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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart;with(DEtools
):\nwith(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 35 "Warning, new definit
ion for adjoint" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definitio
n for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for
 trace" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "R stands for the air re
sistance.  When it is small the glider can make loops." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 502 "Vdot:= (theta,v,x,h) ->  piecewise( h>0,  -si
n(theta) - R*v^2, 0);\nThetadot:=(theta,v,x,h) -> piecewise(h>0, (v^2 \+
- cos(theta))/v, 0);\nXdot := (theta, v,x,h) -> piecewise(h>0, v*cos(t
heta));\nHdot := (theta, v,x,h) -> piecewise(h>0, v*sin(theta));\nGlid
er := \{diff(v(t),t) = Vdot(theta(t),v(t), x(t), h(t)), \n           d
iff(theta(t),t) = Thetadot(theta(t),v(t), x(t), h(t)), \n           di
ff(x(t),t) = Xdot(theta(t),v(t), x(t), h(t)), \n           diff(h(t),t
) = Hdot(theta(t),v(t), x(t), h(t))\};\nR:=0.3;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%VdotGR6&%&thetaG%\"vG%\"xG%\"hG6\"6$%)operatorG%&arr
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%HdotGR6&%&thetaG%\"vG%\"xG%\"hG6\"6
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t), v(t), x(t), h(t)], t=0..20, \n [[theta(0)=0, v(0)=2.5, x(0)=0, h(0
)=1],\n  [theta(0)=0, v(0)=3.5, x(0)=0, h(0)=1],\n  [theta(0)=0, v(0)=
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{PARA 0 "" 0 "" {TEXT -1 50 "Can I make my glider fly exactly 8 before
 it hits?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 281 "DEplot(Glider
, [theta(t), v(t), x(t), h(t)], t=0..20, \n [[theta(0)=0, v(0)=1.5, x(
0)=0, h(0)=1],\n  [theta(0)=0, v(0)=2, x(0)=0, h(0)=1],\n  [theta(0)=0
, v(0)=1.8, x(0)=0, h(0)=1]],\n v=0..2, theta=0..4*Pi, x=-1..10, h=0..
3, \n scene=[x,h], stepsize=0.05, linecolor=[black,green,red]);" }}
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+#3b'Fiao7$Fj`wFhboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjb
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jboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjb
oFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboF
jboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjb
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jboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjb
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jboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjboFjb
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jbo-F\\co6&F^coFfesF(F(F_coFcco-%%VIEWG6$;$!++++]:!\"*$\"++++b5Fhes;$!
+++++:!#5$\"++++]JFhaw-%+AXESLABELSG6$Q\"x6\"Q\"hFebw" 2 473 473 473 
2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 1 1 0 0 0 -10688 101 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 72 "Before we can really solve this, we need NUMBERS for th
e final position." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "sol:=
dsolve(\{ op(Glider),theta(0)=0, v(0)=2, x(0)=0, h(0)=1\},\n       [th
eta(t), v(t), x(t), h(t)],type=numeric);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$solGR6#%(rkf45_xG6'%\"iG%(rkf45_sG%)outpointG%#r1G%#r2G6#%aoC
opyright~(c)~1993~by~the~University~of~Waterloo.~All~rights~reserved.G
6\"C&>8&-%&evalfG6#9$@$52-%$absG6#,$F3!\"\"-F<6#,&&%,loc_controlG6#\"
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IFR\"#GFR\"#8FR\"#VFR\"#JFR\"#OFR\"#XFR\"#NFR\"#?FR\"\"%$FG!\")\"#5FR
\"#DFR\"#6FR\"#YFR\"\"($FG!\"*FFFR\"#:FR\"\")\"&++$\"#TFR\"#7FRFGFcp\"
#PFR\"#WFR\"#CFR\"#RFR\"\"$FR\"#;FR\"\"&Fdp\"#BFU\"#QFR>%'loc_y0G-FY6#
=F06#;FGFcpE\\[l%FcpFUFFFSFGFRFfqFR>%'loc_y1G-FY6#=F0F`rE\\[l!@$0F;FRC
$>&FD6#FfqF3@%1%'DigitsG-%'evalhfG6#FasC$>8%-%*traperrorG6#-Fcs6#-%=ds
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5G-Fbt6#%)loc_workG@$/Fgs%*lasterrorGC%>8'-%+searchtextG6$.Fcs-%(conve
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 " 0 "" {MPLTEXT 1 0 8 "sol(99);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'
/%\"tG\"#**/-%&thetaG6#F%$!1.?0,XG)H\"!#;/-%\"vGF*$\"1x\"=9)z;V5!#:/-%
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0 "" {MPLTEXT 1 0 136 "sol:=dsolve(\{ op(Glider),theta(0)=0, v(0)=1.8,
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output=listprocedure);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG7'/%
\"tGR6#F'6\"6#%aoCopyright~(c)~1993~by~the~University~of~Waterloo.~All
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G%#r2GF+F*C&>8%-%&evalfG6#F-@$52-%$absG6#,$F9!\"\"-FA6#,&&%,loc_contro
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "xsol :=subs(sol,x(t));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xsolGR6#%\"tG6&%(rkf45_sG%)outpoint
G%#r1G%#r2G6#%aoCopyright~(c)~1993~by~the~University~of~Waterloo.~All~
rights~reserved.G6\"C&>8%-%&evalfG6#9$@$52-%$absG6#,$F2!\"\"-F;6#,&&%,
loc_controlG6#\"\"#\"\"\"F2F>4-%'memberG6$&FC6#\"\"'<*$!\"#\"\"!F>FFFP
FE$FEFQ$F>FQ$FFFQC%>FC-%%copyG6#=F/6#;FF\"#ZE\\[lP\"#AFQ\"#5FQ\"#:FQ\"
#EFQ\"\"$FQ\"#VFQ\"#7FQ\"\"%$FF!\")\"#CFQFEFQ\"#NFQ\"#YFQ\"#WFQ\"#6FQ
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\"#KFQFMFF\"#@$\"#=F>\"#8FQ\"#SFQ\"#XFQ\"#;FQ\"\"&Fao\"#FFQ\"#QFQ\"#DF
Q\"#<FQ\"#LFQ\"#OFQ\"#PFQ\"#BFT\"#TFQ\"#RFQ\"\"*\"%+5\"#HFQ\"#IFQFfpFQ
\"#UFQ>%'loc_y0G-FX6#=F/6#;FFF`oE\\[l%F]oFQF`oFTFEFepFFFQ>%'loc_y1G-FX
6#=F/F`rE\\[l!@$0F:FQC$>&FC6#F]oF2@%1%'DigitsG-%'evalhfG6#FasC$>8$-%*t
raperrorG6#-Fcs6#-%=dsolve/numeric_solnall_rkf45G6,%&loc_FG-%$varG6#FC
-Fbt6#F\\r-Fbt6#Fdr-Fbt6#%'loc_F1G-Fbt6#%'loc_F2G-Fbt6#%'loc_F3G-Fbt6#
%'loc_F4G-Fbt6#%'loc_F5G-Fbt6#%)loc_workG@$/Fgs%*lasterrorGC%>8&-%+sea
rchtextG6$.Fcs-%(convertG6$-%#opG6$FF7#Fgs%%nameG>8'-Fav6$.%)hardwareG
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"sol[4](2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/--%\"xG6#%\"tG6#\"\"#$
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