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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "
with( linalg ):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name chan
gecoords has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 45 "Warning,
 the name adjoint has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 80 
"Warning, the protected names norm and trace have been redefined and u
nprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Setup the ODE.  D
on't worry about constants, we'll put those in in the next step:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 34 "ode:=diff(v(t),t)=-g+(k/m)*v(t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$odeG/-%%diffG6$-%\"vG6#%\"tGF,,&%\"gG!\"\"*(%\"kG\"\"\"%\"mGF
/F)\"\"#F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "substitute the cons
tants with the \"subs\" routine" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "ode:=subs(\{m=100,k=0.2,g=9.
8\},ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/-%%diffG6$-%\"vG6
#%\"tGF,,&$\"#)*!\"\"F0*&$\"+++++?!#7\"\"\")F)\"\"#F5F5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 91 "we can also add the initial condition so \+
that we don't have to solve for the constant later" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ic:=v(0)=0;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"vG6#\"\"!F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ode;\nic;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%%diffG6$-%\"vG6#%\"tGF*,&$\"#)*!\"\"F.*&$\"+++++?!#7
\"\"\")F'\"\"#F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"vG6#\"\"!F'
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Turn the verbosity up to 3 so
 we can see what \"dsolve\" is doing" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "infolevel[dsolve]:=3;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*infolevelG6#%'dsolveG\"\"$" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Check the odeadvisor.  We know it \+
is separable, but it's a nice command to know" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "odeadvisor(
ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%,_quadratureG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "odeadvisor(ode,[separable]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%+_separableG" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 48 "Solve....and tell it to use separable methods..." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sol:=dsolve(\{ode,ic\},v(t)
);" }}{PARA 6 "" 1 "" {TEXT -1 29 "Methods for first order ODEs:" }}
{PARA 6 "" 1 "" {TEXT -1 37 "--- Trying classification methods ---" }}
{PARA 6 "" 1 "" {TEXT -1 19 "trying a quadrature" }}{PARA 6 "" 1 "" 
{TEXT -1 23 "trying 1st order linear" }}{PARA 6 "" 1 "" {TEXT -1 16 "t
rying Bernoulli" }}{PARA 6 "" 1 "" {TEXT -1 16 "trying separable" }}
{PARA 6 "" 1 "" {TEXT -1 23 "<- separable successful" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%$solG/-%\"vG6#%\"tG*&,&*&\"#q\"\"\"-%$expG6#,$*(\"
\"(F.\"#D!\"\"F)F.F.F.F.F-F6F.,&F.F6F/F6F6" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 71 "Now we can check on the limit of this equation with the \+
\"limit\" routine" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 38 "limit_vel:=limit(rhs(sol),t=infinity);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*limit_velG!#q" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 85 "If you don't substitute at the beginning, substitu
te here and you get the same answer" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "value(subs(\{m=100,k=0.2,g
=9.8\},sqrt(m*g/k)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++q!\")
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "To evaluate the solution at a
 specific time use \"eval\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eval(sol,t=0);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%\"vG6#\"\"!F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "mynumber:=eval(sol,t=10);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)mynumberG/-%\"vG6#\"#5*&,&*&\"#q\"\"\"-%$expG6##\"#9
\"\"&F.F.F-!\"\"F.,&F.F5F/F5F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 
"One may have to force maple to give a numerical value" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "v10:=e
valf(mynumber);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$v10G/-%\"vG6#\"#
5$!+Q:Y(>'!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "We want to inte
grate v(t) to get the distance (see the Fundamental Theorem of Calculu
s)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "sola:=int(rhs(sol),t=0..10);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%solaG,(*&\"$+&\"\"\"-%#lnG6#,&F(F(-%$expG6##\"#9\"\"
&F(F(!\"\"\"$+(F(*&F'F(-F*6#\"\"#F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 39 "Add to 1200 since the value is negative" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "dist1:=ev
alf(sola);\n1200+dist1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dist1G$!
+F#G%HQ!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+t<dq\")!\"(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Evaluate again at t=16" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eva
l(sol,t=16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"vG6#\"#;*&,&*&\"#
q\"\"\"-%$expG6##\"$7\"\"#DF,F,F+!\"\"F,,&F,F3F-F3F3" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%\"vG6#\"#;$!+7.6Vo!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "L
ook what happens to the velocity as t increases.  At t=16 we are almos
t at terminal velocity!!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(rhs(sol),t=0..16);" }}{PARA 
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F@7$$\"3gmmmUg3*\\\"F@$!3GMY;<9k#z'F@7$$\"3/+++H_)G`\"F@$!3aGzOoF66oF@
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ULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Integrate again to fine the \+
distance after t=16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 28 "solb:=int(rhs(sol),t=0..16);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%%solbG,(*&\"$+&\"\"\"-%#lnG6#,&F(F(-%$expG6##
\"$7\"\"#DF(F(!\"\"\"%?6F(*&F'F(-F*6#\"\"#F(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "1200+evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+VvQ4U!\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Now set up
 another ODE for the post-ripcord event" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "ode2:=diff(v2(t),t)=-g
+(k/m)*v2(t)^2;\node2:=subs(\{m=100,k=7*0.2,g=9.8\},ode2);\nic2:=v2(0)
=rhs(v10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode2G/-%%diffG6$-%#v2
G6#%\"tGF,,&%\"gG!\"\"*(%\"kG\"\"\"%\"mGF/F)\"\"#F2" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%ode2G/-%%diffG6$-%#v2G6#%\"tGF,,&$\"#)*!\"\"F0*&$
\"+++++9!#6\"\"\")F)\"\"#F5F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ic
2G/-%#v2G6#\"\"!$!+P:Y(>'!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "
Again solve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 43 "sol2:=dsolve(\{ode2,ic2\},v2(t),[separable]);" }}
{PARA 6 "" 1 "" {TEXT -1 33 "Classification methods on request" }}
{PARA 6 "" 1 "" {TEXT -1 35 "Methods to be used are: [separable]" }}
{PARA 6 "" 1 "" {TEXT -1 28 "----------------------------" }}{PARA 6 "
" 1 "" {TEXT -1 38 "* Tackling ODE using method: separable" }}{PARA 6 
"" 1 "" {TEXT -1 37 "--- Trying classification methods ---" }}{PARA 6 
"" 1 "" {TEXT -1 16 "trying separable" }}{PARA 6 "" 1 "" {TEXT -1 23 "
<- separable successful" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol2G/-%
#v2G6#%\"tG,$*(\"#5\"\"\"\"\"(#F-\"\"#-%%tanhG6#,&**F.F-\"#]!\"\"F.F/F
)F-F-*&F/F--%#lnG6#*&,&*&\"+++++5F-F.F/F-\"+P:Y(>'F-F-,&*&F?F-F.F/F-\"
+P:Y(>'F7F7F-F-F-F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "And find t
he limit velocity because it is interesting" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(rhs(sol2),t=
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45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "limit_vel2:=limit(rhs(sol2),t=infinity);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%+limit_vel2G,$*&\"#5\"\"\"\"\"(#F(\"\"#!\"\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$!+68vXE!\")" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "eval(sol2,t=0):evalf(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%#v2G6#\"\"!^$$!+P:Y(>'!\")$\"+ma%[V#!#<" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 59 "Now find the time it takes to go from a h
eight of AGL+dist1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 15 "First find y(t)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 29 "sol2a:=int(rhs(sol2),t=0..s);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#>%&sol2aG,0*&#\"$]#\"\"(\"\"\"-%#lnG6#,&-%%tanhG6#,&**F)F*\"#]!\"\"F)
#F*\"\"#%\"sGF*F**&F6F*-F,6#*&,&*&\"+++++5F*F)F6F*\"+P:Y(>'F*F*,&*&F?F
*F)F6F*\"+P:Y(>'F5F5F*F*F*F*F5F*F**&F'F*-F,6#,&F/F*F*F*F*F**&#\"%+XF)F
*-F,6#F7F*F**&FIF*-F,6#\"\"&F*F**&F'F*-F,6#F)F*F**&#F(F)F*-F,6#,&*&F?F
*F)F6F5F@F*F*F5*&#F(F)F*-F,6#F=F*F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 58 "Yikes!  Plot it to see if it makes sense for the distance:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "plot(sol2a,s=0..30);" }}{PARA 13 "" 1 "" {GLPLOT2D 366 227 227 
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ULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Maple has trouble evaluatin
g with such large numbers.  Let's see if it can get us the correct ans
wer:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "S
olve for the time of impact:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "t_impact:=solve(int(rhs(sol2),t=0..s)=-(1200+dist1),s);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)t_impactG^$$\"+E4a\\H!\")$!+TH]\"[)
!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Check the real part:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 44 "sol2b:=int(rhs(sol2),t=0..29.4954);evalf(%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&sol2bG^$$!+A$p0<)!\"($\"+H'))G\"**!#@" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#^$$!+A$p0<)!\"($\"+H'))G\"**!#@" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 49 "Besides the (small) imaginary term, it lo
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l2b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)t_impactG$!+A$p0<)!\"(" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "eval(sol2,t=t_impact):\nv_i
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ctG$\"+68vXE!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{MARK "34 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 
2 33 1 1 }