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Etools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "First, lets define a \+
few convenient abbreviations, so we needn't type our equations so verb
osely" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "Theta:=theta(t): \+
 V:=v(t):\nTdot:=diff(theta(t),t):\nVdot:=diff(v(t),t):\nX:=x(t): Y:=y
(t):\nXdot:=diff(x(t),t):  Ydot:=diff(y(t),t):\nS:=s(t): Sdot:=diff(s(
t),t):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 209 "First, let's see what the flight of the glider looks lik
e for a few different initial angles, but all with the same initial ve
locity.  Note that here we are using the coordinates with a rescaled t
ime variable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 389 "R:=.2;\nDEplot([\n  Tdot= V*(V^2 - cos(Theta))/
V,\n  Vdot= V*(-sin(Theta) -R*V^2),\n  Xdot= V*V*cos(Theta),\n  Ydot= \+
V*V*sin(Theta)],\n[theta,v,x,y],\nt=0..20,v=0..2,theta=-Pi..Pi,\nx=-1.
.15, y=-.5..4,\n[[v(0)=2,theta(0)=Pi/4, x(0)=0,y(0)=2], \n [v(0)=2,the
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\n  scene=[X,Y],\n stepsize=0.1,\nlinecolor=[red,green,blue],obsrange=
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^y-%+AXESLABELSG6$%%x(t)G%%y(t)G" 2 752 272 272 2 0 1 0 2 9 0 4 2 
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0 0 -5548 8 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 244 "We wou
ld like the glider to stop flying once it hits the ground.  One way to
 do this is to set the vector field to 0 if Y ever becomes negative.  \+
We can use piecewise to adjust the equations to account for this.  Thu
s, the picture above becomes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 457 "R:=.2;\nDEplot([\n  Tdot= piecewise(Y>0,V*(V^2 - cos(Theta))/V,
0),\n  Vdot= piecewise(Y>0,V*(-sin(Theta) -R*V^2),0),\n  Xdot= piecewi
se(Y>0,V*V*cos(Theta),0),\n  Ydot= piecewise(Y>0,V*V*sin(Theta),0)],\n
[theta,v,x,y],\nt=0..20,v=0..2,theta=-Pi..Pi,\nx=-1..15, y=-.5..4,\n[[
v(0)=2,theta(0)=Pi/4, x(0)=0,y(0)=2], \n [v(0)=2,theta(0)=0, x(0)=0,y(
0)=2],\n [v(0)=2,theta(0)=-Pi/2,\n  x(0)=0,y(0)=2]],\n  scene=[X,Y],\n
 stepsize=0.1,\nlinecolor=[red,green,blue],obsrange=false);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"RG$\"\"#!\"\"" }}{PARA 13 "" 1 "" 
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$\"1FlR\">*oOHF1$\"1*RT)eUki<F17$Fb^s7$$\"1XYZs?r^HF1$\"1*p/)pMQi<F17$
Fh^s7$$\"1-rn\"yZn'HF1$\"1i=)*y;%3w\"F17$F^_s7$$\"1v/q-<%>)HF1$\"1<6AC
O)zv\"F17$Fd_s7$$\"1d\"*HL*Ru*HF1$\"16Sa;Kt`<F17$Fj_s7$$\"1j`#)\\AR8IF
1$\"1&\\aY\"G(zu\"F17$F``s7$$\"1l%RsMb*HIF1$\"1&GEZ\"=aS<F17$Ff`s7$$\"
1c+1&f'HZIF1$\"1[;moZBJ<F17$F\\as7$$\"1EG:$e)flIF1$\"1_WqB$*z><F17$Fba
s7$$\"1#[&ytZ1&3$F1$\"1X0m$RMfq\"F17$Fhas7$$\"1+Mq<m#f5$F1$\"1aj/P\"*G
*o\"F17$F^bs7$$\"1>Yz8DXGJF1$\"1]+>SNYp;F17$Fdbs7$$\"1%zhJ'*eH:$F1$\"1
Qw[!p6gk\"F17$Fjbs7$$\"1MQ%)RT#)zJF1$\"1rKFu\"\\%=;F17$F`cs7$$\"15#)))
*z.&4KF1$\"13$fBUmie\"F17$Ffcs7$$\"1n%zF))[DC$F1$\"1u30$>]*[:F17$F\\ds
7$$\"1'\\[x@B'zKF1$\"1!fm<g:g]\"F17$Fbds7$$\"10<<\\$=:K$F1$\"1qT2e40d9
F17$Fhds7$$\"1&ptOlg\"pLF1$\"1?jC>ox,9F17$F^es7$$\"152JnQgBMF1$\"1T!)Q
$o=,M\"F17$Fdes7$$\"1[Glu%)*f[$F1$\"1xdm8#*Gs7F17$Fjes7$$\"1Z[f9#Hvb$F
1$\"1$H)H)oq))>\"F17$F`fs7$$\"1JRp0ZJROF1$\"19o\\@@)37\"F17$Fffs7$$\"1
J\\7[-FKPF1$\"1(G*[9+\")R5F17$F\\gs7$$\"10X'=UYp$QF1$\"1K9*y$4!ed*F.7$
Fbgs7$$\"1'fkt#*pL&RF1$\"1\"emt]KXw)F.7$Fhgs7$$\"14yrZR#43%F1$\"1uG&>P
T%))zF.7$F^hs7$$\"140S;jI=UF1$\"1@'zFiJ3F(F.7$Fdhs7$$\"1HDK^.fjVF1$\"1
;,F_LkJmF.7$Fjhs7$$\"1'ftvD(Q9XF1$\"1a;fV?K&3'F.7$F`is7$$\"1_0@H*z!oYF
1$\"1qe:W,aRcF.7$Ffis7$$\"1mt*pMy?#[F1$\"1^p-+l*\\H&F.7$F\\js7$$\"1=5+
=[/u\\F1$\"1Qbsf&>i/&F.7$Fbjs7$$\"1OX@i@0A^F1$\"1Vj(o*>1$)[F.7$Fhjs7$$
\"1\"*GT/6mk_F1$\"1$\\$49nZ#z%F.7$F^[t7$$\"1dotbV#4S&F1$\"1?27^v?gZF.7
$Fd[t7$$\"1z%e[pP.`&F1$\"19*RUbb@x%F.7$Fj[t7$$\"1UrLR<w_cF1$\"1lHY\"QC
`\"[F.7$F`\\t7$$\"1v0rPZLodF1$\"17t?)=f$y[F.7$Ff\\t7$$\"1;dMi.RxeF1$\"
1N@BDfv^\\F.7$F\\]t7$$\"1WW:p!)Q!)fF1$\"1XDO\"RQy-&F.7$Fb]t7$$\"1`-BJ'
ey2'F1$\"14@#pm$f+^F.7$Fh]t7$$\"1'*4/:VOqhF1$\"1/%4lHPa;&F.7$F^^t7$$\"
1uNYy:ZeiF1$\"1D`hoy&*=_F.7$Fd^t7$$\"1Y<!ykLFM'F1$\"1'3ZYvr'e_F.7$Fj^t
7$$\"1v;6D/oBkF1$\"1sLKwX!GG&F.7$F`_t7$$\"1qtjdN\"=]'F1$\"1H(y@R1,H&F.
7$Ff_t7$$\"1ENAj\\gxlF1$\"1:WLU<qz_F.7$F\\`t7$$\"1M(33a(\\^mF1$\"1,r@7
D(4D&F.7$Fb`t7$$\"1E*Qv:2Rs'F1$\"1[@H\\,Z._F.7$Fh`t7$$\"1^E/$GD_z'F1$
\"1-*)[$R`o8&F.7$F^at7$$\"18g_V=#e'oF1$\"1G(*fwy%30&F.7$Fdat7$$\"1O6l1
![g$pF1$\"1yVl\"RC_%\\F.7$Fjat7$$\"1i9w'pQi+(F1$\"1w`HKJz>[F.7$F`bt7$$
\"1'o.<p9n2(F1$\"1=n)[j6Wn%F.7$Ffbt7$$\"1[3<yTyZrF1$\"1$H`<X/!4XF.7$F
\\ct7$$\"11jW^Ou>sF1$\"1T6H-FfBVF.7$Fbct7$$\"1S2(f/yGH(F1$\"1'*)e.dJ$=
TF.7$Fhct7$$\"19uB$)*fuO(F1$\"1p9USNb$*QF.7$F^dt7$$\"1m&e6.[PW(F1$\"19
p]6Z\")\\OF.7$Fddt7$$\"1YyI#*Q)>_(F1$\"16bGW4%zQ$F.7$Fjdt7$$\"1#Gp[[)Q
-wF1$\"1hVHj*p!4JF.7$F`et7$$\"11[Wbp:&o(F1$\"1\"R0@(\\o9GF.7$Ffet7$$\"
1o$*4pFXqxF1$\"1\"GSx*\\j1DF.7$F\\ft7$$\"1-'z^:,%eyF1$\"1,=kOs8(=#F.7$
Fbft7$$\"1uGeNB3\\zF1$\"1'zDbHd(e=F.7$Fhft7$$\"12Td/^_U!)F1$\"1r3WCXOC
:F.7$F^gt7$$\"1\"\\0zV,(Q\")F1$\"1:-1X;1(=\"F.7$Fdgt7$$\"1FC$G&G_P#)F1
$\"1*3p=g/4])Fdq7$Fjgt7$$\"1#ohE>R)Q$)F1$\"1QMAMI=n^Fdq7$F`ht7$$\"1>4K
b.WU%)F1$\"1o*\\&Gi7,>Fdq7$Ffht7$$\"1fi>glII&)F1$!1G'[G0SB\\(Fcs7$F\\i
tF\\itFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFait
FaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFa
itFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFait
FaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFa
itFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFait
FaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFaitFait-Fgbn6&FibnF(F
(FjbnF]cnFacn-%%VIEWG6$;$!+++++=!\"*$\"++++!e\"F\\cn;$!++++]s!#5$\"+++
+DUFjit-%+AXESLABELSG6$%%x(t)G%%y(t)G" 2 789 389 389 2 0 1 0 2 9 0 4 
2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 
12020 0 0 0 0 0 0 0 1 1 0 0 0 123 252 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 287 "But, in fact, we \+
want to know what the initial angle should be that will allow the glid
er to fly exactly 10 units.  As a first step toward this, we can let m
aple create a procedure that gives us the location of the glider at an
y time.  To do this, we use dsolve with the numeric option. " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 304 "fly:=dsolve( [\n  Tdot= pie
cewise(Y>0,V*(V^2 - cos(Theta))/V,0),\n  Vdot= piecewise(Y>0,V*(-sin(T
heta) -R*V^2),0),\n  Xdot= piecewise(Y>0,V*V*cos(Theta),0),\n  Ydot= p
iecewise(Y>0,V*V*sin(Theta),0),\n  Sdot= piecewise(Y>0,V,0), \n  v(0)=
2, theta(0)=0, x(0)=0, y(0)=2, s(0)=0],\n  [Theta,V,X,Y,S], type=numer
ic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$flyG:6#%(rkf45_xG6'%\"iG%(r
kf45_sG%)outpointG%#r1G%#r2G6#%aoCopyright~(c)~1993~by~the~University~
of~Waterloo.~All~rights~reserved.G6\"C&>8&-%&evalfG6#9$@$52-%$absG6#,$
F3!\"\"-F<6#,&&%,loc_controlG6#\"\"#\"\"\"F3F?4-%'memberG6$&FD6#\"\"'<
*F?FG!\"#FF$FF\"\"!$F?FR$FPFR$FGFRC%>FD-%%copyG6#=F06#;FG\"#aE\\[lW\"#
GFR\"\"%$FG!\")\"#@FQ\"#ZFR\"\")\"&++$\"#5FR\"#YFR\"#?FR\"\"*\"%+5\"\"
$FR\"#XFR\"#8FRFFFR\"#TFR\"#JFR\"#PFR\"#KFR\"#`FRFG\"\"&\"#EFR\"#7FR\"
#\\FRFNFG\"#_FR\"#WFRFhnFR\"#^FR\"#;FR\"#AFR\"#SFR\"#MFR\"#<FR\"#[FR\"
#]FR\"#RFR\"#QFR\"#FFR\"#6FR\"#VFR\"#IFR\"#LFR\"#HFR\"#OFRF_pF\\o\"\"(
$FG!\"*\"#DFR\"#:FR\"#>FR\"#NFR\"#CFR\"#=FR\"#9FR\"#BFQ\"#UFR>%'loc_y0
G-FY6#=F06#;FGF_pE\\[l&F[oFQFgoFRFFFQFGFRF_pFR>%'loc_y1G-FY6#=F0FgrE\\
[l!@$0F;FRC$>&FD6#FgoF3@%1%'DigitsG-%'evalhfG6#FhsC$>8%-%*traperrorG6#
-Fjs6#-%=dsolve/numeric_solnall_rkf45G6,%&loc_FG-%$varG6#FD-Fit6#Fcr-F
it6#F[s-Fit6#%'loc_F1G-Fit6#%'loc_F2G-Fit6#%'loc_F3G-Fit6#%'loc_F4G-Fi
t6#%'loc_F5G-Fit6#%)loc_workG@$/F^t%*lasterrorGC%>8'-%+searchtextG6$.F
js-%(convertG6$-%#opG6$FG7#F^t%%nameG>8(-Fhv6$.%)hardwareGF[w@%50FfvFR
0FdwFR-Fet6,FgtFDFcrF[sFauFduFguFjuF]vF`v-%&ERRORG6#F^tF]x7$/%\"tGF7-%
$seqG6$/&%$ordG6#,&8$FGFGFG&Fcr6#F]y/F]yFhrF06%FDFcrF[s" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "fly(20);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7(/%\"tG\"#?/-%&thetaG6#F%$!1`bA_/-)z\"!#;/-%\"vGF*$\"1
y6@NDd9)*F-/-%\"xGF*$\"1lP5Uk\"eM\"!#9/-%\"yGF*$!1.DA%z#*R;&!#A/-%\"sG
F*$\"1%QnDdy7\\\"F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 452 " From tha
t, we can see that the middle solution (with initial angle 0) goes  13
.458... before it hits the ground.  \n  Since the initial conditions a
re part of the input to the procedure fly, to find out how the final p
osition depends on the initial angle would mean calling dsolve again. \+
 This is tedious, so we write a little procedure to do it for us.  It \+
looks long, but it really is only  dsolve followed by evaluating the s
olution at a given time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
889 "whereHit:= proc(theta0) \n  local x,y,v,t,s,theta,X,Y,V,S,R,\n   \+
     Theta,Xdot,Ydot,Vdot,Tdot,Sdot,v0,\n        sol,xhit,hittime;\n  \+
R:=0.2;\n  v0:=2;\n   \n  Theta:=theta(t):  Tdot:=diff(theta(t),t):\n \+
 V:=v(t):          Vdot:=diff(v(t),t):\n  X:=x(t):          Xdot:=diff
(x(t),t):\n  Y:=y(t):          Ydot:=diff(y(t),t):\n  S:=s(t):        \+
  Sdot:=diff(s(t),t):\n\n  sol:=dsolve(\n         [Tdot= piecewise(Y>0
,V*(V^2 - cos(Theta))/V,0),\n          Vdot= piecewise(Y>0,V*(-sin(The
ta) -R*V^2),0),\n          Xdot= piecewise(Y>0,V*V*cos(Theta),0),\n   \+
       Ydot= piecewise(Y>0,V*V*sin(Theta),0),\n          Sdot= piecewi
se(Y>0,V,0),\n          x(0)=0, y(0)=2, v(0)=v0, theta(0)=theta0, s(0)
=0],\n         [x(t),y(t),theta(t),v(t),s(t)],numeric);\n  \n  hittime
 := 10;\n  while ((subs(sol(hittime)[3],y(t)) > 0)) do\n    hittime:= \+
evalf(hittime + 10);\n  od;\n\n  xhit:=rhs(sol(hittime)[2]);\n  RETURN
(xhit);\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Thus, we can  ge
t the same answer out of that routine a little simpler:" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "whereH
it(-Pi/2);\nwhereHit(0);\nwhereHit(Pi/4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"1/Mj7E60&)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
11p1Gj\"eM\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1E(4T@y.1\"!#9" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "In fact, if we like, we can ma
ke a picture of how the final x value depends on the initial angle by \+
graphing this function.  Rather than plotting it directly," }}{PARA 0 
"" 0 "" {TEXT -1 50 "it works better to just plot a sequence of values
:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "plot([seq([z/10,whereHit(z/10)],z=-31..31)]);" }}
{PARA 13 "" 1 "" {INLPLOT "6#-%'CURVESG6$7[o7$$!1+++++++J!#:$\"1O#R*G+
65:!#;7$$!\"$\"\"!$\"1#z6&\\0:eDF-7$$!1+++++++HF*$\"1-L;!f$*\\6%F-7$$!
1+++++++GF*$\"14u@!yF%RMF*7$$!1+++++++FF*$\"1vP_Q$4)fNF*7$$!1+++++++EF
*$\"14d!*4$\\mp$F*7$$!1+++++++DF*$\"1M()pFGg`QF*7$$!1+++++++CF*$\"1Ac%
Qd#ePSF*7$$!1+++++++BF*$\"1pXc@AKjUF*7$$!1+++++++AF*$\"1unw=kQtXF*7$$!
1+++++++@F*$\"1h/Re$o$ppF*7$$!\"#F1$\"1oGZhe)pG(F*7$$!1+++++++>F*$\"1:
f]'z9bc(F*7$$!1+++++++=F*$\"1zRh*[+`$yF*7$$!1+++++++<F*$\"1%eZL,H@6)F*
7$$!1+++++++;F*$\"1iTww)Q4T)F*7$$!1+++++++:F*$\"1o`[l(4Jv)F*7$$!1+++++
++9F*$\"12bdi;vy\"*F*7$$!1+++++++8F*$\"1G4l_'zvw*F*7$$!1+++++++7F*$\"1
[i6vRS]5!#97$$!1+++++++6F*$\"1Xu\\3lk36Fhq7$$!\"\"F1$\"1V?LXz4^6Fhq7$$
!1+++++++!*F-$\"1Pv37(fb=\"Fhq7$$!1+++++++!)F-$\"1^EAX\"Gd@\"Fhq7$$!1+
++++++qF-$\"1#\\w/4MLC\"Fhq7$$!1+++++++gF-$\"1#H18@P\"p7Fhq7$$!1++++++
+]F-$\"1-(>)R3;$H\"Fhq7$$!1+++++++SF-$\"1&et4'Hl98Fhq7$$!1+++++++IF-$
\"1&G(emm:K8Fhq7$$!1+++++++?F-$\"1FFpRH(RM\"Fhq7$$!1+++++++5F-$\"1+#3(
)yr([8Fhq7$F1$\"11p1Gj\"eM\"Fhq7$$\"1+++++++5F-$\"1!)*3qC#zM8Fhq7$$\"1
+++++++?F-$\"1jv`0Gf:8Fhq7$$\"1+++++++IF-$\"1r87e(Q#)G\"Fhq7$$\"1+++++
++SF-$\"1B[bH)HID\"Fhq7$$\"1+++++++]F-$\"1cvWDw(3@\"Fhq7$$\"1+++++++gF
-$\"1kj7^%HF;\"Fhq7$$\"1+++++++qF-$\"1^Fv]\\P46Fhq7$$\"1+++++++!)F-$\"
1H[z_rv^5Fhq7$$\"1+++++++!*F-$\"1TqgQ,X=**F*7$$\"\"\"F1$\"1!e\\#Q&=(H$
*F*7$$\"1+++++++6F*$\"1_#4*=o(>y)F*7$$\"1+++++++7F*$\"1pvl.I<)G)F*7$$
\"1+++++++8F*$\"1K#*Hy]GZyF*7$$\"1+++++++9F*$\"1EX]CuX_uF*7$$\"1++++++
+:F*$\"1\")[C`G-&4(F*7$$\"1+++++++;F*$\"1Gk\"Hbw^w'F*7$$\"1+++++++<F*$
\"1gNfC<L]kF*7$$\"1+++++++=F*$\"1)HJ%*\\1(GhF*7$$\"1+++++++>F*$\"1]ZM)
*)*)Ht&F*7$$\"\"#F1$\"1]c^A&o:(RF*7$$\"1+++++++@F*$\"1N.1*=+Be$F*7$$\"
1+++++++AF*$\"1.f9[\"Q!fLF*7$$\"1+++++++BF*$\"1N&)*)*eH\"4KF*7$$\"1+++
++++CF*$\"1r!*f/Fa1JF*7$$\"1+++++++DF*$\"13vE()R>RIF*7$$\"1+++++++EF*$
\"1\\33n/5+IF*7$$\"1+++++++FF*$\"1P#*)RU*o%)HF*7$$\"1+++++++GF*$\"1-&G
5n@(*)HF*7$$\"1+++++++HF*$\"1y\\25Z;fq!#<7$$\"\"$F1$\"1`,(*R,#HP&Fd^l7
$$\"1+++++++JF*$\"1pOa#[tk8*Fd^l-%'COLOURG6&%$RGBG$\"#5F`rF1F1" 2 812 
464 464 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 
10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 255 171 0 0 0 
0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 174 "A good graph would need \+
some more points near -3 and +2, but this is good enough to let us kno
w that we should concentrate near  the angles .8 and .9, or between  -
1.4 and -1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "whereHit(.8);
whereHit(.9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1H[z_rv^5!#9" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1TqgQ,X=**!#:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 160 "We could just try values until we get what we wan
t, but we can do this more efficiently using bisection, trapping the v
alue we are looking for and narrowing in." }}{PARA 0 "" 0 "" {TEXT -1 
67 "Here is a little maple procedure to do that for a general function
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 708 "bisect:=proc(f,xlow_i
n,xhi_in,epsilon)\n   local xlow, xmid, xhi, flow, fmid, fhi;\n   \n  \+
 xlow := xlow_in;   flow:= f(xlow);\n   xhi  := xhi_in;    fhi := f(xh
i);\n\n   if (sign(flow) = sign(fhi)) then\n       printf( `[%g,%g] an
d [%g,%g] dont bracket a zero\\n`,\n               evalf(xlow), evalf(
flow), evalf(xhi), evalf(fhi));\n       RETURN();\n   fi;\n\n   \n   w
hile( evalf(abs(xhi-xlow)) > evalf(epsilon)) do\n      xmid := evalf((
xlow+xhi)/2);\n      fmid := f(xmid);\n   #   print([xlow,flow], [xmid
,fmid], [xhi,fhi]);\n      if (sign(fmid) = sign(fhi)) then\n         \+
    xhi := xmid; \n             fhi := fmid;\n      else\n            \+
 xlow:= xmid;\n             flow:= fmid;\n      fi; \n   od;\n   RETUR
N(xmid);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "bisect(x-
>whereHit(x)-10.0, .8, .9, 0.00001);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#$\"+zn>k))!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "bisect(x
->whereHit(x)-10.0, -1, -1.5, 0.00001);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$!+'H*3o7!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "So now, we
 can see them both flying merrily along, and landing right at 10." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 416 "R:=.2;\nDEplot([\n  Tdot= p
iecewise(Y>0,V*(V^2 - cos(Theta))/V,0),\n  Vdot= piecewise(Y>0,V*(-sin
(Theta) -R*V^2),0),\n  Xdot= piecewise(Y>0,V*V*cos(Theta),0),\n  Ydot=
 piecewise(Y>0,V*V*sin(Theta),0)],\n[theta,v,x,y],\nt=0..20,v=0..2,the
ta=-Pi..Pi,\nx=-1..11, y=0..3.5,\n[[v(0)=2,theta(0)=.88642, x(0)=0,y(0
)=2], \n [v(0)=2,theta(0)=-1.2681, x(0)=0,y(0)=2]],\n  scene=[X,Y],\n \+
stepsize=0.1,\nlinecolor=[red,blue],obsrange=false);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"RG$\"\"#!\"\"" }}{PARA 13 "" 1 "" {INLPLOT "6&-%
'CURVESG6gw7$7$\"\"!$\"\"#F(7$$\"1])G.sj6(=!#;$\"1/_([)oBW;!#:7$F+7$$
\"1f\\J\"psl\"[F.$\"1V>fCS?t8F17$F37$$\"1G$HyV&*>G)F.$\"1TKe4Vx77F17$F
97$$\"1zdR4%GJ<\"F1$\"1\"R&4qvbc6F17$F?7$$\"1zfyfOT\"[\"F1$\"1^Q\\Mo#z
<\"F17$FE7$$\"1z.M:Y**Q<F1$\"1!yQ8\\WiC\"F17$FK7$$\"15(3b5hh%>F1$\"1Nb
4p0rO8F17$FQ7$$\"1'z!f?A$*4@F1$\"1ga3r6*GV\"F17$FW7$$\"1!4\\$*GA!RAF1$
\"1))\\s4@SD:F17$Fgn7$$\"1g$>i))Q9M#F1$\"1'**eWXa'4;F17$F]o7$$\"1jGK_
\")pBCF1$\"1Cn#o.RRo\"F17$Fco7$$\"1-?**)*yz!\\#F1$\"1Vq\"G_@\"[<F17$Fi
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QSFaem7$F\\jp7$$\"1XP13ga-5Ffem$!1Jb488GRVFaem7$FbjpFbjpFgjpFgjpFgjpFg
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