{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } 3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 0 0 255 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "He lvetica" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "Helvet ica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 259 "" 0 "" {TEXT 257 25 "Moderne Physik mit Mapl e " }}{PARA 261 "" 0 "" {TEXT 258 9 "PDF-Buch " }{URLLINK 17 "Moderne \+ Physik mit Maple" 4 "http://mikomma.de/fh/modphys.pdf" "" }}{PARA 258 "" 0 "" {TEXT -1 19 "Update auf Maple 10" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 25 "Kapitel 2.2, S. 60 bis 71" }} {PARA 258 "" 0 "" {TEXT -1 24 "Worksheet newton1_10.mws" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "c International Thomson Publishing, Bonn 1995 filename: newton1.ms " }}{PARA 0 "" 0 "" {TEXT -1 86 "Autor: K omma Datum: 30.4.94" }}{PARA 0 "" 0 "" {TEXT -1 21 "Thema: Newtons Physik" }} {PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Anmelden \+ der Packages" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "restart;with(linalg):with(student):with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 646 "Newtons Bewegungsgl eichung kann als eine m\344chtige Maschine aufgefa\337t werden, die AL LES verarbeitet. Womit wird diese Maschine gef\374ttert und was produz iert sie? Der Input ist ein Kraftgesetz und Anfangsbedingungen, der Ou tput ist die zugeh\366rige Bahn. Wir wollen die Maschine nur in diese Richtung arbeiten lassen, denn die Umkehrung ist entweder trivial (zw eimalige Differentiation der bekannten Ortsfunktion f\374r die Kraft l \344ngs der Bahn) oder nicht eindeutig l\366sbar, wenn man unter Kraft gesetz das Kraftfeld versteht. In der folgenden Animation k\366nnen Si e allerdings durch einen einfachen Klick die Maschine vorw\344rts und \+ r\374ckw\344rts laufen lassen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "step:=0.5: n:=20: nstep:=n*s tep:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "klist:=[seq(display (\{plot(0.5*x*Heaviside(x-2)*sin(x-delta*2*Pi/nstep),x=-2..15)," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "textplot([1,sin(2*Pi*delta/nstep),` F=m*a`],color=red)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "textplot([8* (1-delta/nstep),8*(1-delta/nstep),`Anfangsbedingungen`],color=blue)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "textplot([-2*(1-delta/nstep),5*(1 -delta/nstep),`Kraftgesetz`],color=blue)\})," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "delta=seq(i*step,i=1..n))]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "#p lot(0.5*x*Heaviside(x-2)*sin(x-5*2*Pi/nstep),x=-2..15);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 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Doch daf\374r haben wi r ja Maple. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "sol:=dsolve (sys,\{x(t),y(t),z(t)\},method=laplace);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<%/-%\"xG6#%\"tG,(*&--%\"DG6#F(6#\"\"!\"\"\"F*F3F3-F(F1F3 **\"\"#!\"\"%#FxGF3F*F6%\"mGF7F3/-%\"zGF),(*&--F/6#F " 0 "" {MPLTEXT 1 0 5 "x(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%\"tG " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Die Funktionen sind noch nich t zugewiesen worden, aber das l\344\337t sich mit einem Befehl beheben :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assign(sol);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&--%\"DG6#%\"xG6#\"\"!\"\"\"%\"tGF,F,-F)F*F,**\"\"#! \"\"%#FxGF,F-F0%\"mGF1F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Wie s ieht der Vektor r aus?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "r ;op(r);eval(r); r[1];map(eval,r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %\"rG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'vectorG6#7%-%\"xG6#%\"tG-% \"yGF)-%\"zGF)Q(pprint46\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'vecto rG6#7%-%\"xG6#%\"tG-%\"yGF)-%\"zGF)Q(pprint56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&--%\"DG6#%\"xG6#\"\"!\"\"\"%\"tGF,F,-F)F*F,**\"\"#! \"\"%#FxGF,F-F0%\"mGF1F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'vectorG 6#7%,(*&--%\"DG6#%\"xG6#\"\"!\"\"\"%\"tGF0F0-F-F.F0**\"\"#!\"\"%#FxGF0 F1F4%\"mGF5F0,(*&--F+6#%\"yGF.F0F1F0F0-F=F.F0**F4F5%#FyGF0F1F4F7F5F0,( *&--F+6#%\"zGF.F0F1F0F0-FFF.F0**F4F5%#FzGF0F1F4F7F5F0Q(pprint66\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Oder..." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "v; eval(v); map(eval,v); map(eval,a);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%\"vG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'vect orG6#7%-%%diffG6$-%\"xG6#%\"tGF--F(6$-%\"yGF,F--F(6$-%\"zGF,F-Q(pprint 76\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'vectorG6#7%,&--%\"DG6#%\"xG 6#\"\"!\"\"\"*(%#FxGF/%\"tGF/%\"mG!\"\"F/,&--F*6#%\"yGF-F/*(%#FyGF/F2F /F3F4F/,&--F*6#%\"zGF-F/*(%#FzGF/F2F/F3F4F/Q(pprint86\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'vectorG6#7%*&%#FxG\"\"\"%\"mG!\"\"*&%#FyGF)F* F+*&%#FzGF)F*F+Q(pprint96\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 120 "Man m\366chte nun nat\374rlich gerne de n Output der Newton-Maple-Maschine zu bestimmten Zeiten sehen, also ka nn man versuchen:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 5 "x(7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\" xG6#\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Aber x(7) ist nur ei ne neuer unbekannter Funktionsname und" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(t=7,x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,( *&\"\"(\"\"\"--%\"DG6#%\"xG6#\"\"!F&F&-F+F,F&**\"#\\F&\"\"#!\"\"%#FxGF &%\"mGF2F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 229 "funktioniert zwar, ist aber zum Schreibe n zu schwerf\344llig. Doch es gibt einen praktischen Befehl im " 0 "" {MPLTEXT 1 0 67 "xx:=makeproc(x(t),t); yy:=m akeproc(y(t),t); zz:=makeproc(z(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,(*&--%\"DG6#%\"xG6#\" \"!\"\"\"9$F5F5-F2F3F5*&#F5\"\"#F5*(%#FxGF5F6F:%\"mG!\"\"F5F5F(F(F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yyGf*6#%\"tG6\"6$%)operatorG%&arro wGF(,(*&--%\"DG6#%\"yG6#\"\"!\"\"\"9$F5F5-F2F3F5*&#F5\"\"#F5*(%#FyGF5F 6F:%\"mG!\"\"F5F5F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#zzGf*6#% \"tG6\"6$%)operatorG%&arrowGF(,(*&--%\"DG6#%\"zG6#\"\"!\"\"\"9$F5F5-F2 F3F5*&#F5\"\"#F5*(%#FzGF5F6F:%\"mG!\"\"F5F5F(F(F(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 6 "xx(7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(* &\"\"(\"\"\"--%\"DG6#%\"xG6#\"\"!F&F&-F+F,F&**\"#\\F&\"\"#!\"\"%#FxGF& %\"mGF2F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Das geht auch kompak ter:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "rf:=makeproc(map(ev al,r),t); vf:=makeproc(map(eval,v),t); af:=makeproc(map(eval,a),t);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rfGf*6#%\"tG6\"6$%)operatorG%&arro wGF(7%,(*&--%\"DG6#%\"xG6#\"\"!\"\"\"9$F6F6-F3F4F6*&#F6\"\"#F6*(%#FxGF 6F7F;%\"mG!\"\"F6F6,(*&--F16#%\"yGF4F6F7F6F6-FEF4F6*&F:F6*(%#FyGF6F7F; F>F?F6F6,(*&--F16#%\"zGF4F6F7F6F6-FOF4F6*&F:F6*(%#FzGF6F7F;F>F?F6F6F(F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#vfGf*6#%\"tG6\"6$%)operatorG %&arrowGF(7%,&--%\"DG6#%\"xG6#\"\"!\"\"\"*(%#FxGF59$F5%\"mG!\"\"F5,&-- F06#%\"yGF3F5*(%#FyGF5F8F5F9F:F5,&--F06#%\"zGF3F5*(%#FzGF5F8F5F9F:F5F( F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#afGf*6#%\"tG6\"6$%)operator G%&arrowGF(7%*&%#FxG\"\"\"%\"mG!\"\"*&%#FyGF/F0F1*&%#FzGF/F0F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "rf(TESTZEIT)[2]; vf(TESTZ EIT)[1]; af(TESTZEIT)[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&--%\" DG6#%\"yG6#\"\"!\"\"\"%)TESTZEITGF,F,-F)F*F,**\"\"#!\"\"%#FyGF,F-F0%\" mGF1F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&--%\"DG6#%\"xG6#\"\"!\"\" \"*(%#FxGF+%)TESTZEITGF+%\"mG!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*&%#FzG\"\"\"%\"mG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 207 "Doc h nun ist es Zeit konkret zu werden. Wir k\366nnen als erstes Beispiel den Wurf untersuchen. Damit man wieder von hier aus loopen kann, werd en die rechten Seiten als unausgewertete Ausdr\374cke ('...') gesetzt. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Fx:=0: Fy:=0: Fz:='-m*g ';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FzG,$*&%\"mG\"\"\"%\"gGF(!\" \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 174 "Eine etwas g\344ngigere Sc hreibweise f\374r die Anfangsbedingungen, die den erw\374nschten Neben effekt hat, da\337 die Funktionen in voller Allgemeinheit weiterverarb eitet werden k\366nnen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 " x(0):='x0': D(x)(0):='vx0': y(0):='y0': D(y)(0):='vy0':z(0):='z0': D(z )(0):='vz0';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>--%\"DG6#%\"zG6#\"\"! %$vz0G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Anfangswerte:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "x0:=0: vx0:=4: y0:=0: vy0:=6 :z0:=0: vz0:=7:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " m:=4: g :=10:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Kontrolle der Funktione n (bei der Auswertung (z.B. beim plot) von rf werden die Werte von x0 \+ ... \374bertragen)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "rf(t) ; map(eval,rf(t));vf(t); af(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%, &*&%$vx0G\"\"\"%\"tGF'F'%#x0GF',&*&%$vy0GF'F(F'F'%#y0GF',(*&%$vz0GF'F( F'F'%#z0GF'*&\"\"&F')F(\"\"#F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7%,$*&\"\"%\"\"\"%\"tGF'F',$*&\"\"'F'F(F'F',&*&\"\"(F'F(F'F'*&\"\"&F' )F(\"\"#F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%%$vx0G%$vy0G,&%$v z0G\"\"\"*&\"#5F(%\"tGF(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\" \"!F$!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Graphische Darstellun g: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "myoptions:=axes=norm al,labels=['x','y','z'],orientation=[-48,75],scaling=constrained:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "spacecurve(rf(t),t=0..2,myop tions);" }}{PARA 13 "" 1 "" {GLPLOT3D 542 542 542 {PLOTDATA 3 "6'-%'CU RVESG6#7T7%$\"\"!F)F(F(7%$\"+hIlK;!#5$\"+#fz*[CF-$\"+KU%Qx#F-7%$\"+AhI lKF-$\"+%=fz*[F-$\"+774\"Q&F-7%F5$\"+w(QpM(F-$\"+U4u@yF-7%$\"+WAhIlF-$ \"+m$=fz*F-$\"+U$z&45!\"*7%$\"+/`Ej\")F-$\"+'z*[C7FE$\"+k[K?7FE7%$\"+k $=fz*F-$\"+bxQp9FE$\"+i1T99FE7%$\"+Ur&G9\"FE$\"+9dG9FE$\"+#3.Ev\"FE7%$\"+axQp9FE$\"+K;3/AFE$\"+/(4n*=F E7%$\"+gIlK;FE$\"+\"fz*[CFE$\"+,m:C?FE7%$\"+m$=fz\"FE$\"+]v(Qp#FE$\"+t P%\\8#FE7%$\"+sO=f>FE$\"+4bxQHFE$\"+?72HAFE7%$\"+y*[C7#FE$\"+oMn$=$FE$ \"+U*QlI#FE7%$\"+%G9dG#FE$\"+F9dGMFE$\"+QpMnBFE7%$\"+!fz*[CFE$\"+'QpMn $FE$\"+5_\\6CFE7%$\"+'*[C7EFE$\"+XtO=RFE$\"+dP)*QCFE7%$\"+--^vFFE$\"+/ `EjTFE$\"+yD\")\\CFE7%$\"+3bxQHFE$\"+jK;3WFE$\"+u;)RW#FE7%$\"+93/-JFE$ \"+A71`YFE$\"+Y5\\@CFE7%$\"+?hIlKFE$\"+\"=fz*[FE$\"+#pSBQ#FE7%$\"+E9dG MFE$\"+Sr&G9&FE$\"+81`EBFE7%$\"+Kn$=f$FE$\"+*4bxQ&FE$\"+431aAFE7%$\"+Q ?5bPFE$\"+eIlKcFE$\"+!GJ\\;#FE7%$\"+WtO=RFE$\"+<5bxeFE$\"+E?9f?FE7%$\" +_Ej\"3%FE$\"+y*[C7'FE$\"+XIpO>FE7%$\"+gz*[C%FE$\"+SpMnjFE$\"+RVe(z\"F E7%$\"+oK;3WFE$\"+-\\C7mFE$\"+2f\"=k\"FE7%$\"+w&G9d%FE$\"+kG9doFE$\"+] xQp9FE7%$\"+%)QpMZFE$\"+E3/-rFE$\"+n)*H!G\"FE7%$\"+#>fz*[FE$\"+)yQpM(F E$\"+fAbu5FE7%$\"++XAh]FE$\"+]n$=f(FE$\"+_#\\9_)F-7%$\"+3)*[C_FE$\"+7Z tOyFE$\"+f'y28'F-7%$\"+;^v(Q&FE$\"+uEj\"3)FE$\"+63^tNF-7%$\"+C/-^bFE$ \"+O1`E$)FE$\"+&3dk\\)!#67%$\"+KdG9dFE$\"+)fG9d)FE$!+\\m\"3/#F-7%$\"+S 5bxeFE$\"+glK;))FE$!+ii(y4&F-7%$\"+[j\"3/'FE$\"+AXAh!*FE$!+IJ`@$)F-7%$ \"+c;3/iFE$\"+%[AhI*FE$!+D(y6<\"FE7%$\"+kpMnjFE$\"+Y/-^&*FE$!+jQ'o_\"F E7%$\"+sAhIlFE$\"+3%=fz*FE$!+E(3#**=FE7%$\"+!exQp'FE$\"+P;3/5!\")$!+:L @)G#FE7%$\"+))G9doFE$\"+L9dG5Fb]l$!+Hw(Qp#FE7%$\"+'>3/-(FE$\"+H71`5Fb] l$!+p;?;JFE7%$\"+/Nn$=(FE$\"+E5bx5Fb]l$!+Ma=bNFE7%$\"+7)QpM(FE$\"+A3/- 6Fb]l$!+D*G3,%FE7%$\"+?T?5vFE$\"+=1`E6Fb]l$!+T@8$[%FE7%$\"+G%pMn(FE$\" +9/-^6Fb]l$!+$3&4s\\FE7%$\"+OZtOyFE$\"+5-^v6Fb]l$!+]xrxaFE7%$\"+W+++!) 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Zun\344chst spielerisch in den Befehlen ( Kraft und Anfangsbedingungen) von oben (\"was \344ndert sich wenn ... \+ \") und dann gezielt mit Gleichungen. So kann man sich z.B. fragen:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "1.) Wo l iegt der Auftreffpunkt in der x-y-Ebene?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "print(rf(t));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*&%$vx0G\"\"\"%\"tGF'F'%#x0GF',&*& %$vy0GF'F(F'F'%#y0GF',(*&%$vz0GF'F(F'F'%#z0GF'*&\"\"&F')F(\"\"#F'!\"\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Die z-Koordinate mu\337 Null \+ sein. Das ist der Fall f\374r die Zeiten" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "treff:=solve(zz(t),t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&treffG6$,&#\"\"(\"#5\"\"\"*&F)!\"\"\"#\\#F*\"\"#F*,&F'F**&F)F ,F-F.F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "xx(7/5); yy(7/5) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"(\"\"\"\"\"&!\"\"%$vx0GF& F&%#x0GF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"(\"\"\"\"\"&!\"\" %$vy0GF&F&%#y0GF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "also konkret :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eval(xx(7/5)); eval(yy (7/5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#G\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#U\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Man kann die Elemente der Menge \{treff\} mit dem " 0 "" {MPLTEXT 1 0 12 "op(\{treff\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"\"(\"#5\"\" \"*&F&!\"\"\"#\\#F'\"\"#F',&F$F'*&F&F)F*F+F)" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 83 "Der " 0 "" {MPLTEXT 1 0 99 "punkt:=eval(subs(t=op(2,\{treff\}),rf(t))); # op(1/2, .. ist von Maples internem Zustand abh\344ngig ..." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&punktG7%,&#\"#9\"\"&\"\"\"*(\"\"#F*F)!\"\"\"#\\#F*F,F-,&#\"#@ F)F**(\"\"$F*F)F-F.F/F-,(#F.\"#5F**(\"\"(F*F7F-F.F/F-*&F)F*),&#F9F7F** &F7F-F.F/F-F,F*F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 105 "Eine \"wasserdichte Programmierung\" w\344re zwar m\366glich (Abfrage der Argumente), lohnt sich aber hier nicht." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "1.a) Wie \+ gro\337 ist die Wurfweite?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sqrt(dotprod(punkt,punkt));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$, &*$),&#\"#9\"\"&\"\"\"*(\"\"#F+F*!\"\"\"#\\#F+F-F.F-F+F+*$),&#\"#@F*F+ *(\"\"$F+F*F.F/F0F.F-F+F+F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "norm(punkt,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "1.b) Betrag und Winkel der Geschwi ndigkeit f\374r t = treff? WIEDER VORSICHT MIT op!!!!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "vtreff:=vf(op(2,\{treff\}));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'vtreffG7%%$vx0G%$vy0G,(%$vz0G\"\"\" \"\"(!\"\"*$\"#\\#F*\"\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "norm(vtreff,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"$,\"#\"\"\" \"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "angle(vector(vtref f),vector([1,0,0])); # die Liste vtreff wird nicht akzeptiert!?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'arccosG6#,$*(\"\"%\"\"\"\"$,\"!\"\" F*#F)\"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(%*180 /Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+mieam!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Wer's nicht glaubt, kann es im Plot nach messen (plot geeignet drehen und kontrollieren, ob 1:1 aktiviert ist). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "2.) Steig zeit und Steih\366he bzw. Scheitelpunkt?" }}{PARA 0 "" 0 "" {TEXT -1 51 "Die z-Komponente der Geschwindigkeit mu\337 Null sein:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "vf(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%%$vx0G%$vy0G,&%$vz0G\"\"\"*&\"#5F(%\"tGF(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "steig:=solve(vf(t)[3],t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&steigG,$*&\"#5!\"\"%$vz0G\"\"\"F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Stimmt! Das ist die halbe Wurfdau er ... es ist doch beruhigend, wenn man etwas im Kopf nachrechnen kann ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "eval(zz(steig)); eval( rf(steig)); # wem das eval l\344stig ist, der kann bei D(x)(0) ... ble iben" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#\\\"#?" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7%#\"#9\"\"&#\"#@F&#\"#\\\"#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Zur Erg\344nzung nun noch die Darstellung von Ort und Geschwindigkeit als Funktionen der Z eit." }}{PARA 0 "" 0 "" {TEXT -1 118 "F\374r den, der alles gleich auf einmal sehen will und noch ein paar neue Maple-Befehle kennenlernen w ill, so formuliert:" }}{PARA 0 "" 0 "" {TEXT -1 57 "vtitle:=`Eine Meng e`:vxlab:=t:ytick:=0:pspl(`p2new1.ps`):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "plot(convert(eval(rf(t)),set) union convert(eval(vf(t )),set),t=0..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 542 542 542 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"\"!F)$\"\"%F)7$$\"39LLLL3VfV!#>F*7$$\"3'pmm;H[ D:)F/F*7$$\"3LLLLe0$=C\"!#=F*7$$\"3ILLL3RBr;F6F*7$$\"3Ymm;zjf)4#F6F*7$ $\"3=LL$e4;[\\#F6F*7$$\"3p****\\i'y]!HF6F*7$$\"3,LL$ezs$HLF6F*7$$\"3_* ***\\7iI_PF6F*7$$\"3#pmmm@Xt=%F6F*7$$\"3QLLL3y_qXF6F*7$$\"3i******\\1! >+&F6F*7$$\"3()******\\Z/NaF6F*7$$\"3'*******\\$fC&eF6F*7$$\"3ELL$ez6: B'F6F*7$$\"3Smmm;=C#o'F6F*7$$\"3-mmmm#pS1(F6F*7$$\"3]****\\i`A3vF6F*7$ $\"3slmmm(y8!zF6F*7$$\"3V++]i.tK$)F6F*7$$\"39++](3zMu)F6F*7$$\"3#pmm;H _?<*F6F*7$$\"3emm;zihl&*F6F*7$$\"39LLL3#G,***F6F*7$$\"3F cpF*7$$\"3/++v.Uac>FcpF*7$$\"\"#F)F*-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F $6$7S7$F($\"\"'F)7$F-Fdu7$F1Fdu7$F4Fdu7$F8Fdu7$F;Fdu7$F>Fdu7$FAFdu7$FD Fdu7$FGFdu7$FJFdu7$FMFdu7$FPFdu7$FSFdu7$FVFdu7$FYFdu7$FfnFdu7$FinFdu7$ F\\oFdu7$F_oFdu7$FboFdu7$FeoFdu7$FhoFdu7$F[pFdu7$F^pFdu7$FapFdu7$FepFd u7$FhpFdu7$F[qFdu7$F^qFdu7$FaqFdu7$FdqFdu7$FgqFdu7$FjqFdu7$F]rFdu7$F`r Fdu7$FcrFdu7$FfrFdu7$FirFdu7$F\\sFdu7$F_sFdu7$FbsFdu7$FesFdu7$FhsFdu7$ F[tFdu7$F^tFdu7$FatFdu7$FdtFdu7$FgtFdu-Fjt6&F\\uF(F]uF(-F$6$7S7$F(F(7$ F-$\"3?Y-Q(Ryl&HF67$F1$\"3y!)o#fejWP&F67$F4$\"3,%4G0MU<#zF67$F8$\"3(y< :qf7-.\"Fcp7$F;$\"3U%4%GF@\")[7Fcp7$F>$\"3Xz=[**e;N9Fcp7$FA$\"3TgV)>'4 e6;Fcp7$FD$\"3/!e,kzCjx\"Fcp7$FG$\"30&pJJDCE#>Fcp7$FJ$\"3#eg*[`'[W0#Fc p7$FM$\"3U=f\\VK)[:#Fcp7$FP$\"3!*ykP\\*z.D#Fcp7$FS$\"3,s[F`daFBFcp7$FV $\"3-z)HG_dTQ#Fcp7$FY$\"3cMG+%Hr/U#Fcp7$Ffn$\"3`Mi%o[^\\W#Fcp7$Fin$\"3 9SMXcZz\\CFcp7$F\\o$\"3#oeX!\\`3PCFcp7$F_o$\"3tB,&f\"eP4CFcp7$Fbo$\"3Q IN/*[\">hBFcp7$Feo$\"3a3DdLS,)H#Fcp7$Fho$\"3)*yK6U%4T@#Fcp7$F[p$\"3f\\ PSb1)37#Fcp7$F^p$\"3Hkk)[mcH+#Fcp7$Fap$\"33/Z0,cQh=Fcp7$Fep$\"3t9Pq0yJ A-$f\"F67$F]r$!3C!Qyvb3A8\"F67$F`r$!3?*[!HQ@#oG%F67$Fcr$! 3Dk9'*)e\">AuF67$Ffr$!3!G/?mnws3\"Fcp7$Fir$!3X#Rqp!)\\5E$F67$F4$\"3MLLLLA Kn\\F67$F8$\"3=LLLLc$\\o'F67$F;$\"3)emmm^&Q%R)F67$F>$\"3wKLL$Qk#z**F67 $FA$\"3))*****\\YJ?;\"Fcp7$FD$\"3?LLL=\"\\qM`Fcp7$Fjq$\"3%)*******HSu ]&Fcp7$F]r$\"3'HLL$ep'Rm&Fcp7$F`r$\"3')******R>4NeFcp7$Fcr$\"3#emm;@2h *fFcp7$Ffr$\"3]*****\\c9W;'Fcp7$Fir$\"3Lmmmmd'*GjFcp7$F\\s$\"3j*****\\ iN7]'Fcp7$F_s$\"3aLLLt>:nmFcp7$Fbs$\"35LLL.a#o$oFcp7$Fes$\"3ammm^Q40qF cp7$Fhs$\"3y******z]rfrFcp7$F[t$\"3gmmmc%GpL(Fcp7$F^t$\"3/LLL8-V&\\(Fc p7$Fat$\"3=+++XhUkwFcp7$Fdt$\"3=+++:o$\"3#*** **\\d'*)o\\\"Fcp7$FA$\"3q****\\(>ZIu\"Fcp7$FD$\"3!)****\\xOi(*>Fcp7$FG $\"3]****\\FPQ^AFcp7$FJ$\"3;+++IrS7DFcp7$FM$\"3.+++&o;Bu#Fcp7$FP$\"3x* ******QS6+$Fcp7$FS$\"3#*******\\o-hKFcp7$FV$\"3v******4cZ6NFcp7$FY$\"3 '*****\\xq!*QPFcp7$Ffn$\"31+++!4X$4SFcp7$Fin$\"30+++g:WQUFcp7$F\\o$\"3 #*****\\<_$\\]%Fcp7$F_o$\"3m******fs#3u%Fcp7$Fbo$\"3E++]<#Q'**\\Fcp7$F eo$\"3I++]_u3Y_Fcp7$Fho$\"3P+++v8B.bFcp7$F[p$\"3]****\\n(p$RdFcp7$F^p$ \"3))*****\\#p2%*fFcp7$Fap$\"3-****\\xgkeiFcp7$Fep$\"3%)****\\-V&*)['F cp7$Fhp$\"3P*****\\\\$pPnFcp7$F[q$\"3e******>am%*pFcp7$F^q$\"3w)****\\ JigC(Fcp7$Faq$\"31****\\P " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "Die St\344rke eines CA S liegt im Spiel mit Parametern und der leichten Formulierbarkeit inve rser Problemstellungen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "3.) 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Probleme geben, wenn man das worksheet neu oeffnet un d nicht von vorne abarbeitet !?!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7% ,&*&%$vx0G\"\"\"%\"tGF'F'%#x0GF',&*&%$vy0GF'F(F'F'%#y0GF',(*&%$vz0GF'F (F'F'%#z0GF'*&\"\"&F')F(\"\"#F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ziel:=vector([x1,y1,z1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%zielGK%'vectorG6#7%%#x1G%#y1G%#z1GQ)pprint106\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "zielsys:=equate(rf(t),ziel); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(zielsysG<%/,$**\"#:\"\"\"-%$cos G6#,$*&\"#5!\"\"%#PiGF*F*F*-F,6#,$*(\"#8F*\"$+\"F1F2F*F*F*%\"tGF*F1%#x 1G/,$**F)F*-F,6#,$*(\"#6F*\"#DF1F2F*F*F*-%$sinG6#,$*(\"#ZF*F8F1F2F*F*F *F9F*F1%#y1G/,&*(F)F*-FE6#,$*(\"#RF*F8F1F2F*F*F*F9F*F1*&\"\"&F*)F9\"\" #F*F1%#z1G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Da stehen vom letzt en Plot noch feste Werte drin, also R\374cksetzen von v0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "unassign('vx0','vy0','vz0','v0','x1','y1','z1','t1'); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "zielsys:=equate(rf(t),z iel);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(zielsysG<%/,&*&%$vz0G\"\" \"%\"tGF*F**&\"\"&F*)F+\"\"#F*!\"\"%#z1G/*&%$vy0GF*F+F*%#y1G/*&%$vx0GF *F+F*%#x1G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Man sieht: es fehlt noch eine Bedingung. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "4.a) das Ziel soll zu vorgegebener Zeit t1 erreicht \+ werden." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "tzielsys:=zielsy s union \{t=t1\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)tzielsysG<&/% \"tG%#t1G/,&*&%$vz0G\"\"\"F'F-F-*&\"\"&F-)F'\"\"#F-!\"\"%#z1G/*&%$vy0G F-F'F-%#y1G/*&%$vx0GF-F'F-%#x1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "solt:=solve(tzielsys,\{vx0,vy0,vz0,t\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%soltG<&/%\"tG%#t1G/%$vy0G*&%#y1G\"\"\"F(!\"\"/%$vx0G *&%#x1GF-F(F./%$vz0G*&,&*&\"\"&F-)F(\"\"#F-F-%#z1GF-F-F(F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "4.b) der Betrag der Startgeschwindigkeit ist ge geben (oder die Startenergie)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "vzielsys:=zielsys union \{v0^2=vx0^2+vy0^2+vz0^2\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)vzielsysG<&/,&*&%$vz0G\"\"\"%\"tGF*F**&\"\"& F*)F+\"\"#F*!\"\"%#z1G/*&%$vy0GF*F+F*%#y1G/*&%$vx0GF*F+F*%#x1G/*$)%#v0 GF/F*,(*$)F8F/F*F**$)F4F/F*F**$)F)F/F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "unassign('vx0','vy0','vz0','t','v0','z1','x1','y1'); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "solv:=solve[radical](vz ielsys,\{vx0,vy0,vz0,t\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solvG <&/%$vz0G*&,&*&\"\"&\"\"\")-%'RootOfG6#,,*&\"#DF,)%#_ZG\"\"%F,F,*&,&*$ )%#v0G\"\"#F,!\"\"*&\"#5F,%#z1GF,F,F,)F5F " 0 "" {MPLTEXT 1 0 13 "assign(solv);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "allvalues(t);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6&,$*&\"#5!\"\",(*&\"\"#\"\"\")%#v0GF)F*F**&\"#?F* %#z1GF*F&*&F)F*,**$)F,\"\"%F*F**(F.F*F+F*F/F*F&*&\"$+\"F*)%#x1GF)F*F&* &F7F*)%#y1GF)F*F&#F*F)F*F=F*,$*&F%F&F'F=F&,$*&F%F&,(*&F)F*F+F*F**&F.F* F/F*F&*&F)F*F1F=F&F=F*,$*&F%F&FBF=F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "Zur besseren \334bersicht zweidimensional (y-z-Ebene), und Tre ffpunkt auf der y-Achse, mit der Fragestellung: welche Mindestgeschwin digkeit ist erforderlich?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "z1:=0: x1:=0: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "zeite n:=allvalues(t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'zeitenG6&,$*&\" #5!\"\",&*&\"\"#\"\"\")%#v0GF,F-F-*&F,F-,&*$)F/\"\"%F-F-*&\"$+\"F-)%#y 1GF,F-F)#F-F,F-F9F-,$*&F(F)F*F9F),$*&F(F),&*&F,F-F.F-F-*&F,F-F1F9F)F9F -,$*&F(F)F>F9F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "Es kommen als o vier Zeitpunkte in Frage (zwei in der Zukunft und zwei in der Vergan genheit). Dazu geh\366ren z.B. in y-Richtung die vier Geschwindigkeits komponenten." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "t:='t':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "allvalues(vy0);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&,$*(\"#5\"\"\"%#y1GF&,&*&\"\"#F&)%#v0GF*F&F&*&F *F&,&*$)F,\"\"%F&F&*&\"$+\"F&)F'F*F&!\"\"#F&F*F&#F5F*F&,$*(F%F&F'F&F(F 7F5,$*(F%F&F'F&,&*&F*F&F+F&F&*&F*F&F.F6F5F7F&,$*(F%F&F'F&F " 0 "" {MPLTEXT 1 0 89 "bahn[1]:=subs('vy0 '=allvalues(vy0)[1],'vz0'=allvalues(vz0)[1],rf(t)); #das macht man mit " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%bahnG6#\"\"\"7%,&*&%$vx0GF'% \"tGF'F'%#x0GF',&**\"#5F'%#y1GF',&*&\"\"#F')%#v0GF4F'F'*&F4F',&*$)F6\" \"%F'F'*&\"$+\"F')F1F4F'!\"\"#F'F4F'#F?F4F,F'F'%#y0GF',(*(F4F?F2F@F,F' F'%#z0GF'*&\"\"&F')F,F4F'F?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "bahn[2]:=subs('vy0'=allvalues(vy0)[4],'vz0'=allvalues(vz0)[4],rf(t )); #copy und paste" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%bahnG6#\"\" 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