Displacement and distance traveled with vectorsphysicsleninac@hotmail.com
<Text-field style="Heading 1" layout="Heading 1">Iniciando</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=use DocumentTools in
restart:with(plots):with(VectorCalculus):
end use;
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Ecuaciones Param\303\251tricas en plano y espacio</Font></Text-field>Supongamos que una part\303\255cula se mueve a lo largo de una curva C en el plano xy de tal manera que sus coordenadas "x" y "y", son funciones del tiempo, luego:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzIyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tRiw2JlEiZkYnRi9GMkY1LUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1Ri9GPEYvRjxGL0Y8 y LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEieUYnLyUlc2l6ZUdRIzIyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tRiw2JlEiZkYnRi9GMkY1LUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1Ri9GPEYvRjxGL0Y8Las llamamos ecuaciones param\303\251tricas de movimiento para la part\303\255cula y nos referimos a C como la trayectoria de la part\303\255cula o la gr\303\241fica de las ecuaciones.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=MFNWtKUb<ob<R=MDLCdNFZ>Zy@[<J:XSBPmiR@iYLCTJcDXoXuuV<ktNidpIXlWdOKDB[bweD>slUltwUJ?UWghpOImTIthlXNqwpmnghWaYwbUwaeWAqSgXxsdXguVfQvNQWOQqFUUOPW_@jBXOlxolpJ<DK^UNeDyXavP]RB@q_DNEYyumwMxv`quEiqQXlmyQaiYIQxXLlCaVlQRJ\\rEYkFHc;?dF>[<Pf:?\\NVZ<`[:@]hner@foAqVo_y^ywyxY^oyw]p?veqvGWmDWcD?xa?bVIeFvwvhyQXgof`wwkmAiv`kSYpIgmtfxWqqw?xO_^Lffawf]ys:@qg@]:oZ?ir?yeygxohtm`ddfx[IyIv]yIwa>uIF[GfcEwcAym;FhUG`vGpuw`vowwqvyNegptdiZMv^GOsyvlIvfUhr\\AtJAdvI]JvxNVZ^Gxwq\\M>c[vdJnpJweLvjFFfkHeaWhlW\\LV\\ky`:_uP_[ZB^=RfywxyxhYUWmdTOIpSIuyTNSB\\qh\\=TdcdUmyiUSqytlQhlWFYyEpOY_Av>GrcEwmCxA;RVqTFsysQYI?icGG\\KF\\KI`kFgcIGkcviw=yCuqU`CdBMvAsCXCId]FxABBWx:EUmoVGgCiuhwUupgvesThWYyeYxIimUUpoWWgdKGtLQtFAVjKF\\SFCWCxsyfYHDcbL]TR=IaIrACHoIDaObdUuUiWWqiwwvQAirAyIsHh]ws]vdid<cg@Ui\\IiO?enAcUWc[qhv]hcYVF]X>YFvmB>CRk;FBKc;CRJ]CBKb\\=RJ[D=:>nR>\\>fR:Nj\\;:ZsXdb@m>>keX^gatqHoGFvx^[@fttGvXxhpawCXvAY^]Pnr_aXFmFosIYaQAtJwqgwvH_\\=Nvh_]fFgUIh[HsUPcxIicOdYYpy^]unpcYvGWyIn]ZWpSarKHumovfXeuq_THcBX]L?t\\PjIAiyndYYd?ib?w_y@yLQxOX_thl:?`qIvs?_@^rggrj^_JQkRnuUWyhfjDgmsVgPwo]IcCGcbn:uHT[DTedHAWPSR:sfQIglOtL=xPkrIwEIQVkow^qTbkB>ihcwdyqDciIS]rhiXdaWhmGvmfHgEuubqORk;FhgSK_RgEgP;uvkRiqtZcdTaYDwGZYfR;DI?fRQF@Gx[qIwuS__f^?UFgfloUAirGeSbkB>iCcOWQAemiYYSEC?HJOdLWdyWuqwcR=RJ?vBIv=iR?eRPuW?gyF;cESfPYEU_DTuWdCIK=spOY;qtWSuyaxDQgykIm_wkgRECRLcr?gREkEu]xPmU>Kc;CX[mGCgTWmFiGv;_WdCe^ex<CfJ=iuYgisgVmiVmUXmCPUtYkdfCfLuWSwvK]CBWCMGUgSbIIEKyhkEdAwF_cCqotk?bxSB\\qFRKiFuFeWu[OSRoE_YWMWE\\?EJsT_?fSefk[X??xR[GTcB?Abr]HaUEFiW>EhvmfgoWBOV:]UO_etUhRkj=EPZlmGUoTuUPqtCTQ=hyS=O@\\r]HQTmu@tmnLPc=epNk;Fxboqif`EHfV?]IXba@nR>\\IintheMAoJIrHanBveUnuPfr:@qXvaQwp=y^?`voAbbv[?Ajr_`Sp\\k^^C@avo^Ofr:@ihfodgi_`sVqkNp]GypgimK_[BXkKXgcqeZQtcV`J_;Iv^utSgrJGDe;VUqgWCC?AJCmOChljIQWqtOQJDHJkEPh`Pe`VTTRjtlx\\QJhWmYm]aY@Hj>UWbQObETOupeMUNMypdpX@QBanKXJSaxjtM@]m:HM<YVZMOoavREqNlQaaKlYpAisK]KHpnMuWv`NPevqLOhPSNengemZxjxmO@dTF@K\\aNE@QcHnV=uqAx^XLedXy\\TDEMNLx]DKxAO@tlQHrQdJgTYEdN>=te<j;eNIhK_irD<m[qQFDLaypvpu[hPnTOnenLqsLDktpShhyiejNlPrlQ[\\uaEyBANr@u]ULjYm<EKNhMMTrw]YwHoblJQlnSEoUeYlPq^hTG]YG=Llpv?IyGek\\YxRdyoDkNDSNYyD`NaLo^MyhTdTqkVPqHhdI^jh>cJNiZOZ<OdVn_mvkAOcCNvRIwK>?IDVOgcuUqmfP?gJ;f:wY`cRnub=scMcbfwX`sRv?I`[vUQVCqe<oDGGfeUVpeYMORqOVN=xgof:Yi_sD?QgQuYf?tR]SDav:]D=hvMay@iuQXm[urYhRuuXd]VD@whxYEmQMPyAIL;ukFlxyAOapMnexOEREEn>ey;akbpqk\\vVQLbPLyyvaQMGiXaQkHEpQE_UpdgAk[Xsnqh`gcqP\\pvf>qu=P\\tHju^qyfadGxjtEQtPqyfIvloteiCJmWFEREOWoIcdcSygsF;Um;IucETSrewIOMb_WC`asiqb@[GGMCnGS;myVAwZACYuwgCYGchWKuaoCOogUoDS_VqSh;IEdabGCUooRMiSAcwQuE\\esZuGxksgwB_POW\\S]<PtiWoYLLHS^HqkmYupv?YxwHoJUPv@YfPnu\\KoXmkirOpvQqkQepoMMteMa@ugts\\@MfavjIOIxTSXvUeqwDP\\MYNhwAlK>@XS<VnEYMEstqOMyVQdrhyl:@vWdymaU]XjYLk\\=pwLWwHXgIoTpOMxt\\Hn<ak;MJudQ_lT:aYmMRr\\wvaTcTop`JYiXUtReYrfQqweRLtryLM?AVump`yPoTonQNHdy?Ht[lXw=Tt]KymTf@rbMPw]lMuvXDSa]wldy_Tp`ARkUw:`QWdMbUMMxtZ@uY]WhdtVTtO<xtAqJYs\\evsqR_yPQUUPdKG<XoateutRLt\\]Y[UNxPOnpryhVQiXS@p`=XGQOxYjYUW[yrEHvmtV@AxW<x>mNGPxrhnkhXEIq@uPbEWoYVKAr?QwXQo^MMB]sF@U;YRkyP@mvkEwRQm^QKV`oPHLKUMPhkJDQfInMMNGivb<YLaRnaLqHnpXRuiUk`JdttaTl?uoGXKU`Yw<L@dKuduVMj=dN`awRlr]=LS\\tk<ym@W>=lBXJ_dohalRXuIyPSdSXUN<@q[qQF<PrttGUkVTL<iRmDso<JedvlhVLlTIQQWIPJtS>xLZLVkQq;tL?QsMploxL:qRtisuhNMpuYpM_uksdQblJUYTbEUTXMdAXudSwDwAHK[qQ>AwQYPQQQVYroQuNXUiiVhaW``SG`qkPX[Tk@EyrmVFLk;IQGPnPqJ?ppaEV`iVSmXlXQRxkREoMYMWDtNLJ_pVThVH=xCPopUj=QMY]gGyfUNsTQac^^LyeavblvgXWi;qdEG`Z@]f_c_`pmItwhbih]iPsi_dRXg@VsT^oMa_e@sVGgPgycynDipaxoSysUYw?Oq_HxdXbe?q`QvW>haVoTyqo`xrh_Xn]jn_FxvByb]wa=iqMH`B?py@qoVopXZp`fKvnTofHijQWeNYqh>kea_]ijxNcIHhRYsJooqIgnN_nG`hVaUnoMwgVahPagZfZmGwIw^E`]vahfH]him[Nbp_nix`kxiC_oNGwGYhqoxOFcBf^>Npcqkf>ulXgyWhWXgt^erpfrNdhf^tWcXw]YnrZPlK_[h`rVXeIfaT`\\[YvBy`CQouisoojdOsogy`iq?pf=FgP_oZf`>xvry\\:G`Vy\\jYpWQsc>mfAbk>japcfVf=?qe^jqQvl_isgnspehhxaGtPgkGF]Jx\\:G@VsGQkoPds^[pyfSygXQiY?k[QvWaouivT`ee?hg`lv^ePqp]QbMVk@ouRp]w>dMIeK_[qYsMG_;giInqknaF`cl_u>hy=y`I^rbHgQagxVubQbc?mWw^Ywi@aa`XwCHv;Hxx_o;PiG^\\=iyLNr_x^WaamHaS_rkHvdXiqhZvnstpaMhux?obWoJpht`b<n`efhewcQArHvuUPqjFhZVnZAnR^g:YjqVqKyc`vZtNxcingpdv^Z]Hsey]sg[HymnOi]HwGX\\OOhgH\\VwoypcC`ps?fg_\\=XnSp\\_YwwgqW>`@xsn?hE?cCWoowhsvo;GvvYbOp[PppPPeVwyaPd>Gopo]Axj\\?h>^_TfjbQdZVlFqu<XjYpi;Q`F^rdvwYHomovCo[QVmNXuIy`;qsMI_\\xffHqX?_@vl>ym?WjIImkNj[oxVndSy\\m_]iqvYqZBn[GrHOsZuY`YUMSrasRrAR_etfOtgMsoMT]Ew\\?UoQxIcDDeekEuBCtmWF<_e[idj=I@gWBKh>gCHkvtKIJmxsqF_oiEaHl;emQrbADjwdhohbgRfmrVAIy;u:uFQOeOwVhgXBih@sVEgiQeuhGYuaVO[isAx:mhm[tC;G@SULcVSsdKMu`AwXqgYUwtkUSSEl?WBGY`yFK[gU=s`GDa?ENMHruG=KtOWuy;VsggxCT_StR;fpSUokHNIYX]C?AVUKD>oVMcvDgr\\EX^SsAmBqurkqGj_Sb?C^AtVGI[_ttUvdwEeSWu[sgEricf@OIlgCTcY@sB;kXB=CIGFQQE;IGn]wmEGWIef_sLagNiIp=tCCBEiCN?Rn=bvkIf;V[ADaQcgwc\\=hSSffSvikic[WS;dawEdcWmKe:ic`cCOKhWWHw]cWmfGCUo=TsMtBafyscESYk]SBWhwGFDGr`=D[YEHuUeAF`QyrKDEAYIUhBsF>Wxv]iWGtniF<KdyATccx`OdyCsR[h:qIeAT[wsZaHoSRjsdGufb_X@iUQcrjwwByRaetp=cOovxaBAySyowY;gAkgo=WR_HlIcDqxMytTIfH=yuasRaDKiGeqGoOUGQVWydQCFiAwZIyVYvrEVfGEfmBGORtMvtQvCsIQahuKxKYIi;e]uCASXcYDWSICOfymxVeWSmY^_WwIeTyyCag;cn]Tk@PKien>dlDepgYtIepTYy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yXhRk<NX`V]dOX]uIIUn=nK\\J?A:::CRk::::::BqaD=QeqqmCXkZ@MFAysIy[Mv\\<O@\\JkESxyMTpW:PqIlJ?EO:<J:<j^PNaLNQENjDB4:\"\{\}-%+ANNOTATIONG6'-%)BOUNDS_XG6#$"#]!""-%)BOUNDS_YG6#$""!F*-%-BOUNDS_WIDTHG6#$"%+EF*-%.BOUNDS_HEIGHTG6#$"%5>F*-%)CHILDRENG6"La variable "t" es la figura llamada el par\303\241metro para las ecuaciones.
<Text-field style="Heading 2" layout="Heading 2">Ejemplo: plano</Text-field>1.- Sketch the trajectory over the time interval 0 \342\211\244 t \342\211\244 10 of the particle whose parametric equations of motion are: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 , 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Soluci\303\263n:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Ingrese x(t) = y y(t) = t = use DocumentTools in
restart:with(plots):
x_t:=Do(unapply(%xdet,t)):
y_t:=Do(unapply(%ydet,t)):
gseq:=Do(evalf([seq([x_t(i),y_t(i)],i=0.0..%t_po)],3)):
gp:=Do(pointplot(gseq,color=red,symbolsize=15,symbol=solidcircle)):
gcurv:=Do(plot([x_t,y_t,0..%t_po],view=[-13..13,-13..13],color="SteelBlue",gridlines=true)):
Do(%por=gseq);
Do(%vatp=%t_po);
Do(%Plot0=display(gp,gcurv,scaling=constrained));
end use;
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHuse DocumentTools in
Do(%xdet=NULL);
Do(%ydet=NULL);
Do(%por=NULL);
Do(%vatp=NULL):
Do(%Plot0=plots[display](plots[textplot]([0,0,""]),axes=none));
Do(%t_po(value)=0);
end use;
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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NiI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFH
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2">Ejemplo: espacio</Text-field>2.- Sketch the trajectory over the time interval 0 \342\211\244 t \342\211\244 10 of the particle whose parametric equations of motion are: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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ0RidGL0YyLUY2Ni1RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZWLUYsNiVRJHNpbkYnL0YwRj1GOS1JKG1mZW5jZWRHRiQ2JC1GIzYkRk9GOUY5Rjk= y LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ0RidGL0YyRjk=Soluci\303\263nLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Ingrese x(t) = y(t) = z(t) = t = use DocumentTools in
restart:with(plots):
x_t:=Do(unapply(%xdete,t)):
y_t:=Do(unapply(%ydete,t)):
z_t:=Do(unapply(%zdete,t)):
gseqe:=Do(evalf([seq([x_t(i),y_t(i),z_t(i)],i=0.0..%t_poe)],3)):
gpe:=Do(pointplot3d(gseqe,color="DarkCyan",symbolsize=15,symbol=solidcircle)):
gcrve:=Do(spacecurve([%xdete,%ydete,%zdete],t=0..%t_poe,color="Cyan")):
Do(%pore=gseqe);
Do(%vatpe=%t_poe);
Do(%Plot1=display(gpe,gcrve,scaling=constrained,axes=normal));
end use;
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=use DocumentTools in
Do(%pore=NULL);
Do(%xdete=NULL);
Do(%ydete=NULL);
Do(%zdete=NULL);
Do(%vatpe=NULL);
Do(%Plot1=plots[display](plots[textplot]([0,0,""]),axes=none));
Do(%t_poe(value)=0);
end use;
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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NiI=JSFHJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
JSFHJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Vector posici\303\263n</Font></Text-field>Si r t) es una funci\303\263n de valor vectorial en el espacio-3 dimesional, entonces para cada valor permisible de "t" el vector r = r (t) puede representarse en t\303\251rminos de componentes 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<eM:\\Vr@O^<QrmpixoPYtspS;=JSlOc@U<]Y<yrY`Y<ltA<sjELQXsWep\\evtAjG<Vt=jFtLC]NLxMqpyjYxiTyo@uf=Pw<R@XkHpMxLvpDX\\aNquXjXQ:mjEPkEPkh]UyppjYvI@rbAJSlOdhocHUlIyQDJUPJ;Qv:DMXEoR=rtiKUpPALRy>ZSnetphoQokAa]GwJQlZ>jf?LatNaexCYdEwlivZUW:_CifZY^ijpxUihFnZ=^gG>l[QnWXap__SopFqrPq\\Ff[JLVYnAMVK@YhYqiQjOHJl\\UNeTSepiuWJqvgxu=Hk;=q:mNNLKyYprEt\\xsPtKr]LC=L:mxILQjxyQHkM`uaXYtMjwERr@R@PTklWcEwqXwyXJ]QjjEjF]NLdWyIwahx=hyp=OB<NeqlR]NTUwyPuqmsu<tcPM>xL^ERuUjn@uT=tIQJ>x\\DOkEXlLV\\QaoUwZpfZ[MflLyAy=yY=LJqDj;ERwePBTntAnN]WOQsOpwXTls\\MH<MN\\jExNEdQqmSJQNR=pjENRUL>XK^epodPQuoVEumxOB\\SCLr?LPNPkg=yhey@Ar:IrTLlI\\JJhxP=tXhq``MHHkcHORIL>XK^uX]tw>quyxx?Qt_<V>=Lh<lCdj<iQuxXyqY^AKnEY?Dv?Du;hyy`OtaWkTT?eJf<K<iJlxqh\\UbTO]QyHIm=lv=Dv<DUHmXpyxUxPyup;lRFEJ[YJS@U^LukhgZAlJ>waGZ<Y[lxmqiwUhw]YZiw];@x<`_eV`mF^xFfpxoI>eG^jE^fVo_qYs_qiU>nEY_Bv^BAddnpcWb:?tIYyFasGIr^?^V?^Efaqog<Xs>V_eWgjx\\b?l[Vgf_gVoq^GwYvbHp]t>iJ@c^nynYvyimwfi@OwUQvYvaxH]XFjPYri_u:wb<H_;WZ;`_cF`kN^v^yyXeqplNvlvI]LNf_ggoqwuWxjhe^N`SAr:AriGul^bR`qv`a<?\\FG[tomX_tmn`uV`EVeKitMYauoaNhr\\gxVYo@Qfk?aVFpR_[]?r<pZHFe@of@Fk>OmuvgXQ_yiddwia^[chn_hbKGckHn[HZF@[T>a^gdOhdaxSewW[EtqwdsGeSsDasvqW==r[yuxUXpmX@oG_=f;kv^uBHcEfydjoRxsY?ytYAIG=RgiSe[vCAcjSWr;w?OFnmXpMbO;RrOwyIrZqy\\QX:wHXgG:;B:;RLEdMCde?DR?6>\"\{\}LSUrQU5OT1RBVElPTkc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlUz9GKi0lLkJPVU5EU19IRUlHSFRHNiMkIiVnPEYqLSUpQ0hJTERSRU5HNiI=r medido en unidades de longitud.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Ingrese los componentes del vector posici\303\263n:r = < , , >LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= t = use DocumentTools in
restart:with(VectorCalculus):with(plots):
r:=Do([%cp1,%cp2,%cp3]):
xt:=Do(unapply(%cp1,t)):
yt:=Do(unapply(%cp2,t)):
zt:=Do(unapply(%cp3,t)):
gsq:=Do(evalf([seq([xt(i),yt(i),zt(i)],i=0..%Slider0)],2)):
gseq:=Do(pointplot3d(gsq,color="red",symbolsize=15,symbol=solidcircle)):
pvg:=Do(PositionVector(r,cartesian[x, y, z])):
gcvp:=Do(PlotPositionVector(pvg,t=0..%Slider0)):
pvn:=Do(PositionVector(eval(r,t=%Slider0),cartesian[x, y, z])):
gvpn:=Do(PlotPositionVector(pvn)):
Do(%tpoeval=%Slider0):
Do(%Plot2=display(gseq,gvpn,gcvp,axes=normal,scaling=constrained));
Do(%spts=gsq);
end use;
segundosuse DocumentTools in
Do(%cp1=NULL);
Do(%cp2=NULL);
Do(%cp3=NULL);
Do(%spts=NULL);
Do(%tpoeval=NULL);
Do(%Plot2=plots[display](plots[textplot]([0,0,""]),axes=none));
Do(%Slider0(value)=0);
end use;
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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NiI=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Desplazamiento y distancia</Text-field>Si una part\303\255cula se desplaza a lo largo de una curva C en espacio de 2 o 3 dimensiones, el desplazamiento de la part\303\255cula sobre el intervalo de tiempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EnYHRfXzFgRictSSNtb0dGJDYuUSYmbGVxO0YnLyUlc2l6ZUdRIzE2RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkotRiw2JlEidEYnRjMvJSdpdGFsaWNHUSV0cnVlRicvRjdRJ2l0YWxpY0YnRi8tRiw2I1EnYHRfXzJgRidGNg== se denota com\303\272nmente por \316\224r y se define 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vector de desplazamiento, que describe el cambio de posici\303\263n de la part\303\255cula durante el intervalo de tiempo, se puede obtener integrando la funci\303\263n de velocidad de t1 hasta t2: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Se deduce facilmente que podemos encontrar la distancia s recorrida por una part\303\255cula sobre un intervalo de tiempo 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 integrando la velocidad sobre ese intervalo, ya que: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Ejemplo: Suppose that a particle moves along a circular helix in 3-space so that its position vector at time t is: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 the distance traveled and the displacement of the particle during the time interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JVEiMUYnLyUlc2l6ZUdRIzE2RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEmJmxlcTtGJ0YvRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSi1JI21pR0YkNiZRInRGJ0YvLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJ0Y1LUYsNiVRIjVGJ0YvRjJGL0Yy .
Ingrese los componentes del vector posici\303\263n:r = < , , > Ingrese t1 = y t2 = use DocumentTools in
restart:with(VectorCalculus):with(plots):
t1en:=Do(evalf(%t1e,4)):
t2en:=Do(evalf(%t2e,4)):
r:=Do([%comp1,%comp2,%comp3]):
xt:=Do(unapply(%comp1,t)):
yt:=Do(unapply(%comp2,t)):
zt:=Do(unapply(%comp3,t)):
gsq:=Do(evalf([seq([xt(i),yt(i),zt(i)],i=0..t2en)],2)):
gseq:=Do(pointplot3d(gsq,color="red",symbolsize=15,symbol=solidcircle,labels=[X,Y,Z])):
pvg:=Do(PositionVector(r,cartesian[x, y, z])):
gcvp:=Do(PlotPositionVector(pvg,t=0..t2en)):
pvn1:=Do(PositionVector(eval(r,t=t1en),cartesian[x, y, z])):
gvpn1:=Do(PlotPositionVector(pvn1)):
pvn2:=Do(PositionVector(eval(r,t=t2en),cartesian[x, y, z])):
gvpn2:=Do(PlotPositionVector(pvn2)):
r1e:=Do(arrow(<xt(t1en),yt(t1en),zt(t1en)>)):
r2e:=Do(arrow(<xt(t2en),yt(t2en),zt(t2en)>)):
r12e:=Do(arrow(<xt(t1en),yt(t1en),zt(t1en)>,<xt(t2en),yt(t2en),zt(t2en)>-<xt(t1en),yt(t1en),zt(t1en)>,color="Cyan"));
Do(%Plot3=display(gseq,gvpn1,gcvp,gvpn2,r12e,axes=normal,scaling=constrained));
Do(%vnr1=evalf(<xt(t1en),yt(t1en),zt(t1en)>,3));
Do(%vnr2=evalf(<xt(t2en),yt(t2en),zt(t2en)>,3));
Do(%vnr12=evalf(<xt(t2en),yt(t2en),zt(t2en)>-<xt(t1en),yt(t1en),zt(t1en)>,3));
Do(%distr1r2=Norm(%vnr12));
des:=Do(int(Norm(diff(<%comp1,%comp2,%comp3>,t)),t=t1en..t2en));
Do(%despt1t2=evalf(des,6));
end use;
r01 = r02 = r12 = Desplazamiento: t1 hasta t2 = unidades de longitud Distancia recorrida: t1 hasta t2 = unidades de longituduse DocumentTools in
Do(%comp1=NULL);
Do(%comp2=NULL);
Do(%comp3=NULL);
Do(%t1e=NULL);
Do(%t2e=NULL);
Do(%vnr1=NULL);
Do(%vnr2=NULL);
Do(%vnr12=NULL);
Do(%distr1r2=NULL);
Do(%despt1t2=NULL);
Do(%Plot3=plots[display](plots[textplot]([0,0,""]),axes=none));
end use;
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