\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\froman\fprq2 Times;}{\f4\fmodern\fprq1 Courier New;}{\f5\froman\fprq2 Times New Roman;}{\f6\froman\fcharset1 Times New Roman;}} {\colortbl\red0\green0\blue0;\red255\green0\blue0;\red0\green0\blue255;} \deflang3079\pard\ri4\plain\f3\fs32\b Beobachtung am Radarschirm \par \pard\plain\f3\fs22 \par Die beiden Motorschiffe \plain\f3\fs22\i Albert\plain\f3\fs22 und \plain\f3\fs22\i Bertha\plain\f3\fs22 halten Kurs auf dem Atlantik. \par Ihre Bewegung wird auf einem Gro\'dfbild des Radarschirms der Beobachtungsstation festgehalten. \par \pard\ri4\plain\f3\fs22\i Albert\plain\f3\fs22 erscheint auf dem Bildschirm auf der unteren Kante 900 mm von der linken unteren Ecke entfernt. \par \plain\f3\fs22\i Bertha\plain\f3\fs22 zum gleichen Zeitpunkt 100 mm \'fcber der linken unteren Ecke auf der linken Kante. \par \par Eine Minute sp\'e4ter haben sich die Positionen wie folgt ge\'e4ndert: \par \pard\li500\ri4\plain\f3\fs22\i Albert\plain\f3\fs22 hat sich um 3 mm nach Westen und 2 mm nach Norden bewegt. \par \plain\f3\fs22\i Bertha \plain\f3\fs22 um 4 mm nach Osten und 1 mm nach Norden. Beide Schiffe halten an ihrem geradlinigen Kurs fest.\plain\f4\fs22\cf1 \par \pard\ri4\plain\f4\fs22\cf1 \par \pard\plain\f3\fs22\b Fragen: \par \pard\tx426\plain\f3\fs22 1)\tab \tab Werden die Schiffe kollidieren? \par \pard\ri4\plain\f3\fs22 2)\tab Wenn nein, wo ist der geringste Abstand zwischen den Schiffen? \par 3)\tab In welchem Punkt schneiden sich die Fahrtrouten? Wie schnell sind die Schiffe! \par \plain\f4\fs22\cf1 \par \pard\plain\f3\fs22 Zuerst legen wir die Funktion f\'fcr den Betrag eines 2-dimensionalen Vektors (L\'e4nge desVektors) fest. \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf1 {\pntext\f1\'b7\tab}abs_vek2:=v -> sqrt(v[1][1]^2+v[1][2]^2): \par {\pntext\f1\'b7\tab}va:=matrix([[-3,2]]): \par {\pntext\f1\'b7\tab}vb:=matrix([[4,1]]): \par {\pntext\f1\'b7\tab}anfa:=matrix([[900,0]]): \par {\pntext\f1\'b7\tab}anfb:=matrix([[0,100]]): \par {\pntext\f1\'b7\tab}pa:=t -> anfa+t*va: \par {\pntext\f1\'b7\tab}pb:=t -> anfb+t*vb: \par \pard\ri4\plain\f5\fs22\cf0 Wir stellen die Distanz zwischen 2 beliebigen Punkten auf den Bewegungsgeraden auf. \par Anschlie\'dfend differenzieren wir die Funktion und suchen das Minimum. \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf1 {\pntext\f1\'b7\tab}dist:=t -> abs_vek2([pa(t)-pb(t)]): \par {\pntext\f1\'b7\tab}min_zeit:=solve(diff(dist(t),t)=0,t) \par \pard\li50\ri6\plain\f4\fs22\cf2\protect {\pict\wmetafile8\picw1411\pich827\picwgoal799\pichgoal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}\plain\f4\fs22\cf2\protect \par \pard\li50\ri2\plain\f4\fs22\cf2\protect \par \pard\ri4\plain\f6\fs22\cf0 Mit dem Operator % kann das letzte berechnete Resultat aufgerufen werden - in unserem Fall 128.\plain\f4\fs22\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf1 {\pntext\f1\'b7\tab}min_dist:=float(dist(%1)) \par \pard\li50\ri6\plain\f4\fs22\cf2\protect {\pict\wmetafile8\picw2999\pich827\picwgoal1700\pichgoal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}\plain\f4\fs22\cf2\protect \par \pard\li50\ri2\plain\f4\fs22\cf2\protect \par \pard\ri4\plain\f5\fs22\cf0 Dies bedeutet: nach 128 Minuten ist die Minimaldistanz von ca. 2828,4 Metern erreicht. Daher gibt es keinen Zusammensto\'df. \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf1 {\pntext\f1\'b7\tab}schnitt_zeit:=solve([pa(t)[1]=pb(s)[1],pa(t)[2]=pb(s)[2]],[s,t])[1][1] \par \pard\li50\ri6\plain\f4\fs22\cf2\protect {\pict\wmetafile8\picw1956\pich1134\picwgoal1108\pichgoal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}\plain\f4\fs22\cf2\protect \par \pard\li50\ri2\plain\f4\fs22\cf2\protect \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf1 {\pntext\f1\'b7\tab}float(pb(1500/11)) \par \pard\li50\ri6\plain\f4\fs22\cf2\protect {\pict\wmetafile8\picw5549\pich827\picwgoal3145\pichgoal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}\plain\f4\fs22\cf2\protect \par \pard\li50\ri2\plain\f4\fs22\cf2\protect \par \pard\ri4\plain\f5\fs22\cf0 Der geometrische Schnittpunkt der beiden Routen hat die (gerundeten) Koordinaten \plain\f5\fs22\cf0\b S(545 / 236)\plain\f5\fs22\cf0 . \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf1 {\pntext\f1\'b7\tab}geschw_a:=float(abs_vek2([va]))*6 \par \pard\li50\ri6\plain\f4\fs22\cf2\protect {\pict\wmetafile8\picw2575\pich827\picwgoal1459\pichgoal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}\plain\f4\fs22\cf2\protect \par \pard\li50\ri2\plain\f4\fs22\cf2\protect \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf1 {\pntext\f1\'b7\tab}geschw_b:=float(abs_vek2([vb]))*6 \par \pard\li50\ri6\plain\f4\fs22\cf2\protect {\pict\wmetafile8\picw2575\pich827\picwgoal1459\pichgoal468 0100090000035D01000005001C0000000000050000000B0200000000050000000C023B030F0A03 0000001E00030000001E00050000000C02D501B405050000000B020000000014000000FF060000 0600F602000024000100020000800080FF7FFF7F02000080FF7F0080FF7F0200040000002D0100 0004000000F001000008000000FA0200000000000000000000040000002D01000007000000FC02 0000000000000000040000002D0101001C000000FB0210FF000000000000900100000001070000 00417269616C00C4774000000064070A18B7C5C477C0C5C47720C0C77700003000040000002D01 020005000000020101000000050000000102FFFFFF00050000002E011800000005000000090200 00000004000000080100001C000000FB0210FF0000000000009001000000010700000054696D65 73204E657720526F6D616E00B7C5C477C0C5C47720C0C77700003000040000002D0103000B0000 0026060F000C004D6174685479706500007B000500000009020000FF0007000000210501003200 3B016400070000002105010034003B01DC0007000000210501002E003B01540107000000210501 0037003B019001070000002105010033003B010802070000002105010038003B01800207000000 2105010036003B01F802070000002105010033003B017003070000002105010033003B01E80307 0000002105010037003B016004070000002105010035003B01D80408000000FA02000000000000 00000000040000002D01040004000000F001000007000000FC020000FFFFFF000000040000002D 01000004000000F00101001C000000FB021000070000000000BC02000000000102022253797374 656D00774000000040070A50B7C5C477C0C5C47720C0C77700003000040000002D010100040000 002701FFFF04000000F001020004000000F0010300040000002701FFFF030000000000 }\plain\f4\fs22\cf2\protect \par \pard\li50\ri2\plain\f4\fs22\cf2\protect \par \pard\ri4\plain\f5\fs22\cf0 Die Geschwindigkeit in km/h betr\'e4gt f\'fcr die \plain\f5\fs22\cf0\i Albert\plain\f5\fs22\cf0 ca.\plain\f5\fs22\cf0\b 21,6 km/h\plain\f5\fs22\cf0 , f\'fcr die \plain\f5\fs22\cf0\i Bertha\plain\f5\fs22\cf0 ca. \plain\f5\fs22\cf0\b 24,7 km/h\plain\f5\fs22\cf0 .\plain\f4\fs22\cf1 \par }