VERSION 10@?a4hdelta_t=1 t0=0 t1=t0+delta_t t2=t1+delta_t t3=t2+delta_t t4=t3+delta_t t5=t4+delta_t t6=t5+delta_t t7=t6+delta_t t8=t7+delta_t t9=t8+delta_t s1=m*((y1-y0)/(t1-t0))^2*(t1-t0)-m*g*(y1+y0)/2*(t1-t0) s2=m*((y2-y1)/(t2-t1))^2*(t2-t1)-m*g*(y2+y1)/2*(t2-t1) s3=m*((y3-y2)/(t3-t2))^2*(t3-t2)-m*g*(y3+y2)/2*(t3-t2) s4=m*((y4-y3)/(t4-t3))^2*(t4-t3)-m*g*(y4+y3)/2*(t4-t3) s5=m*((y5-y4)/(t5-t4))^2*(t5-t4)-m*g*(y5+y4)/2*(t5-t4) s6=m*((y6-y5)/(t6-t5))^2*(t6-t5)-m*g*(y6+y5)/2*(t6-t5) s7=m*((y7-y6)/(t7-t6))^2*(t7-t6)-m*g*(y7+y6)/2*(t7-t6) s8=m*((y8-y7)/(t8-t7))^2*(t8-t7)-m*g*(y8+y7)/2*(t8-t7) s9=m*((y9-y8)/(t9-t8))^2*(t9-t8)-m*g*(y9+y8)/2*(t9-t8) s_tot=s1+s2+s3+s4+s5+s6+s7+s8+s9S8+S9+S106+S7+S8+S9+S10+S9+S10=S1+S2+S3+S4+S5+S6+S7+S8+S9+S10+S9+S10=S1+S2+S3+S4+S5+S6+S7+S8+S9+S10+S9+S10=S1+S2+S3+S4+S5+S6+S7+S8+S9+S10+S9+S10=S1+S2+S3+S4+S5+S6+S7+S8+S9+S10+S9+S10=S1+S2+S3+S4+S5+S6+S7+S8+S9+S10+S9+S10=S1+S2+S3+S4+S5+S6+S7+S8+S9+S10+S7+S8+S9+S10S3+S4+S5+S6+S7+S8+S9+S10+S7+S8+S9+S105+S6+S7+S8+S9+S108+S9+S10+S7+S8+S9+S10=.5*m*(v9^2-2*g*y10) t10=#(x^2+y10^2)/v10 S10=(T10-V10)*t10 action=S1+S2+S3+S4+S5+S6+S7+S8+S9+S10)*t10 action=S1+S2+S3+S4+S5+S6+S7+S8+S9+S10}y k!a]ZVQMJFA=:61-*&m ޟ۟ןПmƟ~wsldV,[  C_ޣڣԣϣˣǣC_lb\X QNJD@ =952+ ҹתV˪Ǫyrngc\Qҹ:)=ߩ٩թ Ω˩ǩ   xkga] U5ҹ$V ݨC_è = =[^mͧms\YUQHEA;7 0-)# ʲޮڮծѮͮƮC_|xsolfc]SPLEA>92/+$ ۭ׭ԭϭȭŭC_}ysplhc_\VSNGD@952-&#ݬڬ֬ϬˬȬìC_YVRLC@<6,%!ҹܫЫ= ~ztp ifEA740'޲ײԲвDz̋tjC_VPJD@60*$  Ha,ׁ؁%~gDAD]6؁fؠ  %RDPC. ؁b22%/DSP0qHa,ׁ؁hg%~gDSD_6؁fm m %8&DP* ؁f(  %.DP* ؁fĢ#7#7%.DH* ؁f#7#7%.DH * \ ׁ<ׁ  %.DP ,* u\ ׁ؁ܱ̋ձαȱ±C_njd[OKE?90-*$C_װͰưð̋{xtnh_N:̋ C_C_ޯƯï}zvrmieVɮ nC_жC_C_zvlie\WSJ ŵµyukhd[VRI Ĵrnda]T" ݳٳҳϳ˳³vO4@B4!C_"ӺC_ɮ`ҹKҹ:)ܹعչҹιȹĹ}zvpl b_[UQ LE62+ҹҹٸVȸV-vaб= = ҹVٷOA ball is thrown so that it passes through both grey blobs. What path should it take? You can either work this out with vectors, step by step, or look at the energetics. 4 paths are defined for you. Try these. Then make your own, trying to minimise the action, S._w @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@)e  backtype=0 Color=15 BkColor=1 Sxfact=50 yfact=10 x=40 y=260 tx=24 ty=24 bStretch=0 xVarValue=0 yVarValue=0 xBall=28 yBall=248 bValues=0 bNames=0 bAxis=0 Color=10 BkColor=15 ObjType=7 bConstX=1 bConstY=1 bOrigConstX=0 bOrigConstY=0 OrigConstXVal=0 OrigConstYVal=0 bTrajectory=0 bStrobe=0 StrobVal=10 PenStyle=1 bAutoScale=0 bPen=1 bCallLines=1 Object no. 128 (null) hxfact=50 yfact=10 x=42 y=262 tx=5 ty=5 bStretch=3 xVarValue=400 yVarValue=85 xBall=438 yBall=173 bValues=0 bNames=0 bAxis=0 Color=2 BkColor=15 ObjType=6 bConstX=0 bConstY=0 bOrigConstX=0 bOrigConstY=0 OrigConstXVal=20 OrigConstYVal=20 bTrajectory=0 bStrobe=0 StrobVal=10 PenStyle=1 bAutoScale=0 bPen=0 bCallLines=0 Object no. 136 t8 y8 (null) Object no. 128 kxfact=50 yfact=10 x=42 y=262 tx=5 ty=5 bStretch=1 xVarValue=300 yVarValue=156 xBall=338 yBall=102 bValues=0 bNames=0 bAxis=0 Color=2 BkColor=15 ObjType=6 bConstX=0 bConstY=0 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Object no. 118 t1 y1 (null) Object no. 114 ixfact=50 yfact=10 x=42 y=262 tx=5 ty=5 bStretch=3 xVarValue=150 yVarValue=149 xBall=188 yBall=109 bValues=0 bNames=0 bAxis=0 Color=2 BkColor=15 ObjType=6 bConstX=0 bConstY=0 bOrigConstX=0 bOrigConstY=0 OrigConstXVal=20 OrigConstYVal=20 bTrajectory=0 bStrobe=0 StrobVal=10 PenStyle=1 bAutoScale=0 bPen=0 bCallLines=0 Object no. 122 t3 y3 (null) Object no. 114 Sxfact=50 yfact=10 x=40 y=260 tx=24 ty=24 bStretch=0 xVarValue=0 yVarValue=0 xBall=28 yBall=248 bValues=0 bNames=0 bAxis=0 Color=10 BkColor=15 ObjType=7 bConstX=1 bConstY=1 bOrigConstX=0 bOrigConstY=0 OrigConstXVal=0 OrigConstYVal=0 bTrajectory=0 bStrobe=0 StrobVal=10 PenStyle=1 bAutoScale=0 bPen=1 bCallLines=1 Object no. 164 (null) ixfact=50 yfact=10 x=42 y=262 tx=5 ty=5 bStretch=0 xVarValue=200 yVarValue=164 xBall=238 yBall=94 bValues=0 bNames=0 bAxis=0 Color=2 BkColor=15 ObjType=6 bConstX=0 bConstY=0 bOrigConstX=0 bOrigConstY=0 OrigConstXVal=20 OrigConstYVal=-20 bTrajectory=0 bStrobe=0 StrobVal=10 PenStyle=1 bAutoScale=0 bPen=0 bCallLines=0 Object no. 166 t4 y4 (null) Object no. 164 jxfact=50 yfact=10 x=42 y=262 tx=5 ty=5 bStretch=1 xVarValue=250 yVarValue=162 xBall=288 yBall=96 bValues=0 bNames=0 bAxis=0 Color=2 BkColor=15 ObjType=6 bConstX=0 bConstY=0 bOrigConstX=0 bOrigConstY=0 OrigConstXVal=-20 OrigConstYVal=-20 bTrajectory=0 bStrobe=0 StrobVal=10 PenStyle=1 bAutoScale=0 bPen=0 bCallLines=0 Object no. 168 t5 y5 (null) Object no. 164 Uxfact=1 yfact=1 x=40 y=260 tx=16 ty=16 bStretch=0 xVarValue=0 yVarValue=0 xBall=32 yBall=252 bValues=0 bNames=0 bAxis=0 Color=8 BkColor=15 ObjType=0 bConstX=1 bConstY=1 bOrigConstX=1 bOrigConstY=1 OrigConstXVal=0 OrigConstYVal=0 bTrajectory=0 bStrobe=0 StrobVal=10 PenStyle=1 bAutoScale=0 bPen=1 bCallLines=1 start 0 (const.) 0 (const.) Uxfact=1 yfact=1 x=490 y=260 tx=16 ty=16 bStretch=0 xVarValue=0 yVarValue=0 xBall=482 yBall=252 bValues=0 bNames=0 bAxis=0 Color=8 BkColor=15 ObjType=0 bConstX=1 bConstY=1 bOrigConstX=1 bOrigConstY=1 OrigConstXVal=0 OrigConstYVal=0 bTrajectory=0 bStrobe=0 StrobVal=10 PenStyle=1 bAutoScale=0 bPen=1 bCallLines=1 end 0 (const.) 0 (const.) fxsc=1.000000 ysc=1.000000 x=50 y=270 tx=45 ty=14 Color=0 BkColor=15 bVarNames=1 Digital no. 126 s1 Arialjxsc=1.000000 ysc=1.000000 x=90 y=280 tx=49 ty=14 Color=0 BkColor=15 bVarNames=1 Digital no. 126-141 s2 Arialoxsc=1.000000 ysc=1.000000 x=140 y=270 tx=49 ty=14 Color=0 BkColor=15 bVarNames=1 Digital no. 126-141-144 s3 Arialsxsc=1.000000 ysc=1.000000 x=190 y=280 tx=49 ty=14 Color=0 BkColor=15 bVarNames=1 Digital no. 126-141-144-147 s4 Arialwxsc=1.000000 ysc=1.000000 x=240 y=270 tx=49 ty=14 Color=0 BkColor=15 bVarNames=1 Digital no. 126-141-144-147-150 s5 Arialxxsc=1.000000 ysc=1.000000 x=290 y=280 tx=49 ty=14 Color=0 BkColor=15 bVarNames=1 Digital no. 126-141-144-147-150- s6 Arialxxsc=1.000000 ysc=1.000000 x=340 y=270 tx=49 ty=14 Color=0 BkColor=15 bVarNames=1 Digital no. 126-141-144-147-150- s7 Arialxxsc=1.000000 ysc=1.000000 x=390 y=280 tx=49 ty=14 Color=0 BkColor=15 bVarNames=1 Digital no. 126-141-144-147-150- s8 Arialxxsc=1.000000 ysc=1.000000 x=440 y=270 tx=45 ty=14 Color=0 BkColor=15 bVarNames=1 Digital no. 126-141-144-147-150- s9 Arialrxsc=1.000000 ysc=1.000000 x=11 y=6 tx=162 ty=23 Color=0 BkColor=15 nLines=0 Object no. 88 Parabolae and energy "Arialt_I_I ,{x=74 y=311 tx=411 ty=16 bVert=0 max=10000 min=-10000 bValues=1 bNames=1 Color=9 BkColor=7 Step=0.000000 Bar no. 172 s_tot