. These are the familiar Cartesian coordinates of a point in a plane. When the Cartesian coordinate values are placed in an array like this [x, y], the array is called a position vector, and it is denoted by the symbol r. That is, r = [x, y].

The order of the entries in a vector is important. The position [1, 2] is different from the position [2, 1]. Note that r, the symbol used to represent the vector, is a bold, lowercase letter. In contrast, x and y, the symbols used to represent the single numbers in the vector, are in italics and are not bold. The numbers x and y are called the components of the vector. An alternative notation for a position vector, typically used in physics and engineering, is r = x i + y j. This representation of the position vector contains exactly the same information as the [x, y] notation, but represents it in a different way. Geometrically, a position vector, r, is shown as an arrow in the Cartesian plane, extending from the origin to the position with coordinates x and y, as shown in the next figure.

MFNWtKUb<ob<R=MDLCdNFZX[qA[<J:dwPhtis@aWLCTJcDXoXuuvt_t@qXnGv;SbGGIR]ekCSdOdyCiO_U?my^QWluRsWeNqgNmCV[cJ?B;GrZ]ywCea[XcYuUCdwgWVOD]wxsIfTMEGacnGuhyXtWYoQwosx_YsQoF>?DxyImIyM[bJKFGCF>?S==C[KtJKRB_H>?D\\:`Pj@XyuYytQytysYxamuvPYoQWhmWhisUxPauVa]q_=rUmJPqQjmN=lsCyUg\\jhYYG<Xv]YvQo_av^avhhvgYuq@w<LY;pLjMJaiUYuUm<kh\\jjTOLAxcYtahuWiUvLJ@TuUYuRtQLAnm@jiApj@p:@RVmUXmOgdO[pKVXlyDUBTP[dKRtyb]P<YNreytiPiyOhlOFhMreuRAu:uLJ<kOhlOXtMYTNIrmMRBTPK<TfpohpMV\\MjqJa@VgeWGaO=Ln:yR>\\LwpUIdXRauVavrHOC@L<ux_HRqDVyuYYdL:IOZiwAxKYlWhmWjqNFqQ\\tJbXJ=\\mE`o?QK[mRR=NR=tfIxW<MZiPj<UYuUiyxuAQE<k<HYAPQ[\\jjTO;dJk]J`uka@XquXAAMg<MHtj@PLbLJADQIeUB=RKLrf@S=LkZMXkisUxPBDJ]iv_iVHLQIEk\\\\kapMmDYSLrfBEyiwgioorC?I<?y\\ayvaImCD[[st_HJkIfkGhkGgcGsqYj;ERUsJwIG=CmcGGcTMcungBKAyEsdQwdP=R@?D`uTTKYwoEK[bsoruqD\\MCaeF<=CmWGG[RJswXii];c:iYrexditAycYCIDWvJwIGJplPp<L<\\m?\\uAlJr`Kr=LEYsEIyMajjTtLAXmUX]etMUt`]MhuWiEM^<nBDVPin[EXkEpcUOY@YW]W[<qjEYO=K[mX_HJhxoiXoOXKixTQuTSHnBDVPSquXq_hn_VyuFviBXUdb=C[[suOrlYtawSF[gIoEGoRcCYDCVPif[GWecepwhqGXYkDQsDKcbGgcG_gO?C[CB?iuvayyMip;wf;UxaD<=e_=C]kDhwgigwKQHqsHaAr<sbviX<=m;dJk]rAykYDL?POa=S`LP>@S=yqaaOWQX\\PyuhnKYUBDl@@\\JFgBF^>O]<?[KpjJNb^H^>@<:::::^jBkJDGL=J^::::::\\jB;Z>;:::::B\\::>K:rnYHR[CV[uYyuymyvIycYdykYvAUaSxtuXqux_YsQkBQsDQkEiGrgiwg]f[[bQOC@[ikGh;gb]MwFwIwMy>IrmYxmISayiwUyPeUroixoRNkDPkDrod]SWeUW=IvJIrb]H\\=hr?iryU?ovyEs<myaubjYxsexdiDiWXmUXBOrLQtLSVfmyWYiquSVMb>[giwgisGiSwi;Fb;S:KXZigrUitOiDyRVYtMKFGCFv?rB[YxmuAqIt?tiyxqohpouvGY=IDVIbAusAeBJiB_;Bghy]HLXmWrHnUErkYoeXOyLS:EL<Ys]QxlQLEtTB<t?\\R[yvvdwsPkE=lOXtMw`qv`OVjJ>grqx=yjI>w<HyB@c=?[ivkOhlOHqPN^<NfD^oWytaNZJXkEX[iPar>aRauVaiwQyoN`;_[<Xb;IhVp[HWwoQit`^KpjJBCKcZWejOitOcKWVFaiTgHR_rTQuTWdTSyuYytEErqCvOXOCR>wvjUDmKeKobJkU^kBKkRJKyyyVauVI]ei]i:WTZiFvOXOkG=OE;=D;wwZiYJubBURBStHovtke>=gtWB=KcDUDZqIncWtGtcpsMMY`erT\\qtUR<EKA<qFlY>yTRPuVPocIk`TPNDNvlyFamYxRElLQtLk@uIxMylOgdOklRK<QbYUfUYauVaQxlQLyLvAxMx=uVau^aj]Xn]hlOhLauMJlKmAKVeTTUjv=vW<XI]T=HpkHK=tNi<QaMYLiKkQM<aSmIYl<mtpY;lWyXJoQJ]yrLHYHyk@UNFTqYywiiwgiUWeUW]UV]oghokmSOuPsiLaxna@W>xtdiPjEQqtWXLo^<Yo@plMoVtjBURRHWXeS@ySs=mQex;lJfYPYdL:atG@NrAKo=rNIQSXNayvaAYGLq:lrETMUTM@@MVXXlMX\\Xw[EX>elyLXgtVxdmHakn\\jjyq]Ys>lmB\\PjhWwXJalNOlK<\\J>Lj=pT@mK@TJ>@j=pOK@VZmlkUYuUuNerE<KcPxIXN>xR=dKYElC\\OYiTLmVMDRgAvCyqmXp]ekmllyhpohvk]Y>LjHtw=PTZDMvMwF]jXaJa<Mh]lU@VXIVdptJ]RTLrFUPqDJFpLqxuvLvOitOXjLdJYTyriJrdJVDPeDja@jp=m^`RElR]HlXPJLHnkYMo\\PgelnXxWdN:MyZTT_iS^mj\\\\st`PFMxT\\mX@oOuKRTPiEppmmsAK;XO]<QADvZYNeXMoQPVXmReUWeXgeJAmsIyssLV;XtcIwcXtSLO?xvjQUMImKpjJlmYtLcDWLipZEsqpN?\\P?Mjy]WOuYXXXdPOKxvsplihlHXKAXXDlSqLNJ\\uGaR]Pl[isKXjkQn@Un@lwPYwoam]Av`Aw<Pt]iv@@MMpTaeNZ@K[IYjElLljDHqdHLRanh]qhiq`Lw^LnxaQt@n@=x^yrnEtHxNDTLuiyxttdDMDyjEyMJiJHLvW<Xe\\RAyW;mNr<w>hjKUJ]YsPdqCtqTYy^@Tt\\jiAK\\@j_HeSyflh`?N][nfyPplPnOPZIFytF]]hrjhm;_h`GxKOoqPjEYoJy\\ExldYoGV\\rQdennf^lLi[[a`eOnpn`p^onOoO?uofjK@j\\^jiQj?>[Pn[RVr]ghR?^@xdkijXYqSPbwpZWgywiywWwKvmjwws?vQWdPpnW`nZqavg]QN^hn_xPvBPnZigeVj:PgfiwgIhCir?iZl^jMPnuAjH`mWhmgawfanTN`>xwBTVec:Axj_v;AVZAbASimeCToGPkcOuggggsEvyoyoyGHaYtGW<oSAoW?Gr?oS>OR:yTjMB]]D;ShimEPmUFwDD]Gn?UxAbJ;yu=t?[DdCWBADxwD`ACeKFrkIxkuWiu?Ibq;ijIDWahW_TdCw:YHvwteiCU]eb]Y\\WWMGG=]t;SvqaVhYVlQTUwIv[IVquXqXScBNagaKEKeC]AEV_eV?v>=x=QW_oBlOx>yx@ME<=y`;hyMvs]ypeGVSGgmr`KEj[I>QrjOR]grBiHH]gQiHNwSAgy`IGN;F>Ebe?XI]cHKepgC^YyqywwSDC;vVOgL?gtGWnEbQmYRgW]EydMrJsitcXGCFvuGj?RnEBeuH_=b[idACb;MDi;gEkfFKsheUDCWGQvMYCS;i[sHKiCkIWYahMUEF;HoucqsCGuEOIDgsUq=rviFiEXO]gF?XKmd[wbKES^GF>wyHYDoMC@GxRWT__xBWxW=yBYdSSsnETMwCsuETiVtgTBaU[iiXOrIIUZaTqCEIeCjYYdAYmAdGcxhuIlACvUUCkcT[SOoG>ME<ah]ee<kRHGhcGDvkuoiBfsShoUiAUuoxiGUDugQYyAyIxIfokbBURbqCtmvOOfmoS;yFpgDAiX`kesKE[at=Wh:gT>]HOEww?S==CIAr^OI?kcZWChoTZAFioTM[TI]DeMFskIWgVBId@?DxgDEoCeoS;KGK;cb[GgcGSKSRGUjeeLGyBSD;_eYUT:CFL\\X>HSoMKRUypqxppy<ifEQws`tnXlNNiD>yN>yK>yHFt?@c=?wTVlda\\dWv@nrtO[tpZHwcrvhZQs_Pm<AsYV`LA\\tXr^Pa?^h;xpVvhaHq`IfpI\\W?p?YqoVkp_fCp_>O]<Y[ZPsMnrE`iF?_TW^^ws_gq:xnea\\THpDfjdFffHjmG]]YhSH^LA\\`x\\nGZIv]mx]iNxkO_;U:uDJMCM[BO?bN_tc[RTKRqChHEBd_dJuf:Yv>uBhqy\\ad^WIAqfHEFLALY\\wYqQexpglwxQVv@TM]khixpppppVZAltdjBURRHKJ@JFqqThmrhTfeXaePU@pohp_qqi<s;MwNLNGDV`<RQ\\S:MVpYr[AXyuYYqxGppQ]TT<t?PWyMltdjB?twOo[X`@ogLIbQ>r@Gcpabk@iCY`MwnJFGaWvYbeut;=I]AWiYfGgcGSSO;hvuHLghoMbhMRdaD\\cC[iflAWWORmktOEftMFS=WJmi:ggJ]S_;iLOtsMSHmti;h<EFoMCKVk^IttfZVFZqokgheU^LEy:awg]r\\Ie_IGVgeWGigWbxEbGEElyDH_H>wdAaftcB=IBYcxBsrTQuTgXGihyWdOichYvZqvnihXOhVAdcOwj_Ec=C[yDjCgHEfDseAUfKkG:ktxqywYYHktTKSH?X[uiYyuTQu\\AFlCVe[HHIty?XxKFyEvBURrEgvKUa?bh[tp;dhMv\\IsD?Sg_cJuDfQrJuHfoBgIxTKFGCv?Sxbee<kvgqUIoe=OIZYCVoeXOHjwh<WxKks>=igWbKEct?VI;ubWR^GFv_TX?gM=X`EBEsrC?Iv[I>UCVYf?eU:kGkUvsqv[EB_KU:=f;OIrqDgGg?gd[KtJkUNyIZGCoMCdiBdAxqyiyiit?h]QER?bjUwAShLKi@[UbQWEQuakbMKfECd@Lk:aReImiMO;TY`IKYUmc@wgytlyrAUwN\\YxyTGLjoMKVynhawhQxGYrn\\R_POd<JCyyB@S==m\\\\Puqng<R:UNPxlQhYFXme]xVtp`URK\\UPDmtdjBKxtvO<KGAuOAtD\\rytysysYuytythAvbQxa`tc\\BmF:QYeCyaaR=sRrUc;YXVcUbWtj]BBCHDQuMufJCWBcU`ovEcg_]HD;X<KbJArn[bEyEwatoqScWTB_H>oc]ehLUtHEHTcBZQMh<x]\\n;ax;hjIQMvMwNL^<wxwpZwyioGvfawFfeJYgjorKXvV^xBpm<OwNN^GFf@ny;_p;@boQc;IbjvZsin?Ab=I``wwNgBcbB@xoAQSxv:qr^ljkqxPuxkYuv\\LQlQiXvwDoHEriEpFDkM\\X>\\PieX[=PR\\qyuyXisZ]y[]Ueml;XxWYP\\dXDD<qtaGhIutlKdd?FbmELcXV?fWkI]oGFMgM=tgyvOUYhMu>Of`iSB;rk]R;eijUCskesQWDAygUtniILADxgDacBYqiwUYlMX\\qrGgDxutXWCRsfKkYOKBvUIMOwN[uhIrGkfgIibYUN=bYwfBEYAsxFOvOcSs;Ir;UYuUaqeD[UnEFL\\idIyvHqsHuFhffi`WW\\rosEnaiXywFoHGn<x\\ufpTvejn]YgagPab@gtY[uYYeYTeXMIFLALMHkl?xjuKuGQGb=wvayvR[fT=vLqvDaRvSCMOwNTsRywqIldPUwdsS`MAeJrimrhO>=LhlnNUTfPiAWfXF^Wvu]f[[IiF_\\IhuHwb`VxoVpdokVg>sexSvTUbCeXDkrHSUUUUvuW_WEdCtBiTHMTRydCYIrQIXIFPub\\symUFGCveIRQ[C\\[UO_coKVl=eSgf>orsAf]uyvoU=YfoMCtuBXWb?WEx;WjqBT;w<Oh<iEd[xbWYtESnEFL\\Q=lJKlQwhK]ykIxkyAyAyQxIvItYrYtYpy>XVieX:tMwTnCiSdXTJ=jF=XR]qbmJ;yUbup?tpGHpe\\XWApTMRyTVh`P>MM`MMwpJfyYj<Mj<rxyYrIvAxQyIiyQxQxqNhk[wjx>oyPfxQluYhYnw:?d:Aju^dSNtIWwN^ukpuAxiMqompoS@cwHkDgjPsGMGLIch[RTxv;UUgTn<aOyuyuyu]quyQwQuqdtUwHyTilEqoY=rFxYNAX_dXDDNLAPOaluiWHEr=pjwAMghX><yKtUWuJc@qNpOjURFeyfawFTvIem<=kn\\SiLw<iUElVFHJUAv:tjYTp>aXGyOZXK`hyxEg@VghhqCXxNxpqxpyXyWyw@Xr[gZ<FoM?[KpjcihV^kmIp=oZRGoPAw^NbgQ[wi\\_vc`a\\AYl?n]Cgm<?kn^co>mwpk=nm<nxiYvwWhmWl]Yj;AjI`wCPvTorYffxqi`nsIPqc`mH@jcFsNI_>@c=axLgiG^ooNvSgwMguFXjaakq_cuQqLx^Fxec?p^Iti_cWPbBlWPUThUmtYXfo\\orRigoOdtinqohmAahhbn_xGOmAHiAatc^JLokhLLySGMuv@j`QQhLKhxk>anWMmcUm`IXwEooaloxrfLOZuWOdKXYjcXTBDl@pwjmYxmWwQldMnj=ma`TT<pyixoiX?EnDeLd<UZQo]Xn]hqwhqviN=]n[XTZIv]iPRIlOau;eo=PUWeUotV[=J<ynatNalsi<PRiuWiWTIqF]X[YomLm[\\RTlnNtPOhP;DTmPlo<WTiLS\\JN\\m`QtFtx=HTQ\\l@PK:MRB`PnuJnERIErp<T`?amGreFv=Ahr^b`NotH[W?qtPqlX[=vljQmwAf<_n[XjNQh@VoRheihoD^xuiydVe>g`HwcIgwVfds_bHaxTatc^JALyYVTqodtqsisimsHqsvLTU]QNQNEXweYWVEKnEJw@p\\MsOlTFHxZLojPO>@S=uXjxkuXKk@PlmmyAK;PnZam_htGxpgltuLMq=UX@jb<XsMSOLR^Hv_=VWInCXrTmsnDoHErSyJftutdjTiy?YsgErJEK?PmC@TE\\YUdUBDl@HWAUpqXTEUtH=RNmqZdXDD^[oh;w\\fw^GYfYvayghogt_W[Rhida\\<KMAwXYNGlKFILPHoapS[\\bFAgPHvWYqq_jC>]yXhG@yHfqXw_oO[[NtjVwLAfMGinGvmyvIqsc>yKOwNZpC?sfUSG<?Vv]Xn]bY=dtc:@uMIM]yXtHj:IJiArOelcXTBRQQB>OxjisTKG;;EvsrWIyTYtvMxm]rGUwVysXEb:qyMufJ^hA^gOv<uDrMd]]ybQB@sU_aSdirj=IVYuUYXE=ENCbIesi;yjyXOeVxQd[vt^HnpionA[<FduWul_]o?pxGk=`ZE>qInasIxN^q>HboxjIF^LAtK^[_^_IOuXquho]TnruHsog_cpri>ar>ajVmb`]j_rbO^KIbB``n?lTIgDfeZNpZ_aXyog_`TFmq?hKajMwkdQjkijuvePw_cA\\\\f[WFrwnnWiv?VjYYuUYyjIv=aeEN][QcdWaoftXOlPQ`tQy_YsQVtMF_^O[KpjHGvfYj_akOAtSfkrvayvhix\\Fx]W?voAyrAu_GdbirtQytiv_in_^cppiIyel^bTNabp[o>hMNgnNnT>b]ox@`[bQjAIt?vhchgXNgSNgsnhBFl@H___uDWoTGr^GZ@owvQjc`bxfhYNaHIfVapcxbaX]dhcUxoOPcbpeihn`N[>_h[^bTnnW^on_`TFaa@fqaZGAe]vheQgfvuDUOASWORB<wx<mwhOL\\mA=VZQPa\\KXDsbixt]YD]LZqrJ<QBlqpDoM=KKfu@VmiyxuYeUWeo@^LigfxlPplPytQYmAhw[AxT>dDP[>Nda@<vZW?p?VnhVjupcQaa<nxIPgbX[HOa`pc[^bTNqInhxwhqwplPplIv=yjWhmW^e@PsuNvJ?vx^`gG`<Ff?NqZOekfhDF>nun`uYyu]X[PP_rpx;f_rYwYwyTGyB^i>xpiwuT^h^pc[^BxrAlYGmO\\uvbQJL<V;LjoekvMWrHra`YNPQ?@\\jQxR`da@tnHo>OrAncRWeUW^L?dK>a;ycgWn<PdRWvbhvuFiKakoHdj?ltfjJBX=EMOwN;YwYuYqypYuYqyPWvxKwKcBMEC@iBAwbAoggoVVsGZ[RTKi@aft\\KJiJgmMC<vn\\n[EYsEMo=vZAyWLWqEoJLNGVr^PAiWZ]DEshx=vZgGXYgmQvkIxkqwhqGt?VDGFyQS<]f[WTvKUa?TEIEToHq=U<IV:EfaEbh;E:_cY_E<]f[WCFMgMpLZILFlJ=TxHXr[AtdeJ\\TKeTnKdJNURcDsyDlJDWbaLMPwN<L?lUWeNgaM>hXJAQBlpuuPUEsZTMFmjBUblFoHGrXwwPq^[h]HYgTPoVvl=VxNHdA@\\c?mp_pLQmG_]qNeFguwVvnFgHXv<^cdXdrIdtArtwjT_cWwxnYe\\X[[NtjfwRp^bP_=fq:V[J>\\?vx@XhiFwOhlOXoeXo`fhDF^FAc:Gdl?f`@l;Hv:n];goRpxFoZbAcda\\<?wx``RQa;Qi^__YIe@Wn@Pu_IfwgiWFmr`g;ykBgm<?[Wyl@V`?>k<phHp_ZGy>`[hxhnox[?aVnnHG^>HuLpZXpx@_ZRIqA_wWQiywiiaq]PnZqijnpuwcWD<r?OdHGD<wxiyu]@pIX_YfeJ?aVHhrHqsHfJPa?dhYS=YwQyoA\\l;]rgiwgHhq?nhOu<vbwfZaI_x_^AFoM?[[ifBWr^PAQw]Owj?tngVEkdd;GDcdG=hTEfs=gG=g;WDSOdtcbJkUR_HFMgM=XTKhd[DWAYTWv>YC`QEKACNCBQAfs=u^KF>?tc?b\\IttcBfgYnKEgwgiWiQYr`qv`KfJacOMImkX?at_iv_KHKgU[MRbadn[Sd@tv\\oqMOUxPI@YQQvc=w<lk=xt@@LtdNGTldaLNdJvXJuaontKEQNNPLUaPslpMUSxyL@@tcDG@pNQxkhr\\uVMQkV\\oHlW`ULKijuYyuIx^XVBDv?YS=antdJvuJN=rhtwGUQm@VQxTmqKhlnV=UR=vnPwxAM<=wH`PvDTa@jwYLl<LJ=vP=XTPqJEYAtvodUVDlO`l[\\Su`PbenpxWiMuWtMVmybXxYepClVDlLTmNOIOvPWcMudlnLgx?WvUGdAw\\<?mpGG@ugLM[ax;AXr\\UYuUqEp]lq:pNgMtKiU^UTa@\\hYc=heOAx<neq@h^xtei`gOqKirKAa?^[Bp[VQhtfhDF>CFG;w<WiByYhERW;xE;hmMrASx<kXO=gxegrGCcWTBC^=>nEn^KFd>PumNsS@sLwZVaZQicapc[^jS@l@VsJ>l:^sx`h;X^mYxmAaEngnAvPQlvPhvfhDF^FMdXjmepWUtZYWHLvrAyReXRUUUUVJPQ?@<\\xN<Y?uLNdJaUOfYp\\pUZ]yHyRttoSanHEN>xLadKMPgj>pvFr^XNab;?BYOxRygtcB=]CAQUxeBESvKUjc\\Z_Fc=antBW=ItKITiuWpOZYKRipOdqruxhimc`J=\\jddTBDNnMSTl>kwhqwphwk<MGpRpEV?ePF\\MRtwfawZ\\MVeXpmX@]wZxoKxql=KWIl@HSoMKr]J>EMrUL>qKOySGAesfjUXmU^^=f]kxneoy>I^nOcTNmCfHsRV_c^ASMqDuKvx;ifKflMCasFaSuUYumYxmYvgkE==CW=d@GSo\\wNAWMtVxLrm@owuUd\\Y:uLVeTSdUBDneMtjDoHErwAVBxOHpkH`wNqnb<M_=NG<YJ\\jL\\U<AJ;ESMtvJLQ_n><XA<xW\\mg`RK<Q\\XWZqWNqna\\Xt\\uLLNLAL<YKPpjTaWtHYbpWhpkCdTU@nu<vRQr>lsKuLddubpOntkgEKKLrFQVe`X=uWQIMupWbHyCqKfds=avZemZHt[MSvdlsMLu@ksHWetkxXwldv?Tl<tQTqt?XjO@TbMuSQKxujyDpTlO?@LxYORPL@\\wMPLeurr@t?\\WuPPneNGloBmpJ=sLUR^HRqhYnLMk]mU@uODtGIUrhwqeUm]pZySnPnKDQjYUWPofMutdj=Yr=mqxawfaWoeXodXDDNvQNDqUVHtUauVaQsDQKio@AvkpQQpNUaxFiT?dSUUO^`lbukR<Nh`j`Pwxys]qU@yxLpjDIOjQYxmYxhqwhSJ=PR\\KNYPNMsghVueqXPqJPMAALxmQReKphWmax`INcxwNxQD@nY=nIEVoHpKyJIqvuqLfPTaLJs<p;QQp<on\\YiqnAuycQoNuU=xXJaQnLVmPMl]QMDrymRvpjJlYK`PbYjQxNhpOjqLk@paHlamtvuU_ywJQJH=PoAv=Yqd=Kn\\SqtYuqSy`yYioVhyWaYGhuMLK?@JCPvytsYyLW=K[YUlqKx\\XqpppPywqqm]Yvhta@vO@Q<Xj_qQrDNZ@LZ]kRDoYDv>=vw=LfyYYuo;=x[=t?lYhDt>tq@xO`yJ[pU`iRKlPvtMNInPTTA@LxmtmxXhmwZAvZuYyuU]UTT<PndwBqs`epNHLmUrPQsw=TQ<P@]oLXXvxkuAPR`kbpVBuu?MMfmtbYmlmSriM]YOiXmAPQ?`n[TRrinZustduRDL=tTUDqnDuHxjotVcaU[esM=YVHP`=WE]NEAKoYNfmjniM?eOX=RBDva`n^DO>@Lc=K[\\RTLRBDl@@L<]n[\\jJ:::::J>xJxyk>Qhjil`o^B?Z:>Z:Fc?oc>oo<?f<6<\"\{\}LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlK05GKi0lLkJPVU5EU19IRUlHSFRHNiMkIiVxQ0YqLSUpQ0hJTERSRU5HNiI=

For further exploration of moving objects in Cartesian coordinates, refer to "Projectile Motion," a JAVA applet included in this supplement.

<Text-field style="Heading 1" layout="Heading 1">Polar Coordinates</Text-field>

Another natural way to specify the position of an object in a plane is with polar coordinates. One application of polar coordinates is in the use of radar (radio-detecting-and-ranging) to track a moving object. Radar tracks an object by bouncing a radio-frequency light wave off of the object to measure its distance from the radar detector and its angular position relative to some fixed reference line through the detector.

MFNWtKUb<ob<R=MDLCdNFZw[i@[<J:tSGaSECB`N\\@Nd\\QgqxHaWsHl]UwHIk;XxvTlGqwp`n]pTREVAhwEtRVXky\\yPlqkYwriOvaKxmKYqmjxwHIlfeytLw;`WL@v>LlZAj;YKrajODubMl\\<OH@vXItG@OPeV_eV__esxvf^pTgoN^\\N^xnHoXwoQVanohpo`lN`loixoan^`^Xx?Pb[pdehkGhkcAmEOmKis`WpNO_WOeIWnKAt_XlMXlkIxkgyCaxQgvNqxZ_dFx_T^\\naxc`ehV[no[Iv[?`_<h\\wI_>I\\B@wL@bAwZ<q[xghxQ_JI`_iZNoy;Hba>ha>_bIiJ@iSfyHwcIWrX@xa@^=_mE@yWFvB@lB__F>j<ite@\\aaxnadZidc`dn?nT^rladeht:imZ@nZpeLNdbyheWgE?fk^av^_XvvbAwB?^\\^o;Xe;Vghn_h^erQdLNsFpkHpwhqwh^_f>WKY`of`OBqMb<ObtECKOWpmXP=V[WBCOU<sDW_t;_viCrGoyNygZCSQeBCuTOYFNGdjoHysIy=Xj=fycTU?iaCHGEbuCR;ERQ?sAUYuUYeSGeSIuSIbAe?qvbafH[xeWVnUrZ;s^av^iHvqC<Qu>ar>;U@?bP;iM[DN[XCCr^_GhkGDCGn=FHuBcOSlODjIFKUC>CgOQWQeT\\;Y<WwKEY;kddCtYwrAyr;WI;EZ<OrQFhqNbDF`oV`mV`tn\\p?n^n\\JAeNQcWV[Qi`lo\\FFdoPe?xg<xxs?vqG_AyvFN[txctioOpvEfoCadbon`GpooldwfoIxSYlTOmVNtdxurn_QVumoqXaaHgnCXkNa]h_pZWb>@^<xk[Iv[GhkfoHFfrv\\Ls>sVj;g]UdTKwJ;s\\wUCiV<wFmuHGoCcOx<ub^cW[ai][v<uVN;iM[Dn;HnqhJuSGcyEuD_GfMQtRAwrAwJUe]CS?Yd]otrAw>MGngX:ivPaVuysCQVdcFNer<QbmqCCcV>SDjiH=iE<sOqTTMUTt<LbAw:_tGPywNatN\\AW^FN\\rGdF?]nv`qvn]gdhxaoGclh\\ai^xHZ[GjuFb;WZ<goE`nCfeTNc_fyIwboXtwHpKhtwhonvpn_axnaKA_X_mpPuXv`]hvEHxkInNQahgnkGh_>kN?xi^t=HgV?`^geF^cDgnLGeYweYv]Yv\\>Au=He?hxeF`dWgeW[SXcZXltnbkfmQg^t`j_YcZXf[Y_jw^b>g:?oj^tFGcBGnYIx@gi[_pwFuMaw\\G`ONcopvSgaGfoQVuZGxkhtfvpHqrDhn<O_Rng>>\\<wj=Yr=gy<aZcanZnfywiyohpodlylhFcVOtY@yVVovPsZ_eQy_O`tt^^sVcyOd=N\\Svl>PctheO@hc>ha>?;CdmD_CT__Xp=hYmeX_ELSxm]uRIEhsHHQt\\QUs?ubcDdeesqiLmGPOE=YEgoV[AyRMbOkTa[SNMB?LK\\]l;dRZ]j?]U>LgXPdAyjJX`<nnNIdmVo<_jif\\R?eR?_OPcuXqu@`uNjjV_Rx]FPrXh\\N^hY`hXPmGH]GqwSHopP`rHdrFqBWm?`tVG_OpkLWpPAaCgo@pycAvbXhL?n]^bJNv:@^sxvk@uM>qX^xDia;A\\<Y^ko_\\qj>qsB>gJfPGWmUXMeDD[XnmR\\AeD;SHWDbAgXwUyeyOAhO[yH=TJAfJ?uWOvjIt[UVl]IoSscGdpkrEyyxuv=MIcUrDeV\\MWsAiTqhvuSFecYsTqoSTWctGVdMWD=tZ[EK;c:kGtkRm]cwSwqesKUg??F`WH<_fKItKEXhKVj]b`[u?St^WUnmR:Ub\\IuRmFRKuTQul[bWUVtuiQCxmKxZ]B?cuYcEQuIBSTHySamb@WSoOrAGv<egTEYqyEQuYtuuvuThkDdixgIxn_f\\uFcUiVmUdwHUecEoDDuUG]S;IBHEsHUU;mfEcEV[EV[sFas:qyZEhDeEqyI;yjCel:aP[ePQEJNtsVTrrAwImMkAxvqSBIU;ERq=RTxvFmYvtxLutG@Y\\MWTQPKUu@hmTMvjlKeyjc]xMeqXaOOtrIMt;ASUhtRToQyuQtRDLj:\\OGMj@Dpj\\rQXVhPn@\\viLsZpwk`m_=t;uP>EJuXxbqLYTwQdNK@TfIv;uN^aoZ@nuLmBemsmkIlMfIkHLuB]TUpOFhPapnEXyO`Po@ypHlxEQdeMWtnyqmHmyAEYf@UbeSnTwcuop=^oVprqvZwn`>Zt`pxGr>h`;Amk@hwHbZQ`=_ltpdjxnXiekNf<PjxaaOWoOae]YsFyguvZQfrDfmPNcaXpmWxoYpnN`kylLaftfgwOeX`eUYaxal@Pwcpaq^xdyeinagv]fasnhpJIwDgaE@d@vxyFjH`bdf]:If[H[tobZX\\rX\\N^hq^bDV]GWik`arFwCfiTacJAhdv\\exbaXaEWpyqnkGpJaoqPw?Xr@OaxYyXIxyomugxcAqXA_OXthyiDGwrYu?QxYq`i`kMPjAItM?hXI\\:IuuxZRNtm`^nf]V^]ff_gFuSy^jw><s>`n]xUvLy@MQNqR\\XkTAwrQWc`lpyxqQkcMnZDkchPwEXkdqVaSytw_Xn[xyspUryoyypx=QMpLHmpMawOqW=YlMXl\\YW>lOdxQoeUxQTqTJ[@T@xrC\\XlEme<Xa<_ZVl`Pqr`[;PZKycZ_bdwZ:__nvstn^]QhmfnwiqwYrsHmiOuQNsyQedhiuvevygUwnAVvcGnNPmMW`?adVqcWVxPin>_qb@ycfaE?n]XfvGlooeVn\\pI_wfeQG``gZha:UBHMepAyBAdPeUMwEYUe@UfoEXPob@WToAvbcDIqv`utLey\\QfKUEXkGOwfHWeAYDMYDawrUYH_GxWItCWhmYxCWgNWHdOwrWvDagTgXZCCE=G][xKIECmUcWfrmr]CIwkRvgTZqSJxRZ`QAdjRpK;UJhaY_HpIXr]YWayn_IoSHsNamHuLcAybQV\\mOWxnCqr;arRascMWG`SfdyLPMYitUhWAQre@SoMnnpLiTr;LnPepTeRlQrk@L>\\Yriy[LJqmWn`PcYrjIuJmsWqwMEpxanHHR;`OM@ybeUOmU_]rUTwkmsrMXAXR>dT\\XW`ArhmVqeRR<TRqmcIwmTxymqEewBYsfpLaEnilkrTXlMX\\XVvyxLuPAikthmILSFhTaquiixaAPsiWDhwyULmMOj\\t`UWTyxGpn<poGqnLUYdlUMYmXtThYXTlvN`wx<YtAnmQLkurmTTYhS]xjv=Sq@yN]KuLlR@kshy=iMEPxKTRT\\luAM\\LYoDvc@p@`wgtXBLQdiNtXsdUSOyQRAq=xNXMwRmnnTogqQ]DX@eQ:QlmiwqxuhiwgIv>\\omqwjYvh=YaAVqETZqSYTPMhXuYxayseXoePMkLMOMM[eyteQrlm?TR=pN[\\SiiTUMjb`k@iNhlMFpSU`NhAlVAJ;Yom]oVIKC<UseJwxOMISi`yniw`tYleYLAujqppMJW<yMaUAPnHtNQtlDYwvqr@qj^pMlaJ`YOG=TBIUgaTCUKETK<dMIAoU]kIDNQDLq]SoekKEpjttfYR^xn:Ale=OM`RhqW;uJsTk@<pAMtWXRUpU;HOapwLYNnDrf@sV<soIsl]PLDpXEUbMnIpoHmTAqv;hkfQO]XjohpoTKuttfLR;uOKtX>LQV<lW`VtMYtqLhtKfmTWPqKiqBpSQandhtreNKmosimnunchW?hJ\\Av`qv`Ak@\\NGpwWLjHDosUTqtpEpvOlMeUjt<SZqv>tPf]L=qkk\\N;IJk=XE\\RoxnxmTiUneItRMRAaQb@kvuNnQv\\QRK]ofawfAjrLNG@mU=sE<Sa=Oq=jjdO@xt:@KIEXNtM;UjqUmIUY_eV?TxdQv\\QNR@YAXykHpyhOrTkJDR`Xq;eMa\\RKMQm=OcYtClyNyoRENoYjtaR:av;YrlMr@lm^lPxpVyDm:imQlx<ir\\pV]yKy\\yZ=QrhuvINiPLUePL`LHDJdQmsUo;QK=@VptvThYfXy_ljrTUH\\qyuyXUKiiudyUBtk@HpLYqjXKDlLVIR_DSKPYeeoeLQZdMt<WH`YWPQCAOn=r:<sQELvLyLipuIs>lwv`O<InmuWiuKhEmgewiEPy]XQ\\Uq<M^\\vLajvpUIXokLs@LpfmKeHTuujpqNGDXGyJFAUR]YiYN<uJIaLpHPJYyttv=AvK@LA]y]puNumCqTgLxPTtkPwa=rWpWqikBpVBtw;xO=ToYIpE\\k=PVn\\Ln\\Uvo]>^bZW`J>yag[`huWie^?ivxgRxdf`gXIZPntTf_lV`Gg^OobqNcVHyjIv=fsJNh[_co?cXNwOFrMhwnQinFqrh`ZHlS_oU`hpn^lag]xq>wbMPa;w`ywyuyuoftkG\\LA_QidGVgeAmZf\\:IkX`cK?qS@tI^sPnw?at_P`Xqx>xnb@mhGbaGnMptCWod^`\\?awFo<Oi=`gx`ftXju>eWgeON_SYcZIpBQutp]qA_pFgJWpgap_xaOi\\wnriOaYym@YgCXk[xv\\ayQY_Gxmr>r@qqu>joIZBy`RYZw^[f^ff_^TfyCX[@HeOHysIiGyiJ^bF``uViZVoeAucNub@dewd_Pd[OluVhLViIF[K^dy@xExv<aZ;@d:o]m`tq@jHNinXw[hjdvg:XfD?yPf]ePrQyoYwwfawjoZVFdwNq?Vbinjg_dL>aL@[c>hK>tWOaPYgdI_tvkD`dgPvvv[\\Hftwhl>ydiqXqqwWo<qxAyv?_dR`c]QeaidCpu<@h>glNi^oW^R^b:ox]Qop^tpfpavZ_gpDhi>iwcyfaPcagw]GfjyhI^hBniaQa\\vd]V_LV]pPiCit;qv`qnPIfPviE`^fHyRxskHk<netFqwhqGhdBWcYHe_OwaoltGbMn]c`aS`uR@isQiBXgl^ktpZt^i>p_cOrmXq]NqX>jCAo>v]NXsi>]AoitIwafm>FeZirj>lkfavPwl@j=p[fal>WgoAahwfOwbPVZ[SiEBGMt\\MCNGu^GsP]VWeFFQUOWvOQBAaRwQWmou?eGKur\\qvmAc=GBpcGIaiAKh=yvdii:cENSrxqUvWC`AXuyB@cBhaUAawrMre=wJguTUGpkg_KE@=Co[sPsft]hHyGISW:MFUmUXmb?;D:cx^WT\\UumksRIXXIVP;ERGsFabPCC>UfLkFdSD]isUwHd_UgUBN[b>=uYeTaOUEGTP[SnwCBSH;CsKwt;ut:Ksb?E\\iTdQE:gwjqR@=Er[D^GFquD\\mFJCS<gBX=TmMTW?RwaH=giD[B==VDWr;]TyOegoD\\ktcqRWid=]y=YDh=wjUij;hdQC^CRTCSaGBcoCaOtUIxg[xDCBtCbusHrEbHar:sD>MEUMvi;XXcRHyh<]FaMgg[smmeVMSTkfTORFwTaUxQiT`siQGR>YGZAbk]sNWfpKXxCwcKVoIufcSoGxJMWMSx_;t[qX@iy\\Qv\\XJ_UlHHVR]xEqt<<R^tWDxSZLoSDL?hTFlyHulnMPQxwRyWZdXWlnOiOtEUiINcpOlejbPnKIkheWgEmmXswaXnqwdeOVMLv]MJmoPXN>iv;\\UIhP^\\tL\\X[]pYek>WwO_rHicXy`Dqjavg@A\\lotwPtmxpaG`og`ti`g_`ZquxhieixGHh`Ho_oe=?u;aeOQiH^^caqDYpNYkMFfefbu?`m>yjphc`aMPcMid:Ij^NnC?[M?^LI]uotKWZTVgpfeEgeMYqhFNWX`SC__CmkDnUXU;hVCv]SWEKe]=FnmesSVAgE=uCxYFBeecSSGcfQItsednocCwBciiNmb`=bEahA=f=OfWeemisqsujgE_kwQQIZgyH[H[ysAicI;VkAbo_uRAuBeUYuRogCMTSQ@JJXvd`rrMn`mYPDo<URi`yBQXPlQWmU`Epp@w:DTYulPlLP<ut`oAdoliwbiXFmyTeulMt[DVMXpS<lAmS=PTA\\OxAO=@lj`TFMnMpOCdSnmjv\\S\\ULfYudenLLmQUuO@TaTo`]MB`XiPOaTlMpsU]skPwQdxD\\qRpMdynUMgy_j\\amCqZpIj?FyEQwvPs:P`d>j\\hmjPdOV\\`@oTGhnn`FFrHOoWXweYwdi^jpZm@^PhxNiun@bMYmIPXAY:=xFYRV;vlAbfeYQcgw[COcYiScV?CY=uxqWWweZ;w?CvHKFFShWQB?GevmR[KW`]VMks^QWHSuRgY]mc_meKEYfWdgchNqrnkrhqWKSrHIbBCXxgr`kHiMYWww?_dUMTrkwSGCh]FD_FdgRBQfY[I[=YJGRacV[QUVMf[]enoWUgeO?BP]gcqt;CYiiWtOuwSvPkvWoxBGga_Wk;TMETyIYMGEtOUb_BqWW`=td]GV=d=[X<uimyR<OVASWlIb\\=t[UItegu=uXWcWPPAIMJmZ[Y]mQuffpcOhTWt[QcJYpL^mgxZ[nbkWnwGsJVtZykJ@sva_dIr<QLqUJmWeOXM?WskvjcTbaip]hmstBArLevhcW@iSHMtjcd;WGnsU]yV_ID^[HIStImsmkYWQrnufsqw:gR]gGLgEFuE@=Gvoro]sTMRs=yWkWCyiocUgmTbwBY=GY;fM[yHiEJKyZmdeqVQ_wvkXM?cdedNAIPEHFYRFUH^IdX?BewIFWx>EWGwgGkUpiyDYFPQYlqRsshlEWusg\\;IQ_YHGD`wynsI\\kfbsjpTT]mu^tKxqRadnU=V=dXCAkV\\Lvqs=dKPEjHhU?AsVeV;EsaYXo=rQajX=N_UqlHvePRw<k<=yStUMtW;MKheP@lraqvSQrt=kCxJVmvlXMbYxoUQiPY>HQQ<PEMTU@RaIKtYs[yJJYLrqmEQotqXdySadymLOS]WNylwMvUPq>lwFlK[uP?eT[HRchRitmaAn@TjOAwmeSOEyVxpNMltXMBQTUix;mUrTp[pwKaW;iu`qLu`o;dKfliyVyooocwvOOikVetAfPnmZHxRqi:Foeqq^iooG`dI^DW[R`xCXh`imHPwKoo?fZhauunw:iZfgggGkEArwv`YHmHYl\\gmhi`fnZTV\\GWvWhmWhr?ibIVgV_tpxpqhZd?efpoLXpJ@[xvixaqlPp\\GqZq\\vq[tygn_v>q_Z@Z@isyYwpoe]vf[xj@HZNywIyayayhiuuxnpi]k@]^aachh=awWwnHvh:i_rnwBgaoqZjW\\ZX\\rXx:_nCgjxN\\yYfLHZ=n_>p[sFcjtsesNcRI?BYwcQaELMDfGBu]TicGgcfyKShybxEBJIFR]wpUetKIGcdwcysmwOewvOSiGr^mfE?tKQCnmeZGGAIdqMBQcYccil?RAew>UcIwr]ouQ[Ea;H?kHJuVJOIkQfH_tyuSeMYH;hlur`cWnId^mu[MhCGC<wRJAbxoei;FwuEpUGGWV`wh=ipmuLWuwoQMtUQDlnhMX<qmbMxUpONdvl<qCqTD=YJaUYqse`vfTJJLpXXrVqmTtud<U;ukHHKI@Uj\\tqLVeatfeqi<p<usZ]P>\\lOiwdAo:dm:imALN\\TsS<qTPUf\\kjAQTdm\\Mj:dM\\Ukhxta@JJ<jE`vcLp]PNcAv]yJyyYwMlatNtQNOTJXUpQAOmxoQeL[dYJts;MxGXoTIYGlY[yxBlRepPnQsIewuLoQ`Je\\nj@r[HXquXQIXdmSTTWwtL[lyPeqkiusPMN<qmulNxVEEsamKAAQ<YWfLY?xNbPP_EyY\\OLEXXAS>gov^]N`t<`rgWn[GrUp[hI[i>eh_tMFoY@baWoD@h`NrKFtINssHnNikAwxOguh_k:Xn[IncX]d_bE_dCfw^Hr@IbXpcsWw;_s\\fnhH]Bvhap]FixGpgaxu_xnGiiKokC``@v[=`yoIy:gxHF[@gm\\Yrx>dsP_gvgsxsaIeDWfdxjxwvHHssYbKNxKvojAtKAc?gtDpc>qun@aBIwvpwOAn^`k@Hoj>kJGcNX^bxaRPttvqlPp<vh=_wLNwUytixe]yk=XxeFfJFoePqPaa[>fDffavlj@PWsuKfZGEoWHK;x^kB^yhYqYMUdZ]VjEih[Xcssggb@ceZCCuQCeOOHdTBQmiAVKaQ`MrGTNNlXyuYYeW@AJA=SwPoOitQuoXEw:ySqhNuxoIhRdXN\\mqfAYc]kRAuRALeLkopkADoxeqBlqH]QNMVQDUPQVpqQYMMJtq;MpfXPg=KclRmXnkio[MWMupwEyS`mb<NgUJ<Iv?huTAX:DrbYncPRh=PG=Skay:EkxEpxXWdlsIysiuWiUVAuX`dMATks<lWdog<wjhpohndFgmAdF>fE`lW>y:Wp=P[D^gNQa<ga<vuOw[M@l<qkyxrs>wQyeygeV_eNIcNFpxgfwhZMqu<`kU>jrqn=hpohpYpitI[Iy\\D>lB_j^fkvpaJqhuxw_>n>Ax^P[;@]FYkn?[WwrUPl\\yvfXb^FpH@ckAvqo]Y`mOIkoF]?^^cipBy_;Q^cAu\\vuB?b`pntVnSwfXqqwWsxV\\PA\\k@odOcZV[jVrKOsJH`AgfZohnGwwYyqygXaqkn[dopZA^=O]g>aT>fEWtUV_UFhRp\\c@cW?ou`rcAlCAaogytadCndaNcdibx^irApn`p>>uKf]@^iRQwFpqV@iHGjWFnRan@o`aXZuVv:p^<naNGy`qv`F\\jQr?FjEFyFgxBPc_PjNP[\\au\\wihngk?wCOw`OuawhGatGn_OYs<yatVbZpqffuBPyv^yIXZqo]M@m@^caF`K^fG@tZq^=gZ]AeVfkDavHGsRFnof\\Kf^nYoupykyky?aaYl`n]eApxg`qhmXwjVO[Thh^_sZOx:hZ<f]PHhR>ovV^>vs_NnWV`MFis`dX`rT`vAY^CnZSpbs_pDOwR>f?^hvQ[x_bHIbj_oU?pNH^rOwhNxxovcGh:G`OVsWaoCQpD>ikQmRFnKOn]^i;XdsvnPyoL_eZirMF\\JHsCfkbhZlNa=PgdqnIwnu@iqqmjqkZ@rWxbZvvtYtNG\\MFb@I]PNdgXl^N_AWy=AoOAaphZDY`;Fa<Yu@>lWXcmIrcI\\Zqdb>r@arjg[H?b]?\\NymvnkSHkrAb>^aR@kei[KQocXw\\^[\\>^q?nWApCh]a^\\R^pvfy?x]dPmCqbHIiLY]MNkgglsfo_F[cPgZvuJFeo>\\T`ocO^yqjqpZ_FueX^fYd`AsqWga>fAGjcXnV>eEPtgfhWQZ?fahQknin]f`snj\\fbHYp`xfiws<^hJgyThwQeJUij[dKud:ye=YrUeh[gtXki@cgGSX:sRX]InAXueVuyRuUBSIeGCR:QeW_UOiSTGR\\IFBac=?DAoBDIg<exaSI==BQOWpcc>gG\\uDDato;rUQRAMfvKG;WGCeujKBryG\\kh;;F<aRGyfjwURoCZAyVGYI=X:MiI]dJquM]GqMgpuS\\sb\\SupYSnYbxwdXCfhEtaMVQiCrURj_CM=bq[U`ItdQFlyE?;w>oUsIgj_IjACR?gK=clGvKkfesF@cRd=DN?dcwGqkBMgIAtQuuKrmNZIOEaTADXVeokYKwILgDN^ERgDtnMREYr_LVotlPplHYQ>tR=iTj<KwtPkdvXYuZYwqiNODNC]VNuK<almAsruNiuNWHvc\\NpTlbLoqELXDn@ELwmyvImcpK:DlXHLgYqgIURYsJ]xLuNULuVQQXQLUqPLmmo\\RH\\l=Xl<yOChYDUxfeoHAwPTpq<XaIQKTnPUnsixeytiyuyxpdDNbqL^al]=Vk<QUxoADoUTW\\iM;AltdULQUgAnJywF\\O[lcBijoo^vFbxalZpu;_mNix=N`fxvt^gvWnwX`\\XxSvnh_pa?bI?vHAjMhx;ol\\_rXYvyX\\]HvP^jNnjeYhXxuIqoL@awFimW]pPyRAuRAmWhmO^a[Vo@>w>nwFna[P_@wfVp]uF]jYpQ^tuHqJGkKFhjXoxaeWgeO>wZH_=iZ;ffppohhogHiS>sMgZ@fZha[`FvM^kcXm?FalQbwI]<IsUgqdn_@ffjhco?qTN[e_mP@[sO[wIpmy]WQucIab>haIuJnhJPc_PnSQnKvsXNwkpuXQmfQxawb>alsY\\\\QjFxw<hmFHfs_q=okHojwO_jwf:I[_>w<Qr<nev>atvjbhheOxx?tHHq_^oYnoA_uGVyganHHyY^alXcHFvQnc[^\\N>qm>p>x^ExhNYrIi^nQmqpnOitOwrKHchgpBgvWY\\kie>I`dvqKncAhkYNbBi[VI]vOsqPgEw_RAm\\I[YxZX?ieqqV?g<@fR_gKvh:wqe^sWAnlio>aofosSPcyF`:w^jw^biZKp^vFx\\DgiVRErS?Y`mV`stMSDmcVKOGIwwIKCD[uCKy=KB>=c=wvFKFLmWRqXNAI[]h>;iL;TdkU@er<gTCKCVICdYbBOrQ[dJ?VN_dc=g`=umyViuWIgui[Evig\\=FHCHwQIyUcw=Y<Ob<sXX]HKOe=sFT_tCaV`UiX?B?;tk?SOwWGkFRKyrIx]QytQyYkwEaEN[DrWDniBC[XfqImwiywipohPwWqEbBevbYXasFDLy]EjHMylTpt=TJ=vH=tY<LXYJ^uO@EtMxVyQp[PSadXFqss\\n`IRW`OrMqxUTsAnZ@rW@nipQvuN@]uSXR?TOfqxQPqhxP>XTkXm?XPVDVGuSW`Qr\\vVxkNXn;EjutOQ@s<pOnDN_`qGxucXq;lLkImZpJuHXCXk;xVCYvBINphyCe]natoipkhvEqqsFZtFb;XaiNm>FZjW_Hyn?qgy_fdYrKqZj@^TnlFi\\eV_e^wQnltFcZIp]xbaoZ?nw>fuB@k??s?v]nB;yu^SdaUsdSGk=Vv]iU?ECIV]wF=OBqMbhMSXYU@kt@kVAgin[Xv]Xn]RdyCfovsMw_uVaQcv=F;EbuCRwITXWgR?FFmHAGVUKtvCf\\_vc;UX;DA_Gb=rG=bH=\\;gyv?sH_heIwcIo:wjWPg>N^j?bB>dj>a>^fCxoSV]UVrRo_RN^IgZU_x?O_\\@n:iMqf^;GKKsBST:yDCcSOOS_itewTfae_wUpEHOCDxefM?ROCR^QBf?X@ICD[uCxM[LN[hR`uWilpuPrIujgPpJQQE\\Xj<oMdvZXMSYuEtpW@S?]LN<qmQnCdiCqa\\gdvhjsOylF\\wxx<V^^Gruvllhwi>^\\?_kHoEVjtXnV^mIw^:w^b>hau^CsBCCb?UfKhWGI\\?dHYhsydhcwhqwHysIYbZKFiasm]Xn]rx]fKwhr;GjuNY\\v>\\ofaRv@pj@NJIVeaxNIu\\]YkEXK=nMxwTerpIv<<NiAtJ\\yE=TZqCyWMiTGACn]xaqxpqxLQtLCHSOBWkx=YRuSrHyi^mC>YDAwI=yEa;GjuNYaRXIK;QkRmM;pK\\ulOXQPAvBxNLLsZEpL<w:iKCDn<\\h?nW@NC`T<dM\\mqC]KRPpf\\U^TnZDttDqf\\XXtk^El:uRaijJEpOmRJ=FwC\\SwGsB<wCkWH^Ev\\;yM_weQB`CfLghs;xuCgegy;[i<sX<OBqMbh@PPXKS=u<IRA\\JXEnnIryAmbtJF@w<@M;pKU`vC\\XjxV_YMEaJ?dZsGq^weUhogho_iv_atKqsrFhto]Gos;@\\AXo\\_iJO_qApj^t@hdE@kA`tkvj?wdAfZhaqnOhHi_JFv^av>?n]at\\H_EIffn^?@v@f]iuuAySSwIeENUgOIeSGebHR;XQ;uP\\<RAtSyLpvIQOlKcEX>qpZpPPLRKPUN]uGawfaOC`JshQ:uNb<Xa<oiMLbAkMIN<TLOak:dRlhLZXJIXjRirptKp`V_pWf?w=An:iMqf>OW`cX<;Ig?t;kW>;RbCXHsCIcxeWd^=TQ?u_WH>GXPWs_AX@ICDX\\iOrUpZawauqyCveBGfKgoPGp<XjJ?xx^m[W_eNx>njW_ab>ha>w``dfA\\U_hDffVpfN_qI@]SFrDF[q@pKXfRQfQyZwIcp_fCFe;NAh=MXPKgV;EVkS\\kUkESyos>EVQUVxaTk[HuCDieDN;imQTBUtX=cZucRCxf[w=SfMoxrGFjEb^kYBMTswgjktbKhEofHOcAsu\\mX>kR`QwSYs[sVg]wxyDZWDN;imQFBYikQCG]B:MesqxrGBn]bichTeSaCBQOS_Ohj?h:WXZ[UFMwh=sciU_MTQEX==i=]Xl=TZqki<KHdJnIJ]\\J\\LJTDVDlk<QvM]Jtlul\\mVhLtXSymwiywudSt@RA@N@`UVMkwQo<PJqMjhDNlpx;QxJ@YNuVlaLbQvq=umyLrHjAIpTAMBHlbmnu@rZdT\\iYuTjSIKD\\uCLq>`wqURd\\M>XnmYxmey<HR<qKOHNgpTPTQ?=J:dK:`N\\@NdDOkdKtDK`@w:lq@`pdqoMdYGXWHLKD\\uCLy<@KuqjiHJ>]q[aL:ITb\\LPxlQxppppBUk]<JTiuWiUUUUEeU?<L;=ULDrJlqbhK;iOZxNb<Xa<wlmYADpBYM?AKn]Jh<Ur@X`lVQ<Vftu<hL:uvnaxNTmN@RA`KX]t`DsEeVrYtL\\UJ=TZqcfXjt>u[alTW`mV`RwoJOvfiwgIk[Gt_v^wYj_I\\l@noo\\tAl_vZm?u^wjLwrW?d_VZnOj<vhX_ZKnug^`;gZ]wxS`ejP^C?_H`iV`xh?`Ha`DymYwvuH^XQsoHiIP_wF\\]XidAqmAdXxbZOfIY]JYt^@nZ@RIYMCxA[EZWUKavQIEleSbAEA?RLivBWGKIb:ubjeSksG@]exESoMwpet]wbdoSvKY>SU<CgdehdwBs;WEYIaeB?cBhaCHoi@GIYwVpYHqIbNerHKvR?r?MDf_TX=u>[xJWGawUb=r]=TO?bVMBokhZGT?ODMAWFku@?UeYVbmB?kwFLVAMOJmrFqkZqjrhSGaYkXMbYW`yvaIu;XTJHRb`lDTrrau\\Iq<dmGhkGhpohPL>mR@mRQuTQZTAgC>q`i_:Wj<PZqoZA@lg@rT>]VF\\df\\[NZUnyLpj<h]rf^MFnwioq?cQwhpoh>N^wQqvI[\\_ZtNro^lDoutQwbAwBwZjWb:hmUx\\]X`b>ha?dX?qmvmovy?Na\\Ygpyxqirk?r;Fe=QjY>^t>nNId?W\\UF^[YcB@rPF`wQ\\THgmQctO^K^ndXdy`ZgGw^IxkIpXN]SGu=PZqoZD@clh`sFaCpm_ohpopLGovPioQ[kyjM`nRwvFN_agmnoiV^g[gyd@fNG[u>[=@g:WmdyuXqu`wdCNnq`rUGdZ>_b>ha?hBqwyOfk>`ZWgdXkEApVWvZW_myb>paB>pBCEMxtgByKRLoXvwe;uHLwwvAx=yCResjCsAwEm=TZqsmMv\\=TJQBJKIpoBAqTdSGG=u]UDYmIvQylUwrgDqkBQksrsxGkouiNj]NqisKTJ_\\rs\\Qv\\T=akn=QrajyqSJ=vHYP:]MKlsx=MshoAIrDeRDxQkILrevB<TnYShTKhpQRaQaUkB`tGdWYpmXptS@sEYLnQtb@XeuRrxN?=kd<JDXTyIlM@u_=uItK`XJptjciS:atgAqRXmMlmu<nC@VH]KQItKILxQsYuv_Ix`tY;ER;ywvDT<eS]xUr@MJyNo]U`=Ve\\PqhrfaTehtLujrhjw\\VVpSjquuLqexQTauVara<K?TL`doNDYsMj<tXXxLcmlDhVRhmWhSS=NOUn=UKETK<mLGYXiQPhLuXquXMUToqFIfvwu>vy_n`A`j>PZW>uV`]\\O^ivcJ?vHYxZ^qua`;HsTp_PnblYkH^jc^wyYwcf_PVbfhkD>ZtflDgjgffXnrCwcIibka^n^ncPgmPmAPdsNxl=IcjYHP[YjwCaoVi?EYCcNMeIaWucfmAgGggeItCER:[YA[UIwHjaiQMDDaeXoeDgv[uRJ=tJaIH]v_ICD[usmP]@q:XRVUJdaV<mo>LQtiXBXJ@hORUMHmwiywIom@aZpA`k?mfQih_k]IqKxla>w?Ntiqc_^f`?cWasl>iK@pw`mrXZivt`y\\vwqMWbRYspht_`tGhqv`pvwkWfce^yii`bqk`Ieepn]xudynaGO]HpYutCe@UGoKTmAIwYRAchtIs?MyRosMUfXUFVISMoTgmf@KvEIuxOcNwDZqCsoFlEBGCU@qXhkYOQRhEXy;w?YXYKVCIX<SCQMfLouAwcu?fwgiwsEUqcPOhr]TwsdWybOAYAIgU=vtgRwQgGorugv`stYwtjuuoYTRqySIYsmyFYIvIBdyxgaiomUXycMEvvAyxQwIyRNmuOqiKuE?yWwAwoyuEsi]ueQgwgiwsIys]iT]x<;uneipQdXobVoetuHGqXlOeeAb:eWLAIbySaWhKEBlcxweUXgSuESDogUce:Uwvaht[IlcFZOEDuWMASt[IFUXmOwCgfMYdTAyRAurExoqspIw\\QhTkXxWYJoDQqrMUhWwt@YUbIgH]gQMYtCyh]xbehDeUcWV^EEwCT_cy>mXvOr@gg\\=Y^St=KIxOWyaGwuTwSchQR>YRbYB;yWXuYtQt[GDoSHDExxOwaoP\\PyVeJ>AJLUx]`tIqQy<Pq]ycipGhmQvx:__hOw`UhygxMsZiYhOSVeGyOG`WxrsSkKXB?DxwttIwRatGqEAWUvCydSTAySOsslguWiUpuVu]dtywEeI<oB>QEGMiI_T^QyvQWScW<wF\\gEN_rEkB`QBq=EXQc>_rt;uSIu;UbuEWsuXMGgTegd?eZAwvgHWUGf]sPkc@QCeYT=UvyywfafxETUgEdmUVcGGcXfgT?OxbEfrmdTWrGothUhKquXscZkBUqbB_v>Gd;CwLuHdQGYMwgYI=?s]QBbQgKUSMoEfsHqWD]KeyIcE]XIOdbqboYw@mBkOy>eXt]g^YSeiCeyfKgI\\Ufaas=stiOIoug;SG_mSCmIjaxJYFVvewqrJ_\\v_dUXyCgviHiYixm_cqqxroot>`Jfv:oykp]qpyu`rWV`T?hP_r^W`k^aGhZ:Oai>iLax`fdG>u^iccas]IwX_dRn[vptkid_IiCOh^XwhNyeAxvPpjNdcxnoHnCIoEPykOaWn]e^wIhpPP_gxmMIuuPxb@luAxZwwbOf`_oO?cAx^uWbc_ZVajQHZWA[uaniHjg?rXnlGfbIpk[gsFNwXOwwOlD_kQFyN@wS^ij@iZvgYgxmPrmPu:vxFhlU_tYYrH>[AItsgaDgyFngiFarA\\jQpjhnRq\\c>b[YyhxmUFo>nkifhqyb]vayhiu`o@XqyPywFwBWr<QnHx]cx`vpw;`qAGoyqmfi_?OyN>b;^hBHm`np:xohxp\\WxsfytWgjgm>y_`wiywmt@s<yvJaZ]GrOgvSgdnQfdHeI_vui\\@x\\`Xsy^[GfcW?cfo_vicEVe[OhrA_W_\\@pib^kONwGhqtWx^Of`?otxuhalCHxsNbk@lDFt_Vte?cX_nsHjiwjgNnqFsl`[HqyvYxIa`OgZN_[JhvXWk>HwEiym@_fXfkahpncSoccYoxO`YpvUodbYuY^]V^xUwaiAgGapxadAXvF?xbfxmne;W\\N>hFAedFtYHi\\Ggwquqgyf_Z\\QcJOZJhk>w[<hpqOxMfyswuoYw:g]EnnMg\\myuFHkxN[CqmbhwkPwyTi_SXKXAcIDSIoosjwsVKDZovUSrKGwA=V;wu;SiogXQMIckG@uxdytP[eEUsaoW@WVX=eq[tUqgNsIiqtewDT[y=ybcGyYMtbqxseY\\aIgqEhahhKeUubaseTOw:QrS_byWhhOII_dcaxfOiLug:StHAG[ssHcEGkH_ocuEFt=WwqiuuxHuxtswYOC=id:Qb_YyaEXZoykuXKWcAuS>]UlkEWaixmwOIb>aWoSeq?hCyy`ixeOUJGdk?gyCTVkIPeyfeWBYW<QS]wcQCwZcX]=trOSHwruegYwT@gwOeT^=FJevfOFt]ERsr<SR<IsuegYeFOIhhaVRKWyMBhixAKxaSYrQyiItXgBh;hm\\vtTw=ixfepFDsEUPAhjT<v:TTKyMeyLPyTAQOi<JxIxAyMqtPqDYyYSEynh=VqyPYaY=@o?inmQvDAuRaTdmw_XlkHkKykcaXxpYhlYuevGauyIQEUV]QUaMkT`yQdjLMMhav^QT=ajkEY:\\JXYjbQoFynZdsTXjEEu`pKbDXnEKsQrq@lc=t_YYpysLIx@qr@ukG@rZQXmQyS<Lt<NMAVhtjqiTyxL@lV?PUbPrtTxRexeqUAEssuMklNd\\q=<W<PRJdugiqPyW@pSyPoDdxVlMUpTeXxbMLiyuqeydxrrTWWLKEUVIyRkyVt]ltywEtXRMs:mLaXy>Qy=ISkLPdTn`@wvLk[=TJUtIPJNxR^=vF@mqhwgwoXgq\\OpWgdxfhmHfKNnsYqCvaW`thniSpmd^kaHomXj_IlnnoMgpmahNQyqhknFan^aMPec`edW`>hnlad[fmeX_K_`lP\\THpNgjfHmRWtnahoVby@_dI_FnxeqhwG^Q>d[wUUi;;W`owxIUjWByEyEyUZmBoMDxSC?cWYexUMh:cFW;ITyWwAHw=e[isFEHMwE_oYWYgdeG[?g`ERpcYbWBJiCceR?ADmywooY@YeoyRPOSlmtRYEIUgksxWuit_TWSC=KrbiX`qDw=CwmCXog\\MyFmdIoExoyuQhaEDHEgX[FuEIDEU^MURyELMBMqWPAR;QCF?G=]BwkdxERMSEBaEMGwG?gnuGSyFJ[cSUGjoTTUBwcIN=W`WdnLwfawvdkdEMsYUttyx`URpqrxUyMjuTNneXbtvfEw>uv_UrAXvHiuE]l;\\xsYtTauVEyA@j`ipE`MoaY;mPvQtkhPYqSKtWOdlePRULoUIuBalghS[YRwpxjIjletlxpXTSWikFdwjDle@PmmlDQmrqMedlQDQall_AQq<nrPOY`VkMv_eWQHpqdkwQRseSYesoaohXPqlWpAoQEuG]rqEu:yk?`PIDVkYNf]XmhKutxeeqnpM^<NjLrpmUQlwkhOHlUxMjlTrBhJAIum@JETkZmpZXNYQnhDta\\lBeNxprsdNlEY>qTLLqRdvBlOj<Ow=yvqlRIvj`XjHuDuuHpTpusn<q@YSMtyedMxYSchNhpRVTvmAuTiJ@IRZXjopK=IjXuU]HMx`Kr<xfXrA\\JXusPYLvMmbARviYjPNm<yxdYtaNXMLgUoAMOmhJUaQ@aT=pkaYmOhyjEtrpWieWgEw@=TIlp<xW[ut;DP^PoDHPZ=SDYlVDuoeNoUtUHwIESSQYY=p@HV:UVJIM=ayluOZ`sbXv?dQHDSU<jUalTIS<=UF@j>ipSxxNuNr\\uQqUK=uCekSEqSAmuEXGtXRApgURvXQctxSMYcXpalWHus_HJh]qgDq:]NxXVQpt]PrdaJvmKsAtXYlPYOnluXmTexKNlwj\\kjHV;DMXAPM\\N`lrLHxgetFYyqygsWx@_kIVxDx[Fx]TpvOPw[op:yw[G_]?dtgmqhyXFbN^vLAjUfmhnylOaK_c_ht<WnRGyQX^kgpgHm<Oaxo_rXrRptUFuc?jNx\\tYlvQsFAb:wjH?xTxgSVur`s:Xtin]S^iBFjHpfoQwdQ_=w]VIrV?k`qaYFrrNg=Hwd?]@xa^WexhZuPiCnpfA`gog<f]SNoA_p@ne;@T[S`]iM;RU=ymeYweuOYvoyfKQRO=rSccn[X\\ASCUImUdD;E^wU:ubKWVuOd?wGpAC<[HseBdCBc[eo]Bf;vWMXYMGmKHrUH;kRB=Uretl?EjicnQFd]cHGg;kIWIBq;VPoeb;uOUrhOgAwGeWBOwrluVeyBFWeSIHawVT;grCCvqCn;DK]fKiEdAIr;F`?dOAY:GdDsyoygyeH=WbEIEL?h:uRMUCxMdKqV?WC\\MBdwGweDoiHZusQurfEfhwEPyVd]GGgf]yUdWYUigTIW]mvnAY]]GguIQoCOui[SGcwrnCeKwycYRnYwr;GZgi]]ecmBdWSOyhdqXV_CRaUNOJgDuGLlI\\JbAYi<PO=KeLsf@V>ToEEwgUXPlW^PU\\xOeUTWhsk<PvAml]PPiP_\\XfUn@LTppUD`nnAVV<l;aU?My@mUjprGeTwmyvIMMIKphLPUNTDO`drEHQVtJmpSIpRfXjkIrUMRFpMjPn`DW^Hrcn_?Nap`t:xk`owf_o[Gv>qe@XoQpm<P];hihIxDIn;nfsidLfxGnujPqd>tkPy=_eRf\\tXiOWjTqZiWoqXqdp]wNbYGrUNkjilgYo;wmehadFk]yu:HvHyw@NhantNYyeXababRh]>vdyHqtncaFmW?_LHcWxn<>xV@aiArjY`BadmVu?AiH>wKNyLQ]HPfT@]Z`o=fwOYqr^hxoir^atfud^qi_`QX[?XkLWhlhgpWiPG]AHrS?m<VwN>kt@jU@pL^joxtyOu]W\\HGZyx^w_\\D>vjXk>QkxibZinX`fSvtQPdyVgP?engf>VpKwk_Ha;pamp[=yeX_sKIid?jxpyDQsDFlwAZYNdIpcdYr_InX^`Rp\\ToiKWmoIhoYo[Ir@_s>^fbhjeYweQ]jNfShdO^eIhrqnbWVasAoA^dFn\\Uwh[hZAFfEgeLYvxnivAs;Njdpfcfm=O\\rNuCVnaw`kwiKf]BNlNQ\\:?tHVlF_ZoaaePZfqZDQcdInophRYbrpZr^q]V]RNoiXhcgatXstVdIH]O>veOiW@fwxytnmkIu]qkwIkgh`n_\\_oo=`so^fjW_XX`rWws^j<>wkpj<Fj<YnDHh`PfbGiVF^;?`HWeRxoggkFv_QqcLAuBOaxvuinp`wr@YtXqfTXeyo]WNokyZYWnX>tb^pXar]WsmaeE_p`X\\BxwF^jnaafwqdIt^P^IxZBQnDFlQ^kM?ir@g<VrU@\\Txg_I\\]x]:wfM?b=vm]VfSWiOqZw>c=peYIdO@f_FpYXo_IbAaxlP]kviJiaUQmCWaK`u;n[APZSirHHqApcDhxYPoJviRwj>F`FQgOoo=FjW>w:VsZXlBngBf`:?jThcTHZ\\^aNi_jYajv\\dVleOpGpyP__:Y\\;AsbX\\>qeGwhBgxMvr^xrKImd?jXpytixeWxfqu>FsYpZ=VsOgxJIpZx[Yn[xYk^GvTgenY[eabYg[RnwNHxfX``@yU>eKw[bAasq^E_bDqh:PrrQkVnc[O^g@uyY]wxkthdYgeAQww_d^pvdFhr`p>o`vNqvGdIgvCy[cayoar^GaoVt[iryqmUwyqgw:GtaG^rph^aiAQm:Acu^Z;AxgHu_HdbqgQ>vlfwkwwPwfdosvYxYxynPaFVybV\\LQaHnx=AdYQov^a`p^RXvg`o^nrqh[wp^HgljwZFA\\\\Xh[wiBNjKaakvgI`n=a_L^gSWwNYpKAeh?`lqtianbFZo>kYGr;@rtf]p_t:go]pc]gj]o]`qmP`jEfvsww[?wnVi[Ajv^jna\\u`y[hhLyo]QeKNbyxyxYiCvbdHZZgpsW[_nw`qv`NjaV_bVjSftFpeY^gLyjMYicIwdXdOXwvomnN\\NOny_dZQot?ly@dCFf<vbxft;NinWquYcRW]vFae`meP\\OicrQxUWrAw^\\`ebIt\\?oqfw?ogon]JVrrOjQi`Pxvkp`cVs[okux^IXgeW\\Lqu^AtDIy?acs_b\\htPIcg_gdOvy^pThr<Y`CwqQxhNh_G?cK@pX^honyGp]ixnWNj;aflVgjWeWOrRfsG^pj@w:nlEAh@I]y^a`hr]OdFquOp`=VuyfrKWqjIetFwjydJikoxtc`arW[^y\\f?[bGfRHsv?gU_[JYhSwa_ar=ohG_bOp[vIrwwv`W^[nkr@]uoi[G^_gl<gf=gj:xo=QsRQ\\Pwc;yyBpcMIpdPnUIdi@`<Frr>unV^low>v[UXjlFb:IvVy\\rGsl@t\\?dmHyCqntYceInafgginMhwOPrd`kKqZmqs:NvHog=nws>fR_v=@jIOvo?qaVZsVjrvqKYZEixnagAglAQuqXnmNtDWi:w[]_[tqgvOrhgtIYjlGkwvtEHjHntZ>nFxcSO]ZVum@rno_s^iNQw[^ebXxhwbqWv_It:pgpnuHI_sxaEVp<auMXucHvhXxTwaRWcfptkY]snfGPiBHxKax=ibGQlExxEVdUA\\I?eY`]hOjS^\\oNkoN]s?vhvwHymXigOqu[pdl@`mfwCXkIIZvqnuNsrNlevtgyuNYlmIr`NbeO\\fp`po[Z>\\>fopv]Ng^?NaNYxo_b_inoAeUwsUWxJgydHfdY\\u@^:qctfvMAjRYpL?tWWa>GaHnuXg_FHd`y[?>\\Bd[cSaoh=CRa=GEkTBGta?eKsE>CF@KE?oG=adVOWjODgUWZuFdoCUuk=iORAsRYoADt=TPgLYeuR\\XV=lqdaQILV<HumMRIdTuiJeLjI<yQlNvTXCtqReQK=r>tL@`KBtx;xu]mmllJWHluHl@YVwPrOYJwQQIpV\\UU[po\\MLK`qCdQ;]KsxM=\\O[xl>dPU<t;HnEIRAImU`X]Dm[@lBAt;IN=QjEAy]Hj`dyPYr\\dkJEmiLUdHyAyMy]v?dtmqPntKThQe=x>urAiNGDoQyoYUV:qniEXeMpu@JmUucuXNmv]yrIxuwuxWTLOll;Hy]xuIumxDJ_LqdhPJel=QwihogHPpeo[@V\\PWhIwLhpE=oD@VM\\PUdXE<oKPqc\\msXsO@WrXlwqyeYxH]K@tXJXpo=KQAYZEw^yvIymgENPiU<xQ[qk`lMMxMIhLuXQeqUutnbxPjxlMXlFaqmVbjFnlq`PHnBvbd^n]`^]AnKhbvFrFhjtxdQg`ugxnO]@h_oxbA^s]Aea>t:GlWFZK_tfhaDowVW`gPld?m=AghoeIh\\UqdNWal?khNfGHsOPi;Xa;wtKh_:ptA?e]`yoXxup^LFfofaswlwfvBNn[goHY\\f@ea_jyIps^yXf[Hfh\\fl:nhT`^^Ylqor_^wDompvdWxwChh:?n=Xm:flnq_pHa:IocPoDgnlPplPb]?m<?x]n\\m_oDFr@hvv^yHAtrixhooPVkJq__IqD_t<W_<QxFOaspaKAsEnquQdXAZxN^UwaTidG``XQbkfhEXaVqhLpieIjsXnMXx>ihHNdCguB_`ef^<osDNqHan?NyRyd:n_@witvxJnylo]tAoy^`<XxfvacXmsfZ]wZ`?^iOaDqay_bXynhXqANcPo^^XlBAn^gbHQbVVZDf_LN`E@`MaeooeiXfGXZg^puXxiQksHv[g]MxuqvsrpZbAd@gcOgdFWin>qR^nR`jvX[eyaoYnWAbAavhi^T`cMitD@crXc>V\\>OeU?f;Qiuxkkw_@h_XNkLGaoWrYAvl>hCIbDVnk>hQ@xeOwPFj`fx:Ej;bwqt]oh[myaiVg_TlQv^qYJSuU[yH=to]BAwWIyW`gSXORRIiGUXvEhxeu;EcpWbZeD\\er@yYYCR=CcC;Tj]SckyikVr?XviBT;x]=hWWccsbCgIIAwtetsussuRRCIM]DT]CNCHr]W__s;KdWcsJKerSF:]hgsCPefUOvp?uMgvIsT`_tkUy<=V[wHVCW;oStsx<MHuAjJ<kS`YBngXXu;woZpZ]PyCY`ANtdfoGNqQ`rQ>iPavSwuk@bk`^:Vhwqt]FytgiwgmQFt_NbR?q_v[OghnWeZqCUF=qd]YhKehRmUjGSUQXJKRASRPGtf]vDyVxWBiWXgMbOcff_eFiwVWuDYXdsx^CehIiqwVQqvvaIHis\\oeFcswKdWSD@wTZcRf;d>su;gsRwTpeIs?UdifXQHWSTNigl;W\\wcIYuZ=wcId?AVLSYLuv_QYnoESqtcybgQHHWr[se@stGaEyeTOwi<iseIh;Ot;[Us_VGiUv]gvkMBIU;ulMmmqTsdhVPhYr<XHApa@NMETaDtn=uA@lC`n;tjOIuxtjPanDDXjAM^]YMhuOxn]UNAHyamudobLFskNrCx_??sM?wcnotvml@]Q>^cffHWZXI[AphTaZ]gkcXxo?\\I>kshku`a;HqEpn\\Gn=g]hQZt?_HPZdxj`?mU_hhwwNiktwbsQZvWuu^sSXZ?IeZhn@Qn>>ebaqUHnUi^EHrSpcn?kPG\\gwZpPrUQv]nuiga>ylZ>itYh^x_qym;n\\A^s<p]vqcJ?dJ?xuFb;Gb;Xa;GB=TZ:RJ=>:=TZ::Fb;<Z;GJ:Fb;<:jZ@B:@n:;jZ@B:@n:;:RJ=>:=TZ:RJ=>xr^xyIhjlLxO@oElwB<J:<J:`n\\tN\\tT[<P;5;\"\{\}LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlNVFGKi0lLkJPVU5EU19IRUlHSFRHNiMkIiVdPEYqLSUpQ0hJTERSRU5HNiI=

The angle, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==, is positive if it is counterclockwise from the fixed reference line and negative if it is clockwise. Using polar coordinates, we specify positions or points in a plane by a distance r from some fixed reference point, called the pole, and an angle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== from a fixed reference line through the pole. The values of r and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== are called the polar coordinates of the position or point.

In the strict sense, the r-coordinate of a point in polar coordinates can be positive, negative, or zero. (See Section 9.5, "Polar Coordinates and Graphs," in the textbook.) However, in this module we make the assumption that r measures distance from the pole, therefore, we restrict r to be greater than or equal to 0.

<Text-field style="Heading 1" layout="Heading 1">Converting from Polar to Cartesian Coordinates</Text-field>

Given the location of an object specified in polar coordinates, we would like to find its Cartesian coordinates. Therefore, we write a pair of transformation equations that will convert a pair of polar coordinates (r, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== ) into Cartesian coordinates (x, y). The positive x-axis in the Cartesian reference frame is taken to be along the polar reference line with the origin at the pole, and the positive y-axis is in the polar direction, 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. In addition, we treat the variables r and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== as parametric functions of time. First we take care of some housekeeping by loading packages into memory and restricting r to be nonnegative.

restart; with(plots): with(plottools): interface(showassumed=0): assume(r(t)>=0);

If r = r(t) and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSgmdGhldGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RIj1GJ0Y0RjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZORjEvRjhRJ2l0YWxpY0YnRitGUQ==(t), then the transformations to Cartesian coordinates are:

x:=t->r(t)*cos(theta(t)); y:=t->r(t)*sin(theta(t));

Let's consider an example. First, we specify the position of an object in polar coordinates.

r(t):=t: theta:=t->t^2: rvectorpolar:=[r(t), theta(t)];

And we convert the position to a vector in Cartesian coordinates using the transformation equations that we defined above.

rvectorCartesian:=[x(t), y(t)];

<Text-field style="Heading 1" layout="Heading 1">You Try It: Part I</Text-field>

Given the following time functions for the polar coordinates of a moving object, form expressions for the vector position of the object in polar and Cartesian coordinates.

unassign('r','theta','rvectorpolar','rvectorCartesian'); r(t):=sin(t^2); theta(t):=2*t;

<Text-field style="Heading 1" layout="Heading 1">Part II: Position, Velocity, Speed, and Direction</Text-field>

Now that we have a vector expression for the position vector in terms of r(t) and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==(t), we can determine the velocity, speed, and direction of travel for the moving object. Here we express the position vector in Cartesian coordinates.

unassign('r','theta','x','y'); x:=r(t)*cos(theta(t)); y:=r(t)*sin(theta(t)); rvectorCartesian:=[x, y];

If a radar detector is tracking a moving object, then r and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== vary with time. The derivative LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjhRJ25vcm1hbEYnLUYjNiQtRjE2JVEjZHRGJ0Y0RjdGOi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRi8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6 measures the rate of change of the x coordinate of the object. This derivative is called the velocity of the object in the x direction and is denoted by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInhGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ==. Similarly, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjhRJ25vcm1hbEYnLUYjNiQtRjE2JVEjZHRGJ0Y0RjdGOi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRi8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6 is the rate of change of the y coordinate of the object; it is called the velocity of the object in the y direction and is denoted by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInlGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ==. Next we determine expressions for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInhGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInlGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== in terms of r and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== and their time derivatives, and we write an expression for the velocity vector.

vx:=diff(x,t);

vy:=diff(y,t);

vvectorCartesian:=[vx,vy];

Or, more simply, we can take the time derivative of the position vector, rvectorCartesian(t).

vvectorCartesian:=diff(rvectorCartesian,t);

The speed of the object is the magnitude of the velocity vector. First, we define a function to calculate the magnitude of a vector, and then we use it to calculate the speed of the moving object.

magnitude:=vec->sqrt(vec[1]^2+vec[2]^2);

We use magnitude( ) to find the speed of the object.

speed:=simplify(magnitude(vvectorCartesian));

The direction of motion of the object is the velocity divided by the speed.

vdirection:=vvectorCartesian/speed;

The magnitude of the position vector, rvectorCartesian(t), is the distance of the object or point from the pole.

distance:=simplify(magnitude(rvectorCartesian),assume(r(t)>=0));

And the direction of the position vector is the unit vector in the direction of rvectorCartesian(t).

rdirection:=expand(rvectorCartesian/distance);

<Text-field style="Heading 1" layout="Heading 1">Part III: A Specific Trajectory</Text-field>

The path of motion of a moving object is called its trajectory. Let's plot the trajectory of an object. First, we define a position function in terms of the parametric functions r(t) and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvRjNRJ2l0YWxpY0Yn(t) and then graph the trajectory.

r:=.5*t;

theta:=.5*t^(5/4);

x:=r*cos(theta);

y:=r*sin(theta);

rvectorCartesian:=[x,y];

plot([op(rvectorCartesian), t=0..15], labels=["x","y"], scaling=constrained);

<Text-field style="Heading 1" layout="Heading 1">Part IV: Visualizing Position, Velocity, and Acceleration</Text-field>

In Maple, you can create several plot objects and plot them together using the display command. You can then create a series of multiple plots and animate them by setting insequence=true. We create a small loop to generate a series of graphs as time increases by a specified increment. Each graph in the sequence shows the path of motion up to time t and the position vector at time t. To animate the graph, click anywhere on the plot to display the animation toolbar at the top of the worksheet. From the toolbar, click on the play button to view the animation.

window:=12: timel:=12: ax:=unapply(x,t): ay:=unapply(y,t): for i from 1 to 40 do p[i]:=plot([op(rvectorCartesian), t=0..i*15/40], labels=["x","y"]): a1:=evalf(ax(i*15/40)): a2:=evalf(ay(i*15/40)): a[i]:=arrow([0,0], [a1,a2], .1, .6, .2, color=blue): ani[i]:=display({p[i],a[i]}): od: display(seq(ani[i], i=1..40), insequence=true);

Now let's add the velocity vector to the animation. We want the tail of the velocity vector to be at the head of the position vector; therefore, the arrow should be drawn from rvectorCartesian(t) to rvectorCartesian(t) + vvectorCartesian(t).

vvectorCartesian:=diff(rvectorCartesian,t): for i from 1 to 40 do p[i]:=plot([op(rvectorCartesian), t=0..i*15/40], labels=["x","y"]): a1:=evalf(ax(i*15/40)): a2:=evalf(ay(i*15/40)): a[i]:=arrow([0,0], [a1,a2], .1, .6, .2, color=blue): b1:=evalf(subs(t=i*15/40,rvectorCartesian)+subs(t=i*15/40, vvectorCartesian)): b[i]:=arrow([a1,a2], b1, .1,.6,.2, color=red): ani[i]:=display({p[i],a[i],b[i]}): od: display(seq(ani[i], i=1..40), insequence=true);

What the heck, why not add the acceleration?

avectorCartesian:=diff(vvectorCartesian,t): for i from 1 to 40 do p[i]:=plot([op(rvectorCartesian), t=0..i*15/40], labels=["x","y"]): a1:=evalf(ax(i*15/40)): a2:=evalf(ay(i*15/40)): a[i]:=arrow([0,0], [a1,a2], .1, .6, .2, color=blue): b1:=evalf(subs(t=i*15/40,rvectorCartesian)+subs(t=i*15/40, vvectorCartesian)): b[i]:=arrow([a1,a2], b1, .1,.6,.2, color=red): c1:=evalf(subs(t=i*15/40,rvectorCartesian)+subs(t=i*15/40, avectorCartesian)): c[i]:=arrow([a1,a2], c1, .1,.6,.2, color=green): ani[i]:=display({p[i],a[i],b[i],c[i]}): od: display(seq(ani[i], i=1..40), insequence=true);

<Text-field style="Heading 1" layout="Heading 1">Part V: Speeding Up or Slowing Down? Turning Left or Turning Right?</Text-field>

A few things are worth noting at this point:

1) The velocity is always along the tangent to the path of motion.

2) The acceleration has a nonzero component in the direction of the velocity. We show this by plotting the dot or scalar product of the acceleration vector with the unit vector in the direction of the velocity over the interval of time during the motion. This gives the component of the acceleration in the direction of the velocity, which we call Avel.

timel:=40: speed:=sqrt(vvectorCartesian[1]^2+vvectorCartesian[2]^2): vdirection:=simplify([vvectorCartesian[1]/speed, vvectorCartesian[2]/speed]): temp:=linalg[dotprod](avectorCartesian, vdirection): plot(temp, t=0..timel, labels=["t","Avel"],labeldirections=[HORIZONTAL,VERTICAL]);

You should note that Avel is always positive, indicating that the component of the acceleration along the line of the velocity is, in fact, in the same direction as the velocity (instead of opposite it, as would be the case if the dot product were negative). Whenever this situation occurs, the object is speeding up. If the dot product is negative, then the acceleration has a component in the direction opposite that of the velocity, and the object is slowing down.

3) The object turns in the direction of the component of the acceleration that is perpendicular to the velocity. In the preceding example, the object is turning to the left, and the acceleration points to the left. The opposite is true when the object is turning to the right.

<Text-field style="Heading 1" layout="Heading 1">You Try It: Part V</Text-field>

To assist you, we copy the commands you need in the following sections. Change the entries below and execute the commands in all of the sections that follow.

Change the polar coordinate functions r and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEoJnRoZXRhO0YnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0Y1USVib2xkRicvJStmb250d2VpZ2h0R0Y4 to generate the following motions. 1. The same motion in Part IV, with the spiral clockwise instead of counterclockwise. 2. The same motion as in Part IV, with the r-coordinate starting at 10 and decreasing at the same rate it increased in Part IV. 3. For each motion, use a plot of Avel to indicate when the object is speeding up and when it is slowing down.

restart; with(plots): with(plottools):

r := .5*t;

theta:=.5*t^(5/4);

x:=r*cos(theta);

y:=r*sin(theta);

rvectorCartesian:=[x,y]: vvectorCartesian:=diff(rvectorCartesian,t): avectorCartesian:=diff(vvectorCartesian,t): speed:=sqrt(vvectorCartesian[1]^2+vvectorCartesian[2]^2): vdirection:=simplify([vvectorCartesian[1]/speed,vvectorCartesian[2]/speed]):

window:=12: timel:=15: plot(linalg[dotprod](avectorCartesian, vdirection), t=0..timel , labels=["t","Avel"],labeldirections=[HORIZONTAL,VERTICAL]);

ax:=unapply(x,t): ay:=unapply(y,t): for i from 1 to 40 do p[i]:=plot([op(rvectorCartesian), t=0..i*timel/40], labels=["x","y"]): a1:=evalf(ax(i*timel/40)): a2:=evalf(ay(i*timel/40)): a[i]:=arrow([0,0], [a1,a2], .1, .6, .2, color=blue): b1:=evalf(subs(t=i*timel/40,rvectorCartesian)+subs(t=i*timel/40, vvectorCartesian)): b[i]:=arrow([a1,a2], b1, .1,.6,.2, color=red): c1:=evalf(subs(t=i*timel/40,rvectorCartesian)+subs(t=i*timel/40, avectorCartesian)): c[i]:=arrow([a1,a2], c1, .1,.6,.2, color=green): ani[i]:=display({p[i],a[i],b[i],c[i]}): od: display(seq(ani[i], i=1..40), insequence=true, view=[-window..window, -window..window]);

<Text-field style="Heading 1" layout="Heading 1">Part VI: Circular Motion</Text-field>

We put all of the commands into the sections that follow so we can try some different curves without having to scroll back up to re-execute commands. First we generate a parametric plot of the trajectory. From this we select values for the window size and time limit for use in the following sections, which generate the graphs for the animation. The motion we consider as an example is along a circular path; the object moves clockwise, and it is speeding up. First, we plot the trajectory.

restart; with(plots): with(plottools):

unassign('r,theta,x,y,rvectorCartesian, vvectorCartesian, avectorCartesian'); timel:=16*Pi/3: theta:=t->-0.05*t^2: r:=t->5: x:=t->r(t)*cos(theta(t)): y:=t->r(t)*sin(theta(t)): rvectorCartesian:=[x(t),y(t)]; plot([op(rvectorCartesian), t=0..timel], labels=[x,y], scaling=constrained);

window:=12: vvectorCartesian:=diff(rvectorCartesian,t): avectorCartesian:=diff(vvectorCartesian,t): speed:=simplify(sqrt(vvectorCartesian[1]^2+vvectorCartesian[2]^2)): vdirection:=[vvectorCartesian[1]/speed,vvectorCartesian[2]/speed]: prod:=linalg[dotprod](avectorCartesian, vdirection): plot(prod, t=0..15, labels=["t","Avel"],labeldirections=[HORIZONTAL,VERTICAL],view=[-2..15,-0.6..0.6]);

The dot product of the acceleration with the direction vector is positive, so the object is speeding up.

Next, we animate the motion showing the position, velocity, and acceleration vectors on the moving point.

for i from 1 to 40 do p[i]:=plot([op(rvectorCartesian), t=0..i*timel/40], labels=["x","y"]): a1:=x(i*timel/40): a2:=y(i*timel/40): a[i]:=arrow([0,0], [a1,a2], .1, .6, .2, color=blue): b1:=evalf(subs(t=i*timel/40,rvectorCartesian)+subs(t=i*timel/40, vvectorCartesian)): b[i]:=arrow([a1,a2], b1, .1,.6,.2, color=red): c1:=evalf(subs(t=i*timel/40,rvectorCartesian)+subs(t=i*timel/40, avectorCartesian)): c[i]:=arrow([a1,a2], c1, .1,.6,.2, color=green): ani[i]:=display({p[i],a[i],b[i],c[i]}): od: display(seq(ani[i], i=1..40), insequence=true, view=[-window..window, -window..window]);

<Text-field style="Heading 1" layout="Heading 1">You Try It: Part VI</Text-field>

To assist you, we copy the commands you need in the following sections.

Change the polar coordinate functions r(t)and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEoJnRoZXRhO0YnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0Y1USVib2xkRicvJStmb250d2VpZ2h0R0Y4(t) to generate the following motions. 1. A counterclockwise circular path with a radius of 10 and the angle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEoJnRoZXRhO0YnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0Y1USVib2xkRicvJStmb250d2VpZ2h0R0Y4(t) increasing the same way it did in Part VI. 2. A clockwise circular path with a radius of 10 and the angle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEoJnRoZXRhO0YnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0Y1USVib2xkRicvJStmb250d2VpZ2h0R0Y4(t) decreasing at the same rate it increased in Part VI. 3. Try this one: r(t) = 10 and 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. What do you notice about the component of the acceleration in the direction of the velocity? When is the component of the acceleration that is perpendicular to the velocity the largest? When is it the smallest? Is the velocity vector ever equal to the zero-vector? Is the acceleration vector ever equal to the zero-vector? 4. For each motion, use the plot of Avel to indicate whether the object is speeding up or slowing down.

restart; with(plots): with(plottools):

timel:=16*Pi/3: theta:=t->-0.05*t^2: r:=t->5: x:=t->r(t)*cos(theta(t)): y:=t->r(t)*sin(theta(t)): rvectorCartesian:=[x(t),y(t)]; plot([op(rvectorCartesian), t=0..timel], labels=[x,y]);

window:=12: timel:=16*Pi/3: vvectorCartesian:=diff(rvectorCartesian,t): avectorCartesian:=diff(vvectorCartesian,t): speed:=simplify(sqrt(vvectorCartesian[1]^2+vvectorCartesian[2]^2)): vdirection:=[vvectorCartesian[1]/speed,vvectorCartesian[2]/speed]: prod:=linalg[dotprod](avectorCartesian, vdirection): plot(prod, t=0..15, labels=["t","Avel"],labeldirections=[HORIZONTAL,VERTICAL],view=[-2..15,-0.6..0.6]);