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{SECT 0 {PARA 18 "" 0 "" {TEXT 256 21 "The parachute problem" }}{PARA 
19 "" 0 "" {TEXT -1 26 "[ Section 3.4, Example 3 ]" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 43 "A person with mass m fa
lls from a height H " }}{PARA 262 "" 0 "" {TEXT -1 26 "with initial ve
locity 0.  " }}{PARA 259 "" 0 "" {TEXT -1 47 "Without the parachute,  \+
the person experiences " }}{PARA 260 "" 0 "" {TEXT -1 36 "a frictional
 force equal to -b1*v.  " }}{PARA 261 "" 0 "" {TEXT -1 32 "With the pa
rachute, it is -b2*v." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 33 "Re-initialize maple worksheet ..." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 29 "Set up initial parameters ..." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 281 "m := 75.0;   # mass in kg\ng := -9.81;  \+
# choose upwards direction as positive\nv0 := 0.0;   # initial velocit
y\nH := 4000.0; # initial height in meters\nb1 := 15.0;  # friction te
rm without parachute\nb2 := 105.0; # friction term with parachute\nT1 \+
:= 85.0;  # time when parachute opens" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"mG$\"$](!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG$!$\")*!
\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v0G$\"\"!F&" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"HG$\"&++%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#b1G$\"$]\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G$\"%
]5!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T1G$\"$])!\"\"" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 41 "Set up first ode (before parachut
e opens)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ode1 := m*diff(v
1(t),t) = m*g - b1*v1(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode1G/
,$*&$\"$](!\"\"\"\"\"-%%diffG6$-%#v1G6#%\"tGF2F+F+,&$\"']dt!\"$F**&$\"
$]\"F*F+F/F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ic1 := v1
(0) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ic1G/-%#v1G6#\"\"!F)" }
}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "Solve for velocity" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dsolve( \{ode1,ic1\}, v1(t) );" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#v1G6#%\"tG,&#\"$\")*\"#?!\"\"*&#F*
F+\"\"\"-%$expG6#,$*&\"\"&F,F'F/F,F/F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "v1 := unapply(rhs(%),t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#v1Gf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&#\"$\")*\"#?
!\"\"*&#F.F/\"\"\"-%$expG6#,$*&#F3\"\"&F39$F3F0F3F3F(F(F(" }}}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 16 "Solve for height" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 30 "ode2 := diff(y1(t),t) = v1(t);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%ode2G/-%%diffG6$-%#y1G6#%\"tGF,,&#\"$\")*\"#?!
\"\"*&#F/F0\"\"\"-%$expG6#,$*&\"\"&F1F,F4F1F4F4" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "ic2 := y1(0) = H;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$ic2G/-%#y1G6#\"\"!$\"&++%!\"\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 25 "dsolve(\{ode2,ic2\},y1(t));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/-%#y1G6#%\"tG,(*&#\"$\")*\"\"%\"\"\"-%$expG6#,$*&\"
\"&!\"\"F'F-F4F-F4*(F+F-\"#?F4F'F-F4#\"&\")p\"F,F-" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 24 "y1 := unapply(rhs(%),t);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#y1Gf*6#%\"tG6\"6$%)operatorG%&arrowGF(,(*&#\"$\")*
\"\"%\"\"\"-%$expG6#,$*&#F1\"\"&F19$F1!\"\"F1F:*&#F/\"#?F1F9F1F:#\"&\"
)p\"F0F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "y1(T1);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")!****f(!\"'" }}}}}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 41 "Set up second ode (after parachute opens)" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ode3 := m*diff(v2(t),t) = m*
g - b2*v2(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode3G/,$*&$\"$](!
\"\"\"\"\"-%%diffG6$-%#v2G6#%\"tGF2F+F+,&$\"']dt!\"$F**&$\"%]5F*F+F/F+
F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ic3 := v2(T1) = v1(T1
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ic3G/-%#v2G6#$\"$])!\"\"$!+(z
**\\!\\!\")" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "Solve for velocit
y" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "dsolve(\{ode3,ic3\},v2(
t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#v2G6#%\"tG,&#\"$\")*\"$S\"
!\"\"*&#\",z&)**H%H\"*++++(\"\"\"*&-%$expG6#,$*(\"\"(F1\"\"&F,F'F1F,F1
-F46#!$>\"F,F1F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "v2 := u
napply(rhs(%),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2Gf*6#%\"tG6
\"6$%)operatorG%&arrowGF(,&#\"$\")*\"$S\"!\"\"*&#\",z&)**H%H\"*++++(\"
\"\"*&-%$expG6#,$*&#\"\"(\"\"&F59$F5F0F5-F86#!$>\"F0F5F0F(F(F(" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "Solve for position" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ode4 := diff(y2(t),t) = v2(t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode4G/-%%diffG6$-%#y2G6#%\"tGF,,&#
\"$\")*\"$S\"!\"\"*&#\",z&)**H%H\"*++++(\"\"\"*&-%$expG6#,$*(\"\"(F6\"
\"&F1F,F6F1F6-F96#!$>\"F1F6F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "ic4 := y2(T1) = y1(T1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ic
4G/-%#y2G6#$\"$])!\"\"$\")!****f(!\"'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "dsolve(\{ode4,ic4\},y2(t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%#y2G6#%\"tG,(*&#\",z&)**H%H\"*+++!)*\"\"\"-%$expG6#,
&*(\"\"(F-\"\"&!\"\"F'F-F5\"$>\"F-F-F-*(\"$\")*F-\"$S\"F5F'F-F5#\"-@;*
\\uG'F,F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "y2 := unapply(
rhs(%),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2Gf*6#%\"tG6\"6$%)op
eratorG%&arrowGF(,(*&#\",z&)**H%H\"*+++!)*\"\"\"-%$expG6#,&*&#\"\"(\"
\"&F19$F1!\"\"\"$>\"F1F1F1*&#\"$\")*\"$S\"F1F:F1F;#\"-@;*\\uG'F0F1F(F(
F(" }}}}{EXCHG }{EXCHG }}{EXCHG }{SECT 0 {PARA 3 "" 0 "" {TEXT -1 46 "
Find conditions when parachute hits the ground" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "solve(y2(t)=0,t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&*&#\"\"&\"\"(\"\"\"-%)LambertWG6#,$*&#\",z&)**H%H\"++++0\\F(-%
$expG6##!,@;**\\]%F0F(F(F(F(#\"-@;*\\uG'\"++++noF(" }}}{PARA 0 "" 0 "
" {TEXT -1 22 "Find this as a decimal" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "T2 := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#T2G$\"+X*zg:*!\")" }}}{PARA 0 "" 0 "" {TEXT -1 27 "Find the velocity \+
at impact" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "v2(T2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"$\")*\"$S\"!\"\"*&$\"+K>u')*)!#k
\"\"\"-%$expG6#!$>\"F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "
evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+GXX6q!\"*" }}}{EXCHG }
{EXCHG }{EXCHG }{EXCHG }}{EXCHG }{EXCHG }{EXCHG }{EXCHG }{EXCHG }
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 21 "Suprising conclusions" }}{PARA 
264 "" 0 "" {TEXT -1 40 "Final impact velocity is VERY sensitive " }}
{PARA 265 "" 0 "" {TEXT -1 28 "to when you pull the cord.  " }}{PARA 
266 "" 0 "" {TEXT -1 23 "Don't wait to long ...." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "If T1 = 80, final speed i
s 7.007 m/sec, parachute opens 321.25 m from the ground" }}{PARA 0 "" 
0 "" {TEXT -1 156 "If T1 = 85, final speed is 7.011 m/sec, parachute o
pens 76 m from the ground\nIf T1 = 86, final speed is 19.71  m/sec, pa
rachute opens 26.9 m from the ground" }}{PARA 0 "" 0 "" {TEXT -1 53 "I
f T1 = 87, parachute opens after you hit the ground!" }}}}{MARK "20 0 \+
0" 21 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }