{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 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 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 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 "Text \+ Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Out put" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "M aple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 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 256 "" 0 "" {TEXT -1 18 "J\374rgen Krummenauer" } }{PARA 257 "" 0 "" {TEXT -1 21 "Steinstra\337er Allee 38" }}{PARA 258 "" 0 "" {TEXT -1 12 "52428 J\374lich" }}{PARA 0 "" 0 "" {TEXT -1 8 "E- Mail: " }{TEXT 258 24 "krummenauer@fh-aachen.de" }{MPLTEXT 1 0 0 "" }} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 "Der eindimensinoale elastische S to\337" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "Ziel ist es, ausgehend vom Impuls- bzw. Energiesatz, die Geschwin digkeiten der Massenpunkte nach dem Sto\337 zu bestimmen. " }}{PARA 0 "" 0 "" {TEXT -1 46 "Zuerst werden die Geschwindigkeiten definiert." } {MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "v1_vor_dem_sto\337:=v[1 ][x];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0v1_vor_dem_sto|jxG&&%\"vG6 #\"\"\"6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "v2_vor_de m_sto\337:=v[2][x];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0v2_vor_dem_s to|jxG&&%\"vG6#\"\"#6#%\"xG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Nu n zu den Impulsen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "p1_vor_dem_st o\337:=m[1]*v1_vor_dem_sto\337;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0 p1_vor_dem_sto|jxG*&&%\"mG6#\"\"\"F)&&%\"vGF(6#%\"xGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "p2_vor_dem_sto\337:=m[2]*v2_vor_dem _sto\337;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0p2_vor_dem_sto|jxG*&&% \"mG6#\"\"#\"\"\"&&%\"vGF(6#%\"xGF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Damit erh\344lt man den Gesamtimpuls:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "pg_vor_dem_sto\337:=p1_vor_dem_sto\337+p2_vor_dem_sto \337;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0pg_vor_dem_sto|jxG,&*&&%\" mG6#\"\"\"F*&&%\"vGF)6#%\"xGF*F**&&F(6#\"\"#F*&&F-F2F.F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Nun werden die Gr\366\337en nach dem Sto \337 festgelegt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u1_nach_dem_sto \337:=u[1][x];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u2_nach_dem_sto \337:=u[2][x];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1u1_nach_dem_sto|j xG&&%\"uG6#\"\"\"6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1u2_nach _dem_sto|jxG&&%\"uG6#\"\"#6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "p1_nach_dem_sto\337:=m[1]*u1_nach_dem_sto\337;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "p2_nach_dem_sto\337:=m[2]*u2_nach_d em_sto\337;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1p1_nach_dem_sto|jxG* &&%\"mG6#\"\"\"F)&&%\"uGF(6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%1p2_nach_dem_sto|jxG*&&%\"mG6#\"\"#\"\"\"&&%\"uGF(6#%\"xGF*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "pg_nach_dem_sto\337:=p1_nach _dem_sto\337+p2_nach_dem_sto\337;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %1pg_nach_dem_sto|jxG,&*&&%\"mG6#\"\"\"F*&&%\"uGF)6#%\"xGF*F**&&F(6#\" \"#F*&&F-F2F.F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "impulssatz:=pg_vor_dem_sto\337=pg_n ach_dem_sto\337; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+impulssatzG/,& *&&%\"mG6#\"\"\"F+&&%\"vGF*6#%\"xGF+F+*&&F)6#\"\"#F+&&F.F3F/F+F+,&*&F( \"\"\"&&%\"uGF*F/F+F+*&F2F9&&F " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "e1_vor_dem_sto\337:=p1_vor_dem_sto\337^2/(2*m[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0e1_vor_dem_sto|jxG,$*&&%\"mG6#\"\"\"F*)&& %\"vGF)6#%\"xG\"\"#\"\"\"#F*F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "e2_vor_dem_sto\337:=p2_vor_dem_sto\337^2/(2*m[2]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0e2_vo r_dem_sto|jxG,$*&&%\"mG6#\"\"#\"\"\")&&%\"vGF)6#%\"xGF*\"\"\"#F+F*" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eg_vor_dem_sto\337:=e1_vor_ dem_sto\337+e2_vor_dem_sto\337;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0 eg_vor_dem_sto|jxG,&*&&%\"mG6#\"\"\"F*)&&%\"vGF)6#%\"xG\"\"#\"\"\"#F*F 1*&&F(6#F1F*)&&F.F6F/F1F2F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "e1_nach_dem_sto\337:=p1_nach_dem_sto\337^2/(2*m[1]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%1e1_nach_dem_sto|jxG,$*&&%\"mG6#\"\"\"F*)&&%\" uGF)6#%\"xG\"\"#\"\"\"#F*F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "e2_nach_dem_sto\337:=p2_nach_dem_sto\337^2/(2*m[2]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%1e2_nach_dem_sto|jxG,$*&&%\"mG6#\"\"#\"\"\")&& %\"uGF)6#%\"xGF*\"\"\"#F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "eg_nach_dem_sto\337:=e1_nach_dem_sto\337+e2_nach_dem_sto\337;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%1eg_nach_dem_sto|jxG,&*&&%\"mG6#\"\"\"F*)&&%\"uGF)6#%\"xG\"\"#\"\"\" #F*F1*&&F(6#F1F*)&&F.F6F/F1F2F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "energiesatz:=eg_vor_dem_sto\337=eg_nach_dem_sto\337;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,energiesatzG/,&*&&%\"mG6#\"\"\"F+)&&%\"vG F*6#%\"xG\"\"#\"\"\"#F+F2*&&F)6#F2F+)&&F/F7F0F2F3F4,&*&F(F3)&&%\"uGF*F 0F2F3F4*&F6F3)&&F@F7F0F2F3F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Nun werden die Erhaltun gsgr\366\337en zusammengefa\337t. Die triviale L\366sung u1x=v1x und u 2x=v2x soll ausgeschlossen werden." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "erhaltungsgroe\337en:=\{impulssatz,energiesatz\} union \{u[1][x]<> v[1][x]\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%2erhaltungsgroe|jxenG< %0&&%\"uG6#\"\"\"6#%\"xG&&%\"vGF*F,/,&*&&%\"mGF*F+)F.\"\"#\"\"\"#F+F7* &&F56#F7F+)&&F0FF+F+,&*&F4F8F'F+F+*&F;F8FEF+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Gesucht werden die Gr\366\337en nach dem Sto\337;" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "gesuchte_groe\337en:=\{u[1][x],u[2 ][x]\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "gesuchte_groe\337en=solv e(erhaltungsgroe\337en,gesuchte_groe\337en);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1gesuchte_groe|jxenG<$&&%\"uG6#\"\"\"6#%\"xG&&F(6#\" \"#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/<$&&%\"uG6#\"\"\"6#%\"xG&&F' 6#\"\"#F*<$/F,*&,(*&&%\"mGF(F)&&%\"vGF.F*F)!\"\"*&&F6F.F)F7\"\"\"F)*&F 5F=&&F9F(F*F)F/F=,&FF)F;F/F=FAFB" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Diese Gleichungen werden noch beim zweidimensionalen Sto\337 benutzt." }}{PARA 0 "" 0 "" {TEXT -1 29 " N och ein konkretes Beispiel:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "gege bene_groe\337en:=m[1]=2,m[2]=4,v[1][x]=2,v[2][x]=-4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1gegebene_groe|jxenG6&/&%\"mG6#\"\"\"\"\"#/&F(6#F+ \"\"%/&&%\"vGF)6#%\"xGF+/&&F3F.F4!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "loesung:=solve( subs(gegebene_groe\337en,erhaltungsgroe\337en),gesuchte_groe\337en);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(loesungG<$/&&%\"uG6#\"\"#6#%\"xG \"\"!/&&F)6#\"\"\"F,!\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "Der zweidimensionale elastische Sto\337" }}{EXCHG {PARA 0 "" 0 "" {TEXT 256 0 "" }{TEXT -1 46 "Es gilt immernoc h der Energie- und Impulssatz." }}{PARA 0 "" 0 "" {TEXT -1 67 "Da es s ich hierbei um vektorielle Gr\366\337en handelt, ben\366tigt man die \+ " }{TEXT 257 14 "Libary linalg." }}{PARA 0 "" 0 "" {TEXT -1 65 "Zuerst werden die Anfangsbedingungen zum Zeitpunkt t=0 definiert." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "resart:with(linalg):with(plots):" }} {PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "x[1][0]:=0:y[1][0]:=0:x[2][ 0]:=10:y[2][0]:=0:v[1][x]:=1:v[1][y]:=1:v[2][x]:=-1:v[2][y]:=1:m[1]:=1 :m[2]:=2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "m1_zur_zeit_t0:=vector ([x[1][0],y[1][0]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "m2_zur_zeit _t0:=vector([x[2][0],y[2][0]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " v1_vor_dem_sto\337:=vector([v[1][x],v[1][y]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "v2_vor_dem_sto\337:=vector([v[2][x],v[2][y]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%/m1_zur_zeit_t0G-%'vectorG6#7$\"\"!F )" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/m2_zur_zeit_t0G-%'vectorG6#7$ \"#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0v1_vor_dem_sto|jxG-%'v ectorG6#7$\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0v2_vor_dem_st o|jxG-%'vectorG6#7$!\"\"\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Die Impulse der Massenpunkte, der Gesamtimpuls und die Bewegungsgleic hungen der Massenpunkte" }}{PARA 0 "" 0 "" {TEXT -1 17 " werden bestim mt." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "p1_vor_dem_sto\337:=scalarmul(v1_vor_dem_sto\337,m[1]);p2_vor_dem _sto\337:=scalarmul(v2_vor_dem_sto\337,m[2]);gesamtimpuls:=matadd(p1_v or_dem_sto\337,p2_vor_dem_sto\337);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "bewegungsgleichung_von_m1_vor_dem_sto\337:=t->matadd(m1_zur_zeit_t 0,scalarmul(v1_vor_dem_sto\337,t));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "bewegungsgleichung_von_m2_vor_dem_sto\337:=t->matadd(m2_zur_zeit_t 0,scalarmul(v2_vor_dem_sto\337,t));" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0p 1_vor_dem_sto|jxG-%'vectorG6#7$\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0p2_vor_dem_sto|jxG-%'vectorG6#7$!\"#\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%-gesamtimpulsG-%'vectorG6#7$!\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%Gbewegungsgleichung_von_m1_vor_dem_sto|jxGR6# %\"tG6\"6$%)operatorG%&arrowGF(-%'mataddG6$%/m1_zur_zeit_t0G-%*scalarm ulG6$%0v1_vor_dem_sto|jxG9$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %Gbewegungsgleichung_von_m2_vor_dem_sto|jxGR6#%\"tG6\"6$%)operatorG%&a rrowGF(-%'mataddG6$%/m2_zur_zeit_t0G-%*scalarmulG6$%0v2_vor_dem_sto|jx G9$F(F(F(" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Es \374berhaupt zum Sto\337, falls die beiden Mas sen zum gleichen Zeitpunkt am gleichen Ort sind." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "g1:=bewegungsgleichung_von_m1_vor_dem_sto\337(t1)[1]= bewegungsgleichung_von_m2_vor_dem_sto\337(t2)[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g1G/%#t1G,&\"#5\"\"\"%#t2G!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 96 "g2:=bewegungsgleichung_von_m1_vor_dem_sto \337(t1)[2]=bewegungsgleichung_von_m2_vor_dem_sto\337(t2)[2];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g2G/%#t1G%#t2G" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "zeiten:=solve(\{g1,g2\},\{t1,t2\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'zeitenG<$/%#t1G\"\"&/%#t2GF(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Ob es zum Sto\337 kommt, wird in e iner bool'schen Variablen gespeichert." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "es_kommt_zum_sto\337:=is(rhs(zeiten [1])=rhs(zeiten[2]));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2es_kommt_zum_sto|jxG%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 " Nun wird die Bahn des Schwerpunkts besti mmt." }}{PARA 0 "" 0 "" {TEXT -1 32 "Zur Zeit t=0 befindet er sich in " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "s_zur_zeit_t0:=scalarmul(matad d(scalarmul(m1_zur_zeit_t0,m[1]),scalarmul(m2_zur_zeit_t0,m[2])),1/(m[ 1]+m[2]));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.s_zur_zeit_t0G-%'vectorG6#7$#\"#?\"\"$\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "und bewegt sich in Richtung von" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "richtung_von_s:=scalarmul(gesamtim puls,1/(m[1]+m[2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/richtung_vo n_sG-%'vectorG6#7$#!\"\"\"\"$\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Damit lautet seine Bewegungsgleichung" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "bewegungsgleich ung_von_s:=t->matadd(s_zur_zeit_t0,scalarmul(richtung_von_s,t));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%9bewegungsgleichung_von_sGR6#%\"tG6 \"6$%)operatorG%&arrowGF(-%'mataddG6$%.s_zur_zeit_t0G-%*scalarmulG6$%/ richtung_von_sG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " bewegungsgleichung_von_s(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vec torG6#7$,&#\"#?\"\"$\"\"\"%\"tG#!\"\"F*F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 261 "Die Impulse (Geschwindigkeiten) der Massenpunkte werden \+ zerlegt in unabh\344ngige Komponenten, senkrecht bzw. parallel zur Bah n des Schwerpunkts. Die Parallelkomponenten bleiben erhalten, die senk rechten Komponenten werden behandelt wie beim eindimensionalen Sto\337 ." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "richtung_vo n_s_normiert:=scalarmul(richtung_von_s,1/norm(richtung_von_s,2)); \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%8richtung_von_s_normie rtG-%'vectorG6#7$,$*$-%%sqrtG6#\"#5\"\"\"#!\"\"F.,$F*#\"\"$F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "v1_parallel:=scalarmul(richt ung_von_s_normiert,dotprod(v1_vor_dem_sto\337,richtung_von_s_normiert) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,v1_parallelG-%'vectorG6#7$#! \"\"\"\"&#\"\"$F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "v1_sen krecht:=matadd(v1_vor_dem_sto\337,-v1_parallel);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-v1_senkrechtG-%'vectorG6#7$#\"\"'\"\"&#\"\"#F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "v2_parallel:=scalarmul(richt ung_von_s_normiert,dotprod(v2_vor_dem_sto\337,richtung_von_s_normiert) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "v2_senkrecht:=matadd(v2_vor_d em_sto\337,-v2_parallel);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,v2_par allelG-%'vectorG6#7$#!\"#\"\"&#\"\"'F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-v2_senkrechtG-%'vectorG6#7$#!\"$\"\"&#!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "p1_parallel:=scalarmul(v1_parallel, m[1]);p1_senkrecht:=scalarmul(v1_senkrecht,m[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,p1_parallelG-%'vectorG6#7$#!\"\"\"\"&#\"\"$F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-p1_senkrechtG-%'vectorG6#7$#\"\"'\" \"&#\"\"#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "p2_parallel: =scalarmul(v2_parallel,m[2]);p2_senkrecht:=scalarmul(v2_senkrecht,m[2] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,p2_parallelG-%'vectorG6#7$#! \"%\"\"&#\"#7F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-p2_senkrechtG-%' vectorG6#7$#!\"'\"\"&#!\"#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "v1:=norm(v1_senkrecht,2);v2:=-norm(v2_senkrecht,2);" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 41 "u1:=(m[1]*v1+m[2]*(2*v2-v1))/(m[1]+m[2]);" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "u2:=(m[2]*v2+m[1]*(2*v1-v2))/(m[1] +m[2]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G,$*$-%%sqrtG6#\"#5\"\"\"#\"\"#\"\"&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#v2G,$*$-%%sqrtG6#\"#5\"\"\"#!\"\"\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u1G,$*$-%%sqrtG6#\"#5\"\"\"#!\"#\" \"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u2G,$*$-%%sqrtG6#\"#5\"\"\"# \"\"\"\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "Die Orthogonalkom ponenten der Geschwindigkeiten (und damit auch die der Impulse) kehren sich lediglich um. Dies ist nicht nur in diesem Beispiel so. Folglich gilt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "v1_nach_dem_sto\337:=mata dd(v1_parallel,-v1_senkrecht);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%1v1_nach_dem_sto|jxG-%'vectorG6#7$#!\"(\"\"&#\"\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "v2_nach_dem_sto\337:=matadd(v2_para llel,-v2_senkrecht);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1v2_nach_dem _sto|jxG-%'vectorG6#7$#\"\"\"\"\"&#\"\"(F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Damit lauten die Bewegungsgleichungen nach dem Sto\337:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sto\337punkt:=bewegungsgleichung_ von_s(rhs(zeiten[1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*sto|jxpun ktG-%'vectorG6#7$\"\"&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "bewegungsgleichung_von_m1_nach_dem_sto\337:=t->matadd(sto\337punkt,sc alarmul(v1_nach_dem_sto\337,t));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "bewegungsgleichung_von_m2_nach_dem_sto\337:=t->matadd(sto\337punkt,sc alarmul(v2_nach_dem_sto\337,t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% Hbewegungsgleichung_von_m1_nach_dem_sto|jxGR6#%\"tG6\"6$%)operatorG%&a rrowGF(-%'mataddG6$%*sto|jxpunktG-%*scalarmulG6$%1v1_nach_dem_sto|jxG9 $F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%Hbewegungsgleichung_von_m 2_nach_dem_sto|jxGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%'mataddG6$%*sto| jxpunktG-%*scalarmulG6$%1v2_nach_dem_sto|jxG9$F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 166 "Jetzt wird die Bahn der Massenpunkte als absch nittsweise definierte Funktion festgelegt: (in Abh\344ngigkeit vom Wah rheitswert der ober definierten bool'schen Variabeln)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "if es_kommt_zum_sto\337=true then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 161 " bahn1:=t->piecewise(t<=rhs(zeiten[1]),b ewegungsgleichung_von_m1_vor_dem_sto\337(t),t>rhs(zeiten[1]),bewegungs gleichung_von_m1_nach_dem_sto\337(t-rhs(zeiten[1])));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "bahn2:=t->piecewise(t<=rhs(zeiten[1]),bewegungs gleichung_von_m2_vor_dem_sto\337(t),t>rhs(zeiten[1]),bewegungsgleichun g_von_m2_nach_dem_sto\337(t-rhs(zeiten[1])));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " bahn1:=t-> bewegungsgleichung_von_m1_vor_dem_sto\337(t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " bahn2:=t->bewegungsgleichung_von_m2_vor_dem_sto\337( t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&bahn1GR6#%\" tG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6&19$-%$rhsG6#&%'zeitenG6#\" \"\"-%Gbewegungsgleichung_von_m1_vor_dem_sto|jxG6#F02F1F0-%Hbewegungsg leichung_von_m1_nach_dem_sto|jxG6#,&F0F7F1!\"\"F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&bahn2GR6#%\"tG6\"6$%)operatorG%&arrowGF(-%*piec ewiseG6&19$-%$rhsG6#&%'zeitenG6#\"\"\"-%Gbewegungsgleichung_von_m2_vor _dem_sto|jxG6#F02F1F0-%Hbewegungsgleichung_von_m2_nach_dem_sto|jxG6#,& F0F7F1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "und als Punk tmuster gezeichnet:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "data1:=seq( [bahn1(0.1*n)[1],bahn1(0.1*n)[2]],n=1..85):data2:=seq([bahn2(0.1*n)[1] ,bahn2(0.1*n)[2]],n=1..85):data3:=seq([bewegungsgleichung_von_s(0.1*n) [1],bewegungsgleichung_von_s(0.1*n)[2]],n=1..85):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 44 "plot(\{[data1],[data2],[data3]\},style=point );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Die Punktgrafik kann durch \+ entsprechende Tools aus dem INTERNET verbessert werden.. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "polyg:=proc(x::numeric, y::numeric, rx::nume ric, ry::numeric, n::numeric)\n local data, i;\n data:=[];\n for i \+ from 0 to evalf(2*Pi-0.00001) by evalf(2*Pi/n) do\n data:=[ op(data ) , [evalf(x+rx*sin(i),5), evalf(y+ry*cos(i),5)] ];\n od;\n data;\ne nd:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 294 "do tplot:=proc( data::list, rx::numeric,f1,f2,f3::numeric )\n local i, p ol, prop;\n if nargs>=3 then prop:=args[3]: else prop:=1: fi:\n pol: =[];\n pol:=seq( polyg(data[i][1], data[i][2],rx, rx*prop, 25),\n \+ i=1..nops(data));\n pol:=POLYGONS( pol, COLOR(RGB,f1,f2,f3)); \n PLOT(pol):\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "dataneu:=[op([data1]),op([d ata2]),op([data3])]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Die Farbw erte f1, f2, f3 m\374ssen zwischen 0 und 1 liegen. " }{MPLTEXT 1 0 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "dotplot(dataneu,1/50,1/2,0,0); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 182 "Das Punktmuster sieht schon \+ besser aus, nur bewegen tut sich noch nichts. Es wird wohl etwas dauer n, bis die Rechnungen abgeschlossen sind, bzw. bis die Graphik aufgeba ut worden ist." }}{PARA 0 "" 0 "" {TEXT -1 104 "Dem kann abgeholfen we rden. Als erstes wird die Bahn der Massenpunkte zeitlich nacheinander \+ dargestellt." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "datagraph1:=[];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "for i from 1 to nops(dataneu) do" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 71 " datagraph1:=[op(datagraph1),op([dotplot([dataneu[ i]],1/10,1/2,0,0)])]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "od: " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "display(datagraph1,insequence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Nun wird die Bewegung der Massenpunkte (u nd dem Schwerpunkt) gleichzeitig skizziert." }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "datagraph2:=[];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "for i from 1 to nops([data1]) do" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 101 " datagraph2:=[op(datagraph2),op( [dotplot([data1[i],data2[i],data3[i]],1/5,226/256,188/256,24/256)])]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "display(datagraph2,insequence=true);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Allgemeine Betrachtung" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Der Sto\337punkt sei P(xs/y s)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }} {PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {EXCHG {PARA 12 "" 1 "" {TEXT -1 63 "Der Definitionsbereich der phyika lischen Gr\366\337en wird definiert:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "assume(m1>0);assume(m2>0) ;assume(ts>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "assume(x1 0,real);assume(y10,real);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "assume(xs,real);assume(ys,real);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "assume(v1x,real);assume(v1y,real);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "assume(v2x,real);assume(v2y,real);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Die Anfangsbedingungen f\374r den Massenpunkt m 1 werden definiert:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "m1_zur_zeit_t0:=vector([x10,y10]);v1_vor_dem_sto \337:=vector([v1x,v1y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/m1_zur_ zeit_t0G-%'vectorG6#7$%%x10|irG%%y10|irG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0v1_vor_dem_sto|jxG-%'vectorG6#7$%%v1x|irG%%v1y|irG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Dann lauten die Bewegungsgleichung von der Massenpunkt 1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "m1_vor_de m_sto\337:=t->matadd(m1_zur_zeit_t0,scalarmul(v1_vor_dem_sto\337,t)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0m1_vor_dem_sto|jxGR6#%\"tG6\"6$ %)operatorG%&arrowGF(-%'mataddG6$%/m1_zur_zeit_t0G-%*scalarmulG6$%0v1_ vor_dem_sto|jxG9$F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Es mu \337 eine Bedingung f\374r die Bewegungsgleichung des Massenpunkts 2 d efiniert werden, damit es zum Sto\337 kommt. " }}{PARA 0 "" 0 "" {TEXT -1 66 "Da der Sto\337punkt vorgegeben wurde, bieten sich 2 M\366 glichkeiten an:" }}{PARA 0 "" 0 "" {TEXT -1 123 "1. Der Ort zur Zeit t =0 wird in Abh\344ngigkeit vom Sto\337punkt vorgegeben und die Geschwi ndigkeiten k\366nnen frei gew\344hlt werden." }}{PARA 0 "" 0 "" {TEXT -1 138 "2. Die Anfangsgeschwindigkeit vom Massenpunkt 2 wird in Abh \344ngigkeit vom Sto\337punkt vorgegeben und der Ort zur Zeit t=0 wird frei gew\344hlt. " }}{PARA 0 "" 0 "" {TEXT -1 46 "Im folgenden wird d er erste Weg eingeschlagen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "g1: =m1_vor_dem_sto\337(t)[1]=xs;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ts :=solve(g1,t);" }}{PARA 0 "" 0 "" {TEXT -1 74 "Nun k\366nnen die Anfan gsbedingungen f\374r dem Massenpunkt 2 festgelegt werden." }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 32 "x20:=xs-ts*v2x0;y20:=ys-ts*v2y0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "m2_zur_zeit_t0:=vector([x20,y20]);v2_vor_de m_sto\337:=vector([v2x,v2y]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "m2 _vor_dem_sto\337:=t->matadd(m2_zur_zeit_t0,scalarmul(v2_vor_dem_sto \337,t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g1G/,&%%x10|irG\"\"\"* &%\"tGF(%%v1x|irGF(F(%$xs|irG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ts G,$*&,&%%x10|irG\"\"\"%$xs|irG!\"\"\"\"\"%%v1x|irG!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x20G,&%$xs|irG\"\"\"*&*&,&%%x10|irGF'F&!\"\" F'%%v2x0GF'\"\"\"%%v1x|irG!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$y20G,&%$ys|irG\"\"\"*&*&,&%%x10|irGF'%$xs|irG!\"\"F'%%v2y0GF'\"\"\"% %v1x|irG!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/m2_zur_zeit_t0G- %'vectorG6#7$,&%$xs|irG\"\"\"*&*&,&%%x10|irGF+F*!\"\"F+%%v2x0GF+\"\"\" %%v1x|irG!\"\"F+,&%$ys|irGF+*&*&F.F2%%v2y0GF+F2F3F4F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0v2_vor_dem_sto|jxG-%'vectorG6#7$%%v2x|irG%%v2y| irG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0m2_vor_dem_sto|jxGR6#%\"tG6 \"6$%)operatorG%&arrowGF(-%'mataddG6$%/m2_zur_zeit_t0G-%*scalarmulG6$% 0v2_vor_dem_sto|jxG9$F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Di e Parameter m\374ssen so eingegrenzt werden, da\337 es zum Sto\337 kom mt. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m1_vor_dem_sto\337(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vec torG6#7$,&%%x10|irG\"\"\"*&%\"tGF)%%v1x|irGF)F),&%%y10|irGF)*&F+\"\"\" %%v1y|irGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m2_vor_dem _sto\337(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$,(%$xs|i rG\"\"\"*&*&,&%%x10|irGF)F(!\"\"F)%%v2x0GF)\"\"\"%%v1x|irG!\"\"F)*&%\" tGF)%%v2x|irGF)F),(%$ys|irGF)*&*&F,F0%%v2y0GF)F0F1F2F)*&F4F0%%v2y|irGF )F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Zu den Impulsen (eigentlic h unn\366tig):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "p1_vor_dem_sto \337:=scalarmul(v1_vor_dem_sto\337,m1);p2_vor_dem_sto\337:=scalarmul(v 2_vor_dem_sto\337,m2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0p1_vor_de m_sto|jxG-%'vectorG6#7$*&%$m1|irG\"\"\"%%v1x|irGF+*&F*\"\"\"%%v1y|irGF +" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0p2_vor_dem_sto|jxG-%'vectorG6# 7$*&%$m2|irG\"\"\"%%v2x|irGF+*&F*\"\"\"%%v2y|irGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "p_gesamt:=matadd(p1_vor_dem_sto\337,p2_vor_ dem_sto\337);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)p_gesamtG-%'vector G6#7$,&*&%$m1|irG\"\"\"%%v1x|irGF,F,*&%$m2|irGF,%%v2x|irGF,F,,&*&F+\" \"\"%%v1y|irGF,F,*&F/F3%%v2y|irGF,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Zur Bahn des Schwerpunkts: Zur Zeit t=0 befindet er sich in" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "s_zur_zeit_t0:=scalarmul(matadd(sc alarmul(m1_zur_zeit_t0,m1),scalarmul(m2_zur_zeit_t0,m2)),1/(m1+m2));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%.s_zur_zeit_t0G-%'vectorG6#7$*&,&* &%$m1|irG\"\"\"%%x10|irGF-F-*&%$m2|irGF-,&%$xs|irGF-*&*&,&F.F-F2!\"\"F -%%v2x0GF-\"\"\"%%v1x|irG!\"\"F-F-F-F8,&F,F-F0F-F:*&,&*&F,F8%%y10|irGF -F-*&F0F8,&%$ys|irGF-*&*&F5F8%%v2y0GF-F8F9F:F-F-F-F8F;F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "und bewegt sich in Richtung von" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ric htung_von_s:=scalarmul(p_gesamt,1/(m1+m2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/richtung_von_sG-%'vectorG6#7$*&,&*&%$m1|irG\"\"\"%%v 1x|irGF-F-*&%$m2|irGF-%%v2x|irGF-F-\"\"\",&F,F-F0F-!\"\"*&,&*&F,F2%%v1 y|irGF-F-*&F0F2%%v2y|irGF-F-F2F3F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 261 "Die Impulse (Geschwi ndigkeiten) der Massenpunkte werden zerlegt in unabh\344ngige Komponen ten, senkrecht bzw. parallel zur Bahn des Schwerpunkts. Die Parallelko mponenten bleiben erhalten, die senkrechten Komponenten werden behande lt wie beim eindimensionalen Sto\337." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "richtung_von_s_normiert:=scalarmul(richtung_v on_s,1/norm(richtung_von_s,2)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% 8richtung_von_s_normiertG-%'vectorG6#7$*&,&*&%$m1|irG\"\"\"%%v1x|irGF- F-*&%$m2|irGF-%%v2x|irGF-F-\"\"\"*&-%%sqrtG6#,&*&*$)-%$absG6#F*\"\"#F2 F2*$),&F,F-F0F-\"\"#F2!\"\"F-*&*$)-F<6#,&*&F,F2%%v1y|irGF-F-*&F0F2%%v2 y|irGF-F-F>F2F2*$)FA\"\"#F2FCF-F2FA\"\"\"FC*&FIF2*&-F56#F7F2FA\"\"\"FC " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "v1_parallel:=simplify( scalarmul(richtung_von_s_normiert,dotprod(v1_vor_dem_sto\337,richtung_ von_s_normiert)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,v1_parallelG- %'vectorG6#7$*&*&,**&%$m1|irG\"\"\")%%v1x|irG\"\"#\"\"\"F.*(F0F.%$m2|i rGF.%%v2x|irGF.F.*&F-F2)%%v1y|irGF1F2F.*(F8F.F4F2%%v2y|irGF.F.F.,&*&F- F2F0F2F.*&F4F2F5F2F.F.F2,.*&)F-F1F2F/F2F.**F-F2F0F2F4F2F5F2F1*&)F4F1F2 )F5F1F2F.*&F@F2F7F2F.**F-F2F8F2F4F2F:F2F1*&FCF2)F:F1F2F.!\"\"*&*&F+F2, &*&F-F2F8F2F.*&F4F2F:F2F.F.F2F>FI" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "v1_senkrecht:=matadd(v1_vor_dem_sto\337,-v1_parallel) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-v1_senkrechtG-%'vectorG6#7$,&% %v1x|irG\"\"\"*&*&,**&%$m1|irGF+)F*\"\"#\"\"\"F+*(F*F+%$m2|irGF+%%v2x| irGF+F+*&F0F3)%%v1y|irGF2F3F+*(F9F+F5F3%%v2y|irGF+F+F+,&*&F0F3F*F3F+*& F5F3F6F3F+F+F3,.*&)F0F2F3F1F3F+**F0F3F*F3F5F3F6F3F2*&)F5F2F3)F6F2F3F+* &FAF3F8F3F+**F0F3F9F3F5F3F;F3F2*&FDF3)F;F2F3F+!\"\"!\"\",&F9F+*&*&F.F3 ,&*&F0F3F9F3F+*&F5F3F;F3F+F+F3F?FJFK" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "v2_parallel:=simplify(scalarmul(richtung_von_s_normi ert,dotprod(v2_vor_dem_sto\337,richtung_von_s_normiert)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "v2_senkrecht:=matadd(v2_vor_dem_sto\337,-v2 _parallel);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,v2_parallelG-%'vecto rG6#7$*&*&,**(%%v2x|irG\"\"\"%%v1x|irGF.%$m1|irGF.F.*&%$m2|irGF.)F-\" \"#\"\"\"F.*(%%v2y|irGF.%%v1y|irGF.F0F5F.*&F2F5)F7F4F5F.F.,&*&F0F5F/F5 F.*&F2F5F-F5F.F.F5,.*&)F0F4F5)F/F4F5F.**F0F5F/F5F2F5F-F5F4*&)F2F4F5F3F 5F.*&F@F5)F8F4F5F.**F0F5F8F5F2F5F7F5F4*&FDF5F:F5F.!\"\"*&*&F+F5,&*&F0F 5F8F5F.*&F2F5F7F5F.F.F5F>FI" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-v2_s enkrechtG-%'vectorG6#7$,&%%v2x|irG\"\"\"*&*&,**(F*F+%%v1x|irGF+%$m1|ir GF+F+*&%$m2|irGF+)F*\"\"#\"\"\"F+*(%%v2y|irGF+%%v1y|irGF+F1F6F+*&F3F6) F8F5F6F+F+,&*&F1F6F0F6F+*&F3F6F*F6F+F+F6,.*&)F1F5F6)F0F5F6F+**F1F6F0F6 F3F6F*F6F5*&)F3F5F6F4F6F+*&FAF6)F9F5F6F+**F1F6F9F6F3F6F8F6F5*&FEF6F;F6 F+!\"\"!\"\",&F8F+*&*&F.F6,&*&F1F6F9F6F+*&F3F6F8F6F+F+F6F?FJFK" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Nun werden die Betr\344ge der Orth ogonalkomponenten ben\366tigt. " }{MPLTEXT 1 0 51 "v1:=norm(v1_senkrec ht,2);v2:=-norm(v2_senkrecht,2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% #v1G*$-%%sqrtG6#,&*$)-%$absG6#,&%%v1x|irG\"\"\"*&*&,**&%$m1|irGF1)F0\" \"#\"\"\"F1*(F0F1%$m2|irGF1%%v2x|irGF1F1*&F6F9)%%v1y|irGF8F9F1*(F?F1F; F9%%v2y|irGF1F1F1,&*&F6F9F0F9F1*&F;F9FF9F1**F6F9F?F9F;F9FAF9F8*&FJF9 )FAF8F9F1!\"\"!\"\"F8F9F1*$)-F-6#,&F?F1*&*&F4F9,&*&F6F9F?F9F1*&F;F9FAF 9F1F1F9FEFPFQF8F9F1F9" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#v2G,$*$-%% sqrtG6#,&*$)-%$absG6#,&%%v2x|irG\"\"\"*&*&,**(F1F2%%v1x|irGF2%$m1|irGF 2F2*&%$m2|irGF2)F1\"\"#\"\"\"F2*(%%v2y|irGF2%%v1y|irGF2F8F=F2*&F:F=)F? F " 0 "" {MPLTEXT 1 0 33 "u1:=(m1*v1+m2*( 2*v2-v1))/(m1+m2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "u2:=(m2*v2+m1 *(2*v1-v2))/(m1+m2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#u1G*&,&*&%$ m1|irG\"\"\"-%%sqrtG6#,&*$)-%$absG6#,&%%v1x|irG\"\"\"*&*&,**&F(F5)F4\" \"#F)F5*(F4F5%$m2|irGF5%%v2x|irGF5F5*&F(F))%%v1y|irGF;F)F5*(FAF5F=F)%% v2y|irGF5F5F5,&*&F(F)F4F)F5*&F=F)F>F)F5F5F),.*&)F(F;F)F:F)F5**F(F)F4F) F=F)F>F)F;*&)F=F;F))F>F;F)F5*&FIF)F@F)F5**F(F)FAF)F=F)FCF)F;*&FLF))FCF ;F)F5!\"\"!\"\"F;F)F5*$)-F16#,&FAF5*&*&F8F),&*&F(F)FAF)F5*&F=F)FCF)F5F 5F)FGFRFSF;F)F5F)F5*&F=F),&*$-F+6#,&*$)-F16#,&F>F5*&*&,**(F>F)F4F)F(F) F5*&F=F)FMF)F5*(FCF)FAF)F(F)F5*&F=F)FQF)F5F5FDF)F)FGFRFSF;F)F5*$)-F16# ,&FCF5*&*&FeoF)FenF)F)FGFRFSF;F)F5F)!\"#*$F*F)FSF5F5F),&F(F5F=F5FR" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%#u2G*&,&*&%$m2|irG\"\"\"-%%sqrtG6#,& *$)-%$absG6#,&%%v2x|irG\"\"\"*&*&,**(F4F5%%v1x|irGF5%$m1|irGF5F5*&F(F5 )F4\"\"#F)F5*(%%v2y|irGF5%%v1y|irGF5F;F)F5*&F(F))F@F>F)F5F5,&*&F;F)F:F )F5*&F(F)F4F)F5F5F),.*&)F;F>F))F:F>F)F5**F;F)F:F)F(F)F4F)F>*&)F(F>F)F= F)F5*&FIF))FAF>F)F5**F;F)FAF)F(F)F@F)F>*&FMF)FCF)F5!\"\"!\"\"F>F)F5*$) -F16#,&F@F5*&*&F8F),&*&F;F)FAF)F5*&F(F)F@F)F5F5F)FGFRFSF>F)F5F)FS*&F;F ),&*$-F+6#,&*$)-F16#,&F:F5*&*&,**&F;F)FJF)F5*(F:F)F(F)F4F)F5*&F;F)FOF) F5*(FAF)F(F)F@F)F5F5FDF)F)FGFRFSF>F)F5*$)-F16#,&FAF5*&*&FeoF)FenF)F)FG FRFSF>F)F5F)F>*$F*F)F5F5F5F),&F;F5F(F5FR" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "und nun zur Probe. " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "summe:=simplify(v1+u1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&summeG\"\"!" }}}}}{MARK "3" 0 } {VIEWOPTS 1 1 0 1 1 1803 }