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{TEXT -1 260 "Click on a [+] sign to expand a section. Click on a [-] sign to collapse a section. To do these exercises you will have to insert execution groups. That \+ can be done by clicking on the toolbar icon that looks like \"[>\". It can also be done via the Insert menu." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 9 "In this " }{TEXT 257 5 "MAPLE" }{TEXT -1 3 " " } {HYPERLNK 17 "worksheet" 2 "worksheet" "" }{TEXT -1 28 ", you will be asked to use " }{TEXT 266 5 "MAPLE" }{TEXT -1 175 " to solve first o rder differential equations that pertain to motion against resistance. The last exercise concerns the notions of linear dependence and line ar independence. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Background \+ Worksheets" }}{PARA 0 "" 0 "" {TEXT -1 420 "The following worksheets, \+ available for download from the syllabus web page, have examples or d iscussions that will help you do this homework. If they are in the sa me directory as this worksheet, and if you have retained the filename \+ under which they were posted, then clicking on the hyperlink below wil l automatically open them. Use the Window menu to control the view whe n multiple files are opened simultaneously. " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{HYPERLNK 17 "1.8epR4.mws" 1 "1.8epR4.mws" "" }{TEXT -1 81 " (Sections: Resistive Force Proportional to Velocity, Multicase F unctions) \n " }{HYPERLNK 17 "Tutor2R4.mws" 1 "Tutor2R4.mws" "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 40 "Exercise 1 Air Resistance - Lin ear Law" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 18 "Backgroud Reading:" }{TEXT -1 7 " " }{HYPERLNK 17 "1.8epR4.m ws" 1 "1.8epR4.mws" "" }{TEXT -1 112 " (Sections: Resistive Force P roportional to Velocity, Multicase Functions) \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 46 " Detail " }}{PARA 0 "" 0 "" {TEXT -1 440 "As of this writing, former president George Bu sh has been the only American president to have jumped from an airplan e. He has done it three times. During World War II, the future preside nt parachuted from his airplane when it was shot down. On March 25 199 7, at the age of 72, the former president made a recreational skydive/ parachute jump. This problem concerns that jump. (He made his third ju mp in 1999 to celebrate his 75'th birthday.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Assume that the mass of the pre sident and his jumping gear was given by " }{TEXT 365 2 " " } {XPPEDIT 366 1 "m=7.03*lb*s^2/ft" "/%\"mG**$\"$.(!\"#\"\"\"%#lbGF(%\"s G\"\"#%#ftG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 48 "and that gravitationa l acceleration is given by " }{TEXT 367 2 " " }{XPPEDIT 368 1 "g =32* ft/s" "/%\"gG*(\"#K\"\"\"%#ftGF&%\"sG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "rest art; m := 7.03; g := 32; # Execute!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"$.(!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG\"#K" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 44 "Assume that the force of air resistance w as " }{TEXT 284 1 " " }{XPPEDIT 285 1 "-1.4*v(t)*lb*s/ft" ",$*,$\"#9! \"\"\"\"\"-%\"vG6#%\"tGF'%#lbGF'%\"sGF'%#ftGF&F&" }{TEXT -1 34 " dur ing the skydive phase (where" }{TEXT 363 3 " " }{XPPEDIT 364 1 "v(t) " "-%\"vG6#%\"tG" }{TEXT -1 33 " is President Bush's velocity). " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The diffe rential equation of motion is therefore " }{TEXT 369 1 " " }{XPPEDIT 370 1 "m*diff(v(t),t) = - m*g- 1.4*v(t)" "/*&%\"mG\"\"\"-%%diffG6$-%\" vG6#%\"tGF,F%,&*&F$F%%\"gGF%!\"\"*&$\"#9F0F%-F*6#F,F%F0" }{TEXT -1 1 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ode1 := m*diff(v(t),t) = -m* g - 1.4*v(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode1G/,$-%%diffG6$ -%\"vG6#%\"tGF-$\"$.(!\"#,&$!&'\\AF0\"\"\"F*$!#9!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 40 "The president jump ed from a height of " }{TEXT 286 5 "12500" }{TEXT -1 57 " feet and did not open his parachute until he reached " }{TEXT 287 4 "4500" } {TEXT -1 29 " feet. Compute the duration " }{TEXT 288 1 " " } {XPPEDIT 289 1 "tau" "I$tauG6\"" }{TEXT 290 1 " " }{TEXT -1 17 "of thi s skydive. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "dsolve(\{ode1,v(0) =0\},v(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"vG6#%\"tG,&$!+Vr&o g\"!\"(\"\"\"-%$expG6#,$F'$!+]^Y\"*>!#5$\"+Vr&og\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "v1 := unapply(rhs(\"),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G:6#%\"tG6\"6$%)operatorG%&arrowGF(,&$!+Vr&og \"!\"(\"\"\"-%$expG6#,$9$$!+]^Y\"*>!#5$\"+Vr&og\"F/F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y1 := t -> 12500+int(v1(s), s = 0 . . t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G:6#%\"tG6\"6$%)operator G%&arrowGF(,&\"&+D\"\"\"\"-%$intG6$-%#v1G6#%\"sG/F5;\"\"!9$F.F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "tau := solve( y1(t) = 4500, \+ t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tauG6$$\"++mz![&!\")$!+Qai28 F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Ob viously we will use the positive solution!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "tau := 54.80796600;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$tauG$\"++mz![&!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 117 "What was the president's velocity when he opened \+ his parachute? Plot his velocity for the skydive phase of his jump.\n " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "v1(tau);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+>z#og\"!\"(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 172 "This is \+ in feet per second. (If you fall from the tip of the torch of the Stat ue of Liberty at that constant speed then you crash into the ground in just under one second.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot( v1(t), t = 0 .. tau);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7Z7$ \"\"!F(7$$\"1i&e6@Wm)H!#;$!1y&4*)Q!fy#*!#:7$$\"1DrJA%)GtfF,$!1>/;[+9-= !#97$$\"1(ovMjK*f*)F,$!12/QAo$fi#F57$$\"1DMY%odY>\"F/$!1w^#*3X;-MF57$$ \"1u#*=;.R9'4wk%F57$$\"1C^\"z%H7MAF/$!1Gb!*3VfqdF57$$\"1@*H% \\mh=GF/$!1e5P\">W@!pF57$$\"1a['G!4'f\"F]o7$$\"1m!))*e)f8i#F 5$!1eaO]8<)f\"F]o7$$\"1r)*)e.$pPFF5$!1\"G(3Gv'**f\"F]o7$$\"16r%=^I&eGF 5$!1Yj)p1T9g\"F]o7$$\"1Z9*)*))>P'HF5$!1h\\#znkCg\"F]o7$$\"15l$**HFt2$F 5$!1\\U:`SN.;F]o7$$\"1Jh+r[p%>$F5$!1BYO#>%3/;F]o7$$\"16+iTh^4LF5$!1Rs \">*4l/;F]o7$$\"1u'Q\\i61U$F5$!1t%QU'))30;F]o7$$\"1nE!z@kRa$F5$!14y#e+ uag\"F]o7$$\"1B!4!4?![l$F5$!1;K(*ezu0;F]o7$$\"1ty')3]9tPF5$!1?G)p%3)fg \"F]o7$$\"14#ofN\"Q!)QF5$!1([hON\\hg\"F]o7$$\"1D*z\"3!>w*RF5$!1_gBJnH1 ;F]o7$$\"1nsO-0$z5%F5$!1iyvesS1;F]o7$$\"14`g)zPKA%F5$!1,tti&*\\1;F]o7$ $\"1#)G$pvrfL%F5$!1D1?o9d1;F]o7$$\"1fBmjP*RX%F5$!1aD708j1;F]o7$$\"1y5R $)HmnXF5$!1CTPaqn1;F]o7$$\"1qGh?o=*z%F 5$!1GNwxNu1;F]o7$$\"1\"4H%ex60\\F5$!1]I$pz#og \"F]o-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"tG%!G-%%VIEWG6 $;F($\"++mz![&!\")%(DEFAULTG" 2 372 372 372 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 0 0 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "skydive_plot := \":" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "As you can see, the presi dent pretty much achieved terminal velocity!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "President Bush touched down on the ground " }{TEXT 291 3 "540" } {TEXT -1 113 " seconds after he jumped from the plane. Continue to a ssume that air resistance provided a force of the form " }{TEXT 292 1 " " }{XPPEDIT 293 1 "-k*v*lb*s/ft" ",$*,%\"kG\"\"\"%\"vGF%%#lbGF%%\" sGF%%#ftG!\"\"F*" }{TEXT -1 22 " for some constant " }{TEXT 294 1 " " }{XPPEDIT 295 1 "k" "I\"kG6\"" }{TEXT -1 24 " . Find the value of " }{TEXT 296 1 " " }{XPPEDIT 297 1 "k" "I\"kG6\"" }{TEXT -1 194 " - it is different from the coefficient that was used in the skydive pha se: that is the point of a parachute. What was the president's termin al velocity? With what velocity did he touch land?" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ode2 := m*diff(v(t),t) = -m*g - k*v(t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode2G/,$-%%diffG6$-%\"vG6#%\"tGF-$ \"$.(!\"#,&$!&'\\AF0\"\"\"*&%\"kGF4F*F4!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "dsolve(\{ode2, v(tau) = v1(tau)\}, v(t));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"vG6#%\"tG,$*&,&$!%Cc\"\"!\"\"\"*( -%$expG6#,$*&%\"kGF.F'F.$!+2^ZA9!#5F.,&$!+++g\\AF-F.F5$\"+>z#og\"F-F.- F16#,$F5$!+InH'z(!\"*!\"\"$!+++++D!#:F.F5FD$\"+++++S!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "v2 := unapply(rhs(\"), t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G:6#%\"tG6\"6$%)operatorG%&arrowG F(,$*&,&$!%Cc\"\"!\"\"\"*(-%$expG6#,$*&%\"kGF29$F2$!+2^ZA9!#5F2,&$!+++ g\\AF1F2F9$\"+>z#og\"F1F2-F56#,$F9$!+InH'z(!\"*!\"\"$!+++++D!#:F2F9FI$ \"+++++S!#6F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "y2 := t \+ -> 4500 + int(v2(s),s=tau..t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y 2G:6#%\"tG6\"6$%)operatorG%&arrowGF(,&\"%+X\"\"\"-%$intG6$-%#v2G6#%\"s G/F5;%$tauG9$F.F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "k := fsolve(y2(540) = 0, k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG$\"+ -i>\\C!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "Plot the president's height and veloc ity for the parachute phase of the jump." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(v2(t), t = tau .. tau+10);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7[o7$$\"1++++mz![&!#9$!1\"4tY'y#og\"!#87$$\"1M 3FW7_$[&F*$!1v0r6Sjp9F-7$$\"1n;a))eC'[&F*$!1J2G3U'[M\"F-7$$\"1+D\"G`q* )[&F*$!1%40HN$RJ7F-7$$\"1ML3x^p\"\\&F*$!1(eV9B)>G6F-7$$\"1nTN@)>W\\&F* $!1XXM!H[V.\"F-7$$\"1+]ilW9(\\&F*$!15caaC(**[*F*7$$\"1Le*)4\"p)*\\&F*$ 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45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 256 3328 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "parachute_plot := \":" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(y2(t), t = ta u .. 540);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7S7$$\"1++++mz! [&!#9$\"1n@$*)******\\%!#77$$\"1eObcrPQlF*$\"1O'32\"ePfVF-7$$\"1)[eRMs &euF*$\"1-W[$Qb[F%F-7$$\"1GbSs!GM\\)F*$\"1<_NBL!)zTF-7$$\"1(4qj\\V^`*F *$\"1^4Z'>@T3%F-7$$\"1bk(Gy!>d5!#8$\"1kO.IQ*)))RF-7$$\"1dm*z57L:\"FE$ \"1(*[Rreg+RF-7$$\"1(=%fs,%GD\"FE$\"1)fiS!))=4QF-7$$\"1@tPRAxb8FE$\"1p SDU^k9PF-7$$\"1cor,UPe9FE$\"1**z;$o//i$F-7$$\"1NtOJH\"Rc\"FE$\"1,JP(ym M_$F-7$$\"11%*)=]rol\"FE$\"1gD*f$Q3QMF-7$$\"16(>^$3_hM$F -7$$\"1hnco)*fm=FE$\"1$=(H$>ZaC$F-7$$\"14W'z)H'y'>FE$\"1jsrokV_JF-7$$ \"1$od:g>)f?FE$\"1P?@?Q(z1$F-7$$\"1xTN4\\;p@FE$\"1cqth&Rv'HF-7$$\"1Wa! zs%zhAFE$\"1t`#3deC)GF-7$$\"1\")3oF_apBFE$\"1B0<_\"*[$y#F-7$$\"10w%p+moGFE$\"1V S)>Y\\]K#F-7$$\"185ms\\krHFE$\"1'*QlpsXIAF-7$$\"1*G!pVthyIFE$\"15UZmD? K@F-7$$\"1be82rtrJFE$\"1o)=&e:nY?F-7$$\"1\\jv-!4BF$FE$\"1o&Gns&Ha>F-7$ $\"1(Q**=h4iP$FE$\"1Uv]`C')e=F-7$$\"1**ze(3cyZ$FE$\"1MW49'*\\lK)RFE$\"1'y,R$3K,8F-7$$\"13?o=[+(3%FE$\"1Jse*>$*f?\"F-7$$\"1t#Qw(* eY=%FE$\"1(\\`ZK(H;6F-7$$\"1pW^6dt'G%FE$\"1%og3HRD-\"F-7$$\"16n!oPMlQ% FE$\"117A`nt3$*FE7$$\"1k(30L95\\%FE$\"1zM!)4F3\\$)FE7$$\"1#*3mV4k\"f%F E$\"1eOS3=#[U(FE7$$\"18FV*=YXp%FE$\"1sCwU:jzkFE7$$\"10Q\"yK*f'z%FE$\"1 jqtHzEUbFE7$$\"1\"4dJcv.*[FE$\"1jb=?\"G4o%FE7$$\"1EK]6P&y*\\FE$\"1$*=H '\\NPp$FE7$$\"1<.k[M)R4&FE$\"1(os]ky2\"GFE7$$\"1h\">%)zxk>&FE$\"1n0U\" [i$p=FE7$$\"1l=9pzd%H&FE$\"1hn4342$o*F*7$$\"$S&\"\"!$\"1k))=x&***f " 0 "" {MPLTEXT 1 0 27 "parachu te_height_plot := \":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "sk ydive_height_plot := plot(y1(t), t = 0 .. tau):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "Combine plots to produce a plot of his velocity for the entire 540 seconds. Do the same for the height function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "display(\{skydive_plot,parachute_plot\});" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7Z7$\"\"!F(7$$\"1i&e6@Wm)H!#;$!1y&4 *)Q!fy#*!#:7$$\"1DrJA%)GtfF,$!1>/;[+9-=!#97$$\"1(ovMjK*f*)F,$!12/QAo$f i#F57$$\"1DMY%odY>\"F/$!1w^#*3X;-MF57$$\"1u#*=;.R9'4wk%F57$$ \"1C^\"z%H7MAF/$!1Gb!*3VfqdF57$$\"1@*H%\\mh=GF/$!1e5P\">W@!pF57$$\"1a['G!4'f\"F]o7$$\"1m!))*e)f8i#F5$!1eaO]8<)f\"F]o7$$\"1r)*)e.$ pPFF5$!1\"G(3Gv'**f\"F]o7$$\"16r%=^I&eGF5$!1Yj)p1T9g\"F]o7$$\"1Z9*)*)) >P'HF5$!1h\\#znkCg\"F]o7$$\"15l$**HFt2$F5$!1\\U:`SN.;F]o7$$\"1Jh+r[p%> $F5$!1BYO#>%3/;F]o7$$\"16+iTh^4LF5$!1Rs\">*4l/;F]o7$$\"1u'Q\\i61U$F5$! 1t%QU'))30;F]o7$$\"1nE!z@kRa$F5$!14y#e+uag\"F]o7$$\"1B!4!4?![l$F5$!1;K (*ezu0;F]o7$$\"1ty')3]9tPF5$!1?G)p%3)fg\"F]o7$$\"14#ofN\"Q!)QF5$!1([hO N\\hg\"F]o7$$\"1D*z\"3!>w*RF5$!1_gBJnH1;F]o7$$\"1nsO-0$z5%F5$!1iyvesS1 ;F]o7$$\"14`g)zPKA%F5$!1,tti&*\\1;F]o7$$\"1#)G$pvrfL%F5$!1D1?o9d1;F]o7 $$\"1fBmjP*RX%F5$!1aD708j1;F]o7$$\"1y5R$)HmnXF5$!1CTPaqn1;F]o7$$\"1qG< yh!Ro%F5$!1\\c=qUr1;F]o7$$\"1Y>h?o=*z%F5$!1GNwxNu1;F]o7$$\"1\"4H%ex60 \\F5$!1]I$pz#og\"F]o-%'COLOURG6&%$RGBG$\"#5!\" \"F(F(-F$6$7in7$Ff\\l$!1\"4tY'y#og\"F]o7$$\"1-BR#*f%Q^&F5$!1+6,t'*p3dF 57$$\"1/Yy%Q&*oa&F5$!1bQMxH6LCF57$$\"10pcVgk&F5$!1XdF5$!1< 1#QC*=+#*F/7$$\"1Awq3$*QxeF5$!1D`+Kb?&=*F/7$$\"1HoFyoe4gF5$!1'o@!od0&= *F/7$$\"1V_Td5F]oFfal7$$\"1dm*z57L:\"F]oFfal7$$\" 1(=%fs,%GD\"F]oFfal7$$\"1@tPRAxb8F]oFfal7$$\"1cor,UPe9F]oFfal7$$\"1NtO JH\"Rc\"F]oFfal7$$\"11%*)=]rol\"F]oFfal7$$\"16(>^$3_hF]oFfal7$$\"1$od:g>)f?F]oFfal7$$\"1xTN4\\ ;p@F]oFfal7$$\"1Wa!zs%zhAF]oFfal7$$\"1\")3oF_apBF]oFfal7$$\"10w%p+moGF]oFfal7$$\"185ms\\krHF]oFfal7$$\"1*G!pVthyIF]oFfal7$ $\"1be82rtrJF]oFfal7$$\"1\\jv-!4BF$F]oFfal7$$\"1(Q**=h4iP$F]oFfal7$$\" 1**ze(3cyZ$F]oFfal7$$\"1L6QzU?wNF]oFfal7$$\"1L(f%yJS&o$F]oFfal7$$\"1(4 *4YK_$y$F]oFfal7$$\"18KhhtG))QF]oFfal7$$\"1z\"y7A>K)RF]oFfal7$$\"13?o= [+(3%F]oFfal7$$\"1t#Qw(*eY=%F]oFfal7$$\"1pW^6dt'G%F]oFfal7$$\"16n!oPMl Q%F]oFfal7$$\"1k(30L95\\%F]oFfal7$$\"1#*3mV4k\"f%F]oFfal7$$\"18FV*=YXp %F]oFfal7$$\"10Q\"yK*f'z%F]oFfal7$$\"1\"4dJcv.*[F]oFfal7$$\"1EK]6P&y* \\F]oFfal7$$\"1<.k[M)R4&F]oFfal7$$\"1h\">%)zxk>&F]oFfal7$$\"1l=9pzd%H& F]oFfal7$$\"$S&F(FfalFj\\l-%+AXESLABELSG6$%\"tG%!G-%%VIEWG6$;F(Fcjl%(D EFAULTG" 2 953 584 584 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 461 142 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 24 "Here is the height plot:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "display(parachute_heigh t_plot,skydive_height_plot);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVES G6$7S7$$\"1++++mz![&!#9$\"1n@$*)******\\%!#77$$\"1eObcrPQlF*$\"1O'32\" ePfVF-7$$\"1)[eRMs&euF*$\"1-W[$Qb[F%F-7$$\"1GbSs!GM\\)F*$\"1<_NBL!)zTF -7$$\"1(4qj\\V^`*F*$\"1^4Z'>@T3%F-7$$\"1bk(Gy!>d5!#8$\"1kO.IQ*)))RF-7$ $\"1dm*z57L:\"FE$\"1(*[Rreg+RF-7$$\"1(=%fs,%GD\"FE$\"1)fiS!))=4QF-7$$ \"1@tPRAxb8FE$\"1pSDU^k9PF-7$$\"1cor,UPe9FE$\"1**z;$o//i$F-7$$\"1NtOJH \"Rc\"FE$\"1,JP(ymM_$F-7$$\"11%*)=]rol\"FE$\"1gD*f$Q3QMF-7$$\"16(>^$3_ hM$F-7$$\"1hnco)*fm=FE$\"1$=(H$>ZaC$F-7$$\"14W'z)H'y'>FE $\"1jsrokV_JF-7$$\"1$od:g>)f?FE$\"1P?@?Q(z1$F-7$$\"1xTN4\\;p@FE$\"1cqt h&Rv'HF-7$$\"1Wa!zs%zhAFE$\"1t`#3deC)GF-7$$\"1\")3oF_apBFE$\"1B0<_\"*[ $y#F-7$$\"10w%p+moGFE$\"1VS)>Y\\]K#F-7$$\"185ms\\krHFE$\"1'*QlpsXIAF-7$$\"1*G!p VthyIFE$\"15UZmD?K@F-7$$\"1be82rtrJFE$\"1o)=&e:nY?F-7$$\"1\\jv-!4BF$FE $\"1o&Gns&Ha>F-7$$\"1(Q**=h4iP$FE$\"1Uv]`C')e=F-7$$\"1**ze(3cyZ$FE$\"1 MW49'*\\lK)RFE$\"1'y,R$3K,8F-7$$\"13?o=[+(3%FE$\"1Jse*>$*f ?\"F-7$$\"1t#Qw(*eY=%FE$\"1(\\`ZK(H;6F-7$$\"1pW^6dt'G%FE$\"1%og3HRD-\" F-7$$\"16n!oPMlQ%FE$\"117A`nt3$*FE7$$\"1k(30L95\\%FE$\"1zM!)4F3\\$)FE7 $$\"1#*3mV4k\"f%FE$\"1eOS3=#[U(FE7$$\"18FV*=YXp%FE$\"1sCwU:jzkFE7$$\"1 0Q\"yK*f'z%FE$\"1jqtHzEUbFE7$$\"1\"4dJcv.*[FE$\"1jb=?\"G4o%FE7$$\"1EK] 6P&y*\\FE$\"1$*=H'\\NPp$FE7$$\"1<.k[M)R4&FE$\"1(os]ky2\"GFE7$$\"1h\">% )zxk>&FE$\"1n0U\"[i$p=FE7$$\"1l=9pzd%H&FE$\"1hn4342$o*F*7$$\"$S&\"\"!$ \"1k))=x&***f\"!#:$\"1Ih2%G()yC\"Fh[l7$$\"1C^\"z%H7MAF\\\\l$ \"1!)QIju2V7Fh[l7$$\"1'*)*QoC7Fh[l7$$\"1)[a)\\**)4v&F\\\\l$\"1rt_Myg77Fh[l7$$\"1. V$yy*yOoF\\\\l$\"1d>'p,_,?\"Fh[l7$$\"1t73GE2hzF\\\\l$\"1\\pP,WB'=\"Fh[ l7$$\"19zw\\v!Q7*F\\\\l$\"1j#z`hn4<\"Fh[l7$$\"1W9ec8GG5F*$\"1xX5&yY]: \"Fh[l7$$\"1ZmKr$*\\Z6F*$\"1>;u[-4Q6Fh[l7$$\"1Rfgjm]_7F*$\"12bKWjwA6Fh [l7$$\"1!*G!Q+?2P\"F*$\"1:PxP$o^5\"Fh[l7$$\"1#RK$R(=%*[\"F*$\"1'*3t!G. s3\"Fh[l7$$\"16fNlp!Qg\"F*$\"1Dvc*Gp'p5Fh[l7$$\"1hJf]NV.\"Fh[l7$$\"1eb%4MOe$>F*$\"1Taldx\"z ,\"Fh[l7$$\"1)=T>!Gbd?F*$\"1*f#QDNF()**F-7$$\"1VH)R\\#Hl@F*$\"1ka)RwRn \")*F-7$$\"1Q`(>,+NG#F*$\"1q0!R!f1H'*F-7$$\"1bquA:1'R#F*$\"1YMRD@\"*\\ %*F-7$$\"1W%f2l2N^#F*$\"199IP'HF*$\"16\\6?$F*$\"1+TY\\61s\")F-7$$\"16+iTh^4LF*$\"1?l&\\QWy)zF-7$$\"1u'Q\\ i61U$F*$\"1a#*[)o\\&4yF-7$$\"1nE!z@kRa$F*$\"1!\\v%*3L:h(F-7$$\"1B!4!4? ![l$F*$\"11qtc,dLuF-7$$\"1ty')3]9tPF*$\"1e-90o_VsF-7$$\"14#ofN\"Q!)QF* $\"1'*yBHzHrqF-7$$\"1D*z\"3!>w*RF*$\"1$H\\\\x()H)oF-7$$\"1nsO-0$z5%F*$ \"1L\\;j%)y0nF-7$$\"14`g)zPKA%F*$\"1i([.Y_0_'F-7$$\"1#)G$pvrfL%F*$\"1? i'4?T%RjF-7$$\"1fBmjP*RX%F*$\"1'p/JuE)\\hF-7$$\"1y5R$)HmnXF*$\"1aU5=&* >nfF-7$$\"1qGV!y&F-7$$\"1Y>h?o=*z%F*$\"1Tft!>2_f&F- 7$$\"1\"4H%ex60\\F*$\"1m%3D:-]U&F-7$$\"1RxYnl_E]F*$\"1KSoea#*H_F-7$$\" 1Dof`g6N^F*$\"1_r,>ZWb]F-7$$\"1)QeX4&*3D&F*$\"1-7'\\N5%p[F-7$$\"1`8$)4 /rh`F*$\"1G9Un0N\"p%F-7$F($\"1!pc&*******\\%F-F[[l-%+AXESLABELSG6$%\"t G%!G-%%VIEWG6$;FgzFez%(DEFAULTG" 2 993 624 624 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 0 508 231 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 259 " " 0 "" {TEXT -1 68 "Exercise 2 Linear Resistance - The Hill-Keller T heory of Sprinting" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 18 "Backgroud Reading:" }{TEXT -1 7 " " }{HYPERLNK 17 "1.8epR4.mws" 1 "1.8epR4.mws" "" }{TEXT -1 55 " (Sections: Resistiv e Force Proportional to Velocity" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 285 "Since the 1920's, mathematical models h ave been developed to study running performance. These models rely on \+ physical theory, empirical evidence, and statistical analysis. In 1973 J.B. Keller modified an existing mathematical theory of running to ma ke it more appropriate for sprints." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 6 "\nLet " }{TEXT 307 1 " " }{TEXT 319 4 "v (t)" }{TEXT -1 50 " denote the speed of a runner at time t. Let \+ " }{TEXT 308 1 " " }{TEXT 320 4 "p(t)" }{TEXT -1 59 " denote the hor izontal component of the propulsive force " }{TEXT 317 13 "per unit ma ss" }{TEXT -1 40 " that is exerted by the runner at time " }{TEXT 309 1 "t" }{TEXT -1 7 ". Let " }{TEXT 310 1 " " }{TEXT 321 4 "R(v)" } {TEXT 322 1 " " }{TEXT -1 30 " be the total resistive force " }{TEXT 318 13 "per unit mass" }{TEXT -1 253 ". There are two components to t he resistive force. There is an internal resistance to motion and the re is also air resistance. We will neglect the air resistance in our a nalysis because it is a far weaker force than the internal resistance \+ to running." }}{PARA 0 "" 0 "" {TEXT -1 14 "\nAssume that " }{TEXT 311 1 " " }{XPPEDIT 328 1 "R(v)=v/tau" "/-%\"RG6#%\"vG*&F&\"\"\"%$tauG !\"\"" }{TEXT 327 3 " " }{TEXT -1 27 " for some positive constant" } {TEXT 325 1 " " }{TEXT 312 1 " " }{XPPEDIT 326 1 "tau" "I$tauG6\"" } {TEXT 324 1 " " }{TEXT -1 23 " (the unit of which is " }{TEXT 316 1 "s " }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Suppose also that " }{TEXT 313 8 "p(t) = P" }{TEXT -1 28 " for a positive constant " }{TEXT 323 1 "P" }{TEXT -1 25 " (th e unit of which is " }{TEXT 315 3 "s^2" }{TEXT -1 33 " ). Obviously t his assumption on " }{TEXT 314 4 "p(t)" }{TEXT -1 123 " is not tenabl e for races longer than a sprint. With the preceding assumptions, the differential equation for velocity is" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "vel_de : = diff(v(t), t) = P - v(t)/tau; # Execute this line!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'vel_deG/-%%diffG6$-%\"vG6#%\"tGF,,&%\"PG\"\"\"* &F)F/%$tauG!\"\"F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 158 " In this exercise we w ill use the Hill-Keller model to compare the performances of Ben J ohnson and Carl Lewis at the 1987 World Championship meet in Rome." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "a) Maximum Velocity" }}{PARA 0 "" 0 "" {TEXT -1 78 "\nSolve the velocity differential equation with appropria te initial condition:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "d solve(\{vel_de,v(0)=0\},v(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"vG6#%\"tG,&*&%\"PG\"\"\"%$tauGF+F+*(-%$expG6#,$*&F,!\"\"F'F+F3F+F*F+ F,F+F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "v := unapply(rhs( \"),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG:6#%\"tG6\"6$%)operat orG%&arrowGF(,&*&%\"PG\"\"\"%$tauGF/F/*(-%$expG6#,$*&F0!\"\"9$F/F7F/F. F/F0F/F7F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The velocity function " }{TEXT 333 4 "v(t)" }{TEXT -1 26 " increases to a limit " }{TEXT 329 4 "vMax" }{TEXT -1 7 " as " }{TEXT 334 1 "t" }{TEXT -1 20 " te nds to infinity." }}{PARA 0 "" 0 "" {TEXT -1 126 "Although the model t hat has been developed is only applicable over a limited time range (a nd is not valid for large values of " }{TEXT 339 1 "t" }{TEXT -1 6 " ) , " }{TEXT 340 4 "v(t)" }{TEXT -1 48 " is already very close to th e limit velocity " }{TEXT 335 4 "vMax" }{TEXT 330 1 " " }{TEXT -1 17 " for values of " }{TEXT 338 1 "t" }{TEXT -1 42 " near the end of t he range of the model. " }}{PARA 0 "" 0 "" {TEXT -1 18 "The limit velo city" }{TEXT 331 2 " " }{TEXT 336 4 "vMax" }{TEXT -1 27 " is therefo re called the " }{TEXT 332 16 "maximum velocity" }{TEXT -1 26 " of t he runner. Compute " }{TEXT 337 4 "vMax" }{TEXT -1 30 " in terms of \+ the parameters " }{TEXT 341 1 "P" }{TEXT -1 11 " a nd " }{TEXT 342 1 " " }{XPPEDIT 344 1 "tau" "I$tauG6\"" }{TEXT 343 1 " " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "assume(tau > 0);\ninter face(showassumed = 0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "v Max := limit(v(t), t = infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%vMaxG*&%\"PG\"\"\"%%tau|irGF'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "b) Displacement" }}{PARA 0 "" 0 " " {TEXT -1 6 "Let " }{TEXT 345 4 "x(t)" }{TEXT -1 49 " denote the \+ distance a runner has run at time " }{TEXT 346 1 "t" }{TEXT -1 34 ". \+ Use your explicit formula for " }{TEXT 347 4 "v(t)" }{TEXT -1 62 " \+ to state and solve a first order initial value problem for " }{TEXT 348 4 "x(t)" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 18 "Show tha t " }{TEXT 349 10 " " }{TEXT 351 8 " " } {XPPEDIT 350 1 "x(t) = v[infinity]*(t-tau*(1-exp(-t/tau)))" "/-%\"xG6# %\"tG*&&%\"vG6#%)infinityG\"\"\",&F&F,*&%$tauGF,,&F,F,-%$expG6#,$*&F&F ,F/!\"\"F6F6F,F6F," }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "dsolve(\{diff(x(t),t) = v(t) , x(0) = 0\}, x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG ,(*(%\"PG\"\"\"%%tau|irGF+F'F+F+*(F,\"\"#-%$expG6#,$*&F,!\"\"F'F+F4F+F *F+F+*&F,F.F*F+F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x := u napply( rhs(\"), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG:6#%\"tG 6\"6$%)operatorG%&arrowGF(,(*(%\"PG\"\"\"%%tau|irGF/9$F/F/*(F0\"\"#-%$ expG6#,$*&F0!\"\"F1F/F9F/F.F/F/*&F0F3F.F/F9F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "'x(t)' = factor(x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG*(%%tau|irG\"\"\"%\"PGF*,(F'F**&F)F*-%$ex pG6#,$*&F)!\"\"F'F*F3F*F*F)F3F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "This is pretty obviously the required for mula." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 " For convenience, redefine:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "x := t -> vMax*(tau*exp(-1/tau*t)-t au+t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG:6#%\"tG6\"6$%)operato rG%&arrowGF(*&%%vMaxG\"\"\",(*&%$tauGF.-%$expG6#,$*&F1!\"\"9$F.F7F.F.F 1F7F8F.F.F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "c) The 1987 Worl d Championships" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 188 "At the 1987 World Championships in Rome, split times for the Men's 100 Meters competition show that the maximum velocities of \+ Ben Johnson (Canada) and Carl Lewis (United States) were both " } {TEXT 352 8 "11.8 m/s" }{TEXT -1 30 ". Johnson's winning time was " } {TEXT 353 6 "9.83 s" }{TEXT -1 35 " and Lewis's second place time was \+ " }{TEXT 354 6 "9.93 s" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "What were the approximate values of \+ " }{TEXT 356 1 "P" }{TEXT 355 1 " " }{TEXT -1 3 "and" }{TEXT 357 1 " \+ " }{XPPEDIT 358 1 "tau" "I$tauG6\"" }{TEXT -1 16 " for these men?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "xJ := t -> 11.8*(tau*exp(-t/ tau)-tau+t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xJG:6#%\"tG6\"6$%)o peratorG%&arrowGF(,(*&%$tauG\"\"\"-%$expG6#,$*&F.!\"\"9$F/F5F/$\"$=\"F 5F.$!$=\"F5F6F7F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "xL : = t -> 11.8*(tau*exp(-t/tau)-tau+t);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#xLG:6#%\"tG6\"6$%)operatorG%&arrowGF(,(*&%$tauG\"\"\"-%$expG6#,$* &F.!\"\"9$F/F5F/$\"$=\"F5F.$!$=\"F5F6F7F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "tauJ := fsolve(xJ(9.83) = 100, tau); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tauJG$\"+B(*Qc8!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "tauL := fsolve(xJ(9.983) = 100, tau);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tauLG$\"+)ff/^\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "PJ := 11.8/tauJ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#PJG$\"+(ek&*p)!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "PL := 11.8/tauL;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#PLG$\"+V<>7y!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 479 "According to the Hill -Keller model, can Johnson's victory be attributed to greater propulsi ve force, lesser resistance to motion, or to both factors? (One year later, the international governing body of track and field decided Jo hnson's victory could be attributed to the anabolic steroid stanozolol , for which he tested positive after winning a gold medal at the Seoul 1988 Olympic Games. Johnson did not receive the gold medal and his pr evious world records were disallowed.)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 378 114 "Answer: Johnson's victory be attrib uted to both factors: greater propulsive force and lesser resistance t o motion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "d) In the same coordinate plane, plot " }{TEXT 359 4 "v(t)" }{TEXT -1 20 " for both runners. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "xJ := t -> 11.8*(tauJ*exp(- t/tauJ)-tauJ+t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xJG:6#%\"tG6\"6 $%)operatorG%&arrowGF(,(*&%%tauJG\"\"\"-%$expG6#,$*&9$F/F.!\"\"F6F/$\" $=\"F6F.$!$=\"F6F5F7F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "xL := t -> 11.8*(tauL*exp(-t/tauL)-tauL+t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xLG:6#%\"tG6\"6$%)operatorG%&arrowGF(,(*&%%tauLG\"\" \"-%$expG6#,$*&9$F/F.!\"\"F6F/$\"$=\"F6F.$!$=\"F6F5F7F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot([D(xJ)(t), D(xL)(t)], t = 0 .. 9.83, color = [red, blue]);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG 6$7W7$\"\"!F(7$$\"1n;HF,Lr5!#;$\"1wT'RVQ:'*)F,7$$\"1LLea-mU@F,$\"1oO@n !\\Us\"!#:7$$\"1](o*p)=[2$F,$\"1#RrAOKMR#F47$$\"1nTN&[xp+%F,$\"1$fTipr \"=IF47$$\"1]7yRtGb]F,$\"1ff)4qD8n$F47$$\"1L$3U>(f.hF,$\"1qzsl4!fF%F47 $$\"1ML$o#f&)erF,$\"1;'\\qgx!R[F47$$\"1N$e%fY69#)F,$\"1AlHR15g`F47$$\" 1Z$F4$\"1jm2ycv)3 \"Fjp7$$\"1](ocw#H!p$F4$\"1vIXy+K-6Fjp7$$\"1mm\"QmFN)QF4$\"1X,%RUME6\" Fjp7$$\"1](oJ(p`&4%F4$\"1qA'zG#QA6Fjp7$$\"1]i]r*>uH%F4$\"1gFdS.NI6Fjp7 $$\"1oTN,P13XF4$\"1!Qw(>>\\P6Fjp7$$\"1bF4$\"1s7gj1$)f6Fjp7$$\"1++bcnz HdF4$\"1o.$e,IF;\"Fjp7$$\"1+v.8FtNfF4$\"1TZ^7F;l6Fjp7$$\"1^(o*Rj)\\8'F 4$\"1_R*yx*=n6Fjp7$$\"1+D'3_BiN'F4$\"1h\\$or<\"p6Fjp7$$\"1nm\"y'\\,blF 4$\"1PC!)[7gq6Fjp7$$\"1,]iosEnnF4$\"1l-<(pi><\"Fjp7$$\"1$3_+S*ffpF4$\" 1y%)>]_-t6Fjp7$$\"1,]F@#p)prF4$\"14Q%z%o-u6Fjp7$$\"1;aLOnrntF4$\"1&))3 \"\\v$[<\"Fjp7$$\"1]Pu'RCXd(F4$\"1\\9P&fnb<\"Fjp7$$\"1n;z&o;nx(F4$\"1f !o5V\"=w6Fjp7$$\"1](=UF$R))zF4$\"1-uV!=Ln<\"Fjp7$$\"1NLB()HE#>)F4$\"10 4uu!*=x6Fjp7$$\"1NeM9#\\2S)F4$\"1sXhl&*ex6Fjp7$$\"1=aBq!4vg)F4$\"1kWWj .$z<\"Fjp7$$\"1,+b/**\\(z)F4$\"1J#=@(3?y6Fjp7$$\"1o\"H6%3D:!*F4$\"10wj 2xYy6Fjp7$$\"1MLju)4+@*F4$\"1$yfyls'y6Fjp7$$\"1_(ocijwT*F4$\"19g;u5')y 6Fjp7$$\"1^7VhZT;'*F4$\"1m%*[4j,z6Fjp7$$\"1++++++I)*F4$\"1yqTF'f\"z6Fj p-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7WF'7$F*$\"1Et\\[(>&z!)F,7$F 0$\"1:0))oIeg:F47$F6$\"1`\"GzM!Rt@F47$F;$\"1E9Y$)F47$Fgo$\"1@ta8G7z()F47$F\\p$\"1&Gh]\"*\\K8*F47$ Fap$\"1U$Q\\<\"\\#[*F47$Ffp$\"1vV(Gybry*F47$F\\q$\"1;L=b(zU+\"Fjp7$Faq $\"1Oi*[AqY-\"Fjp7$Ffq$\"1.r'3,ge/\"Fjp7$F[r$\"1P>dZ3VS<\"Fjp7$F^y$\"1f\")\\-_zu6Fjp 7$Fcy$\"1&[#=OiYv6Fjp7$Fhy$\"1,n$)Ri/w6Fjp7$F]z$\"1-`Y^N^w6Fjp7$Fbz$\" 1%Hngh\")p<\"Fjp7$Fgz$\"1z0dgnMx6Fjp7$F\\[l$\"1Bu'zb(ox6Fjp7$Fa[l$\"1` +ZnE(z<\"Fjp7$Ff[l$\"1?lz%**R#y6Fjp-F[\\l6&F]\\lF(F(F^\\l-%+AXESLABELS G6$%\"tG%!G-%%VIEWG6$;F($\"$$)*!\"#%(DEFAULTG" 2 805 580 580 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 3367 22223 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot([xJ(t), xL(t)], t = 0 ..9.83, color = [red, blue]);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURV ESG6$7S7$\"\"!F(7$$\"1LLea-mU@!#;$\"15&>&fQ&e*=F,7$$\"1nTN&[xp+%F,$\"1 2VW?O;WjF,7$$\"1L$3U>(f.hF,$\"1`R>jlX-9!#:7$$\"1N$e%fY69#)F,$\"1Z#4u() pAU#F97$$\"1Fgn7$$\"1 $3_w/)yiIF9$\"1+E6%4%*3=#Fgn7$$\"1n;*G&=K%G$F9$\"1S835w3Z$F9$\"1gn6S7uH %F9$\"1W!p6)*fx`$Fgn7$$\"1oTN,P13XF9$\"1_EvbF9$\"1 crwMWeR\\Fgn7$$\"1++bcnzHdF9$\"195xG^/%=&Fgn7$$\"1+v.8FtNfF9$\"1$pWE() \\PU&Fgn7$$\"1^(o*Rj)\\8'F9$\"1)fvhl&Fgn7$$\"1+D'3_BiN'F9$\"1&Hk') \\kX\"fFgn7$$\"1nm\"y'\\,blF9$\"1hfbeh7ZhFgn7$$\"1,]iosEnnF9$\"1Ra$*=s t&R'Fgn7$$\"1$3_+S*ffpF9$\"17>,)*yC@mFgn7$$\"1,]F@#p)prF9$\"1Gn;Ix+ooF gn7$$\"1;aLOnrntF9$\"1\"Q47IyFgn7$$ \"1NLB()HE#>)F9$\"1h&o?;V,2)Fgn7$$\"1NeM9#\\2S)F9$\"1ghN)o8cJ)Fgn7$$\" 1=aBq!4vg)F9$\"1A=jj!G\"f&)Fgn7$$\"1,+b/**\\(z)F9$\"1$)z$HL]Hy)Fgn7$$ \"1o\"H6%3D:!*F9$\"1@_90X`R!*Fgn7$$\"1MLju)4+@*F9$\"1r]Is@2p#*Fgn7$$\" 1_(ocijwT*F9$\"1qaeK![Q^*Fgn7$$\"1^7VhZT;'*F9$\"1hy=8Q;[(*Fgn7$$\"1+++ +++I)*F9$\"0([t**********!#8-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7 SF'7$F*$\"1XNl]UT6v&F,7$F5$\"1VOEG#puF\"F97$F;$\"1M9 /(\\Xi@#F97$F@$\"1\")*f#\\B]^LF97$FE$\"1j[$oRQ.c%F97$FJ$\"1hK\\%3W1&fF 97$FO$\"1,1)f-J$=vF97$FT$\"1d&)*G+Cc>*F97$FY$\"1)*zYtN[-6Fgn7$Fin$\"1U n?8&G7F\"Fgn7$F^o$\"18w()Q.mo9Fgn7$Fco$\"1GDUkD&Qn\"Fgn7$Fho$\"1!Q>O&o Kx=Fgn7$F]p$\"15v$*>pOm?Fgn7$Fbp$\"1\"\\9X\\qdH#Fgn7$Fgp$\"1$*er:ia$\\ #Fgn7$F\\q$\"1,=WaO1FFFgn7$Faq$\"10!\\?C%[OHFgn7$Ffq$\"17wqM2\")oJFgn7 $F[r$\"1xRwjs@#R$Fgn7$F`r$\"1Vf@*z!HFOFgn7$Fer$\"1$\\O%*f8Z%QFgn7$Fjr$ \"1!*Hl[%)o!3%Fgn7$F_s$\"1#Q[N,&>FVFgn7$Fds$\"14m9*>!zUXFgn7$Fis$\"1TT g#Qilx%Fgn7$F^t$\"1dJ(\\t`*=]Fgn7$Fct$\"129@$oUoD&Fgn7$Fht$\"1S$Rb&Rj( [&Fgn7$F]u$\"1^a/P?^WdFgn7$Fbu$\"1cf`Ln\"e(fFgn7$Fgu$\"1\"p>PhFKA'Fgn7 $F\\v$\"1#eMl1lxW'Fgn7$Fav$\"1.*[3'Gd$p'Fgn7$Ffv$\"19s9_P8DpFgn7$F[w$ \"1fg\"pOIu;(Fgn7$F`w$\"1*o%)HhMXS(Fgn7$Few$\"13HQ<%fHl(Fgn7$Fjw$\"1E# z_j*Q#*yFgn7$F_x$\"1*p-S\")*QP\")Fgn7$Fdx$\"1Xe'=P!\\!Q)Fgn7$Fix$\"1NA -,P(Rg)Fgn7$F^y$\"1W\\^8G@g))Fgn7$Fcy$\"10,.PpZ*3*Fgn7$Fhy$\"1eFb-O*RL *Fgn7$F]z$\"1#pnjH*3o&*Fgn7$Fbz$\"1lEv!4;(>)*Fgn-Fhz6&FjzF(F(F[[l-%+AX ESLABELSG6$%\"tG%!G-%%VIEWG6$;F($\"$$)*!\"#%(DEFAULTG" 2 936 552 552 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 0 182 205 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "e) What was Johnson's Winning Mar gin?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "xJ(9.83)-xL(9.83);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*4RG!=!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 41 "Exercise 3 Air Resis tance - Quadratic Law" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 18 "Backgroud Reading:" }{TEXT -1 6 " " }{HYPERLNK 17 "1.8epR4.mws" 1 "1.8epR4.mws" "" }{TEXT -1 54 " Sections: Resist ive Force Proportional to Velocity" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 15 " " }}{PARA 0 "" 0 "" {TEXT -1 545 "\nOn December 7 1941, during the attack on Pearl Harbor, an 8 00 kg bomb was dropped from a Nakajima BN52 Kate bomber flying at an a ltitude of 3170 meters. The bomb struck the battleship USS Arizona, \+ igniting its black powder magazine which in turn set off a series of c atastrophic explosions. The ship sank in nine minutes with a death tol l of 1177. An analysis of the attack that was published in 1997 asse rted that the flight of the bomb lasted 26 seconds. Let us accept tha t figure. Assume the Quadratic Drag Law in which air resistance " } {XPPEDIT 360 1 "K*v^2" "*&%\"KG\"\"\"*$%\"vG\"\"#F$" }{TEXT 361 2 " \+ " }{TEXT -1 49 " is proportional to the square of the velocity. \n" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "a) What was the value of " } {TEXT 362 1 "K" }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 54 "Execute the next block of commands to get you started." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ode := m*diff(v(t),t) = -m*g + K*v(t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/*&%\"mG\"\"\"-%%diffG6$-%\"vG6#%\"tGF/F(,&*&F'F (%\"gGF(!\"\"*&%\"KGF(F,\"\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqn := dsolve( ode, v(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$eqnG/,&*&*(%\"mG\"\"\"%\"gGF*%\"KGF*#!\"\"\"\"#-%(arctanhG6#*(F,F*- %\"vG6#%\"tGF*F(F-F*F**&F)F.F7F*F*%$_C1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "with(student): \neqn2 := isolate(eqn, v(t));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%\"vG6#%\"tG,$*(-%%tanhG6#*( ,&*&%$_C1G\"\"\"%\"mGF3!\"\"F)F3F3*(F4F3%\"gGF3%\"KGF3#F3\"\"#F4F5F3F8 F5F6F9F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "_C1 = solve(sub s(t = 0, rhs(eqn2)), _C1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$_C1G \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(\", eqn2);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"vG6#%\"tG,$*(-%%tanhG6#*(F'\"\" \"*(%\"mGF.%\"gGF.%\"KGF.#F.\"\"#F0!\"\"F.F2F5F/F3F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "v := unapply(rhs(\"), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG:6#%\"tG6\"6$%)operatorG%&arrowGF(,$*(-%%tan hG6#*(9$\"\"\"*(%\"mGF3%\"gGF3%\"KGF3#F3\"\"#F5!\"\"F3F7F:F4F8F:F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 "This \+ gives you the downward velocity function. Next comes the height funct ion. Replace the question marks in an appropriate way before executing ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "y := t -> 3170 + Int(v(s), s = 0 .. t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG:6#%\"tG6\"6$%)op eratorG%&arrowGF(,&\"%qJ\"\"\"-%$IntG6$-%\"vG6#%\"sG/F5;\"\"!9$F.F(F( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Now \+ make the definitions:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "m := 800: g := 9.81:" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 " " 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 72 "The height that \+ the bomb reaches at 26 seconds depends on the value of " }{TEXT 371 1 "K" }{TEXT -1 47 " . We can write that height as a function of " } {TEXT 372 1 "K" }{TEXT -1 14 " as follows:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f := z -> evalf(subs(K=z,y(26))); # Execute" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"zG6\"6$%)operatorG%&arrowG F(-%&evalfG6#-%%subsG6$/%\"KG9$-%\"yG6#\"#EF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Find a value of " } {TEXT 373 1 "K" }{TEXT -1 14 " for which " }{TEXT 374 8 "f(K) < 0" }{TEXT -1 19 " and a value of " }{TEXT 375 1 "K" }{TEXT -1 14 " f or which " }{TEXT 376 8 "f(K) > 0" }{TEXT -1 16 " . (Experiment!)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "Then exe cute the following function. (Read about it in the worksheet to which \+ you have been referred.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1069 "bisection := proc()\noption `Copy right: Brian E. Blank, 1996-2000`;\nlocal a,b, f, epsilon;\na := args[ 2]; b := args[3]; f := args[1];\nif nargs < 3 or nargs > 4 then ERROR( `bisection expects three or four arguments`);\nelif not type(f, \{name ,procedure\} ) then \n ERROR(`bisection expects its first argumen t to be a name or procedure`);\nelif not type(a, realcons) or not type (args[3], realcons) then\n ERROR(`bisection expects its second an d third arguments to be real constants`);\nelif not type(f(a),realcons ) or not type(f(b),realcons) then\n ERROR(`bisection expects its \+ first argument to return a real number when evaluated at its second an d third arguments.`);\nelif evalf((args[1])(a)*(args[1])(b)) > 0 then \n ERROR(`No root found`);\nelif args[1](a) = 0 then RETURN(a);\n elif args[1](b) = 0 then RETURN(b);\nelse\n if nargs = 4 then eps ilon := args[4];\n else epsilon := 1.0e-9;\n fi;\n while a bs(a-b) > epsilon do\n if evalf(f(a)*f((a+b)/2)) > 0 then a := evalf ((a+b)/2);\n else b := evalf((a+b)/2);\n fi;\n od;\nRETURN((a+b) /2);\nfi;\nend;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*bisectionG:6\"6& %\"aG%\"bG%\"fG%(epsilonG6#%ECopyright:~Brian~E.~Blank,~1996-2000GF&C& >8$&9\"6#\"\"#>8%&F26#\"\"$>8&&F26#\"\"\"@1529#F92\"\"%FB-%&ERRORG6#%J bisection~expects~three~or~four~argumentsG4-%%typeG6$F;<$%*procedureG% %nameG-FF6#%inbisection~expects~its~first~argument~to~be~a~name~or~pro cedureG43-FK6$F0%)realconsG-FK6$F7FW-FF6#%`obisection~expects~its~seco nd~and~third~arguments~to~be~real~constantsG43-FK6$-F;6#F0FW-FK6$-F;6# F6FW-FF6#%iqbisection~expects~its~first~argument~to~return~a~real~numb er~when~evaluated~at~its~second~and~third~arguments.G2\"\"!-%&evalfG6# *&-F-F-FF6#%.No~root~foundG/FjoFeo-%'RETURNGF\\o/F[pFeo-F apF`oC%@%/FBFD>8'&F26#FD>Fhp$\"#5!#5?(F&F>F>F&2Fhp-%$absG6#,&F0F>F6!\" \"@%2Feo-Fgo6#*&F[oF>-F;6#,&F0#F>F4F6F^rF>>F0-FgoF\\r>F6F`r-FapF\\rF&F &" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 64 "F ill in and execute the next line to find the actual value of " } {TEXT 377 1 "K" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "K := bisection( f , 0.001 , \+ 0.5 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG$\"+U-^?M!#6" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 " b) What was the theoretical terminal velocity of the bomb." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "terminal_velocity = limit(v(t), t = infin ity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2terminal_velocityG$!+dQ)** y%!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 76 "c) What was the actual velocity of the bomb when it struc k the USS Arizona. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "v(26);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$! +i&fSL#!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 70 "d) Plot the height and velocity of the bomb during its 26 second fall." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(v(t), t = 0 .. 26);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7S7$\"\"!F(7$$ \"1LLL$3gsm&!#;$!14J#e]K$fb!#:7$$\"1nm\"zFJ)f5F/$!1E%QqBJ&R5!#97$$\"1L L$eszVh\"F/$!11[.'oHJe\"F57$$\"1ML$33/E<#F/$!1?`&R:>*H@F57$$\"1mm\"HHv \"GFF/$!1h64y$eNn#F57$$\"1LLeC4EVKF/$!1R2C,\"op<$F57$$\"1***\\7E-mx$F/ $!1mR.]lZ(p$F57$$\"1LLeMY=GVF/$!1R\"=s>j[B%F57$$\"1++Dw!)*z([F/$!1*38` Vf%pZF57$$\"1mmm\"y[NW&F/$!1r:)*zn5=`F57$$\"1LL$3:'oTfF/$!1\"HVR0$>+eF 57$$\"1+++X3Z-lF/$!1<70@)z9M'F57$$\"1+++v\"eb1(F/$!1qIEb`L$)oF57$$\"1+ ++br>3wF/$!14u/3(=QS(F57$$\"1KLeM`'45)F/$!1b'o;8C\\(yF57$$\"1mmmhV\"po )F/$!1_*G9kvIV)F57$$\"1mmmY+H$=*F/$!1b([SsnS!*)F57$$\"1)**\\7(Hpg(*F/$ !1=PHq@q\\%*F57$$\"1nmmR#zr-\"F5$!1#f'=>jcI**F57$$\"1+]7Z\\D$3\"F5$!1= 5LL^dX5!#87$$\"1+]P\"G_m8\"F5$!1(p]X@T`4\"Fjq7$$\"1nm\"zzmB>\"F5$!16BN x?+Z6Fjq7$$\"1n;H;,`V7F5$!1'oy+<&>%>\"Fjq7$$\"1LL3nmr)H\"F5$!1]w+jL#[C \"Fjq7$$\"1L$eM)*RgN\"F5$!1GOK5x4(H\"Fjq7$$\"1+]())4SfS\"F5$!1oY'zMLBM \"Fjq7$$\"1LLe!fL)f9F5$!1OuQm')*3R\"Fjq7$$\"1+++T3^::F5$!1he&eTX2W\"Fj q7$$\"1++Do,)*p:F5$!1NQ=m5=*[\"Fjq7$$\"1+]7V?oA;F5$!1p$=LYEd`\"Fjq7$$ \"1++v*[)>\"o\"F5$!1m[js2.(e\"Fjq7$$\"1mmm>\"yPt\"F5$!1/H\"*HTyK;Fjq7$ $\"1++](4=**y\"F5$!1AV&f-m7o\"Fjq7$$\"1L$e9E*yS=F5$!1,det9'[s\"Fjq7$$ \"1++]!)[S'*=F5$!1&\\4]s\\@x\"Fjq7$$\"1m;zV[t[>F5$!1.&)>H?G;=Fjq7$$\"1 +]iLZV.?F5$!1NM\">0J?'=Fjq7$$\"1mm;uQ\"p0#F5$!1e>\"Gfvj!>Fjq7$$\"1+]7y :!H6#F5$!1')Q%4r(Q_>Fjq7$$\"1LLL\"RCo;#F5$!12C>O#)H'*>Fjq7$$\"1LL3c#o> A#F5$!16[1\\pyS?Fjq7$$\"1n;z,blwAF5$!1&etB]'[%3#Fjq7$$\"1+++,u!pK#F5$! 1*=7)R#oU7#Fjq7$$\"1mmT[<]%Q#F5$!1!e/[$yTp@Fjq7$$\"1LLL>[,OCF5$!1q40;Q R4AFjq7$$\"1+]7(\\Q4\\#F5$!1qTH4+f^AFjq7$$\"1+]([Y2Na#F5$!1m?wu9c\"H#F jq7$$\"#EF($!16T_h&fSL#Fjq-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABEL SG6$%\"tG%!G-%%VIEWG6$;F(Fcz%(DEFAULTG" 2 372 372 372 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 191 135 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(y(t), t = 0 .. 26);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7S7$\"\"!$\"%qJF(7$$\"+%3gsm&!#5$\"+bYUoJ!\"'7 $$\"+y7$)f5!\"*$\"+H4\\kJF17$$\"+E(zVh\"F5$\"+5)=s:$F17$$\"+\"3/E<#F5$ \"+8]&o9$F17$$\"+$Hv\"GFF5$\"+d8^LJF17$$\"+D4EVKF5$\"+oMW=JF17$$\"+hAg wPF5$\"+x46+JF17$$\"+NY=GVF5$\"+@QByIF17$$\"+x!)*z([F5$\"+:)zM0$F17$$ \"+#y[NW&F5$\"+LS&\\-$F17$$\"+^hoTfF5$\"+c8E(*HF17$$\"+X3Z-lF5$\"+)H;K 'HF17$$\"+v\"eb1(F5$\"+f=)f#HF17$$\"+br>3wF5$\"+Ys@()GF17$$\"+N`'45)F5 $\"+JAd\\GF17$$\"+iV\"po)F5$\"+NGz,GF17$$\"+Z+H$=*F5$\"+OMweFF17$$\"+s Hpg(*F5$\"+WYx0FF17$$\"+S#zr-\"!\")$\"+cvCcEF17$$\"+Z\\D$3\"Feq$\"+)\\ (3*f#F17$$\"+\"G_m8\"Feq$\"+.o#>a#F17$$\"+)zmB>\"Feq$\"+&)*f%zCF17$$\" +;,`V7Feq$\"+Xsc>CF17$$\"+nmr)H\"Feq$\"+%RlAN#F17$$\"+%)*RgN\"Feq$\"+l \"3%zAF17$$\"+*4SfS\"Feq$\"+\\Jb8AF17$$\"+\"fL)f9Feq$\"+m+!*R@F17$$\"+ T3^::Feq$\"+H%p51#F17$$\"+o,)*p:Feq$\"+nBF\")>F17$$\"+V?oA;Feq$\"+D=c, >F17$$\"+!\\)>\"o\"Feq$\"+TN>5=F17$$\"+?\"yPt\"Feq$\"+_PaDFeq$\"+_&RZN\"F17$$\"+MZV.?Feq$\"+(4OTD\"F17$$\"+uQ \"p0#Feq$\"+@)oL:\"F17$$\"+y:!H6#Feq$\"+*=X`/\"F17$$\"+\"RCo;#Feq$\"*m 7))Q*F17$$\"+c#o>A#Feq$\"*!3pv#)F17$$\"+-blwAFeq$\"*Psw9(F17$$\"+,u!pK #Feq$\"*rn,4'F17$$\"+\\<]%Q#Feq$\"*b%o`[F17$$\"+>[,OCFeq$\"*%y$es$F17$ $\"+(\\Q4\\#Feq$\"*T\\2]#F17$$\"+lu]VDFeq$\"*h(e18F17$$\"#EF($!\"%F1-% 'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"tG%!G-%%VIEWG6$;F(Fez %(DEFAULTG" 2 372 372 372 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 -18776 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 97 "e) Wha t would the time of the fall have been had there been no air drag? The velocity on impact?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "v_ freeFall := t -> - g*t;\ny_freeFall := t -> 3170 + int(v_freeFall(s), \+ s = 0 .. t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+v_freeFallG:6#%\"tG 6\"6$%)operatorG%&arrowGF(,$*&%\"gG\"\"\"9$F/!\"\"F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+y_freeFallG:6#%\"tG6\"6$%)operatorG%&arrowGF(,& \"%qJ\"\"\"-%$intG6$-%+v_freeFallG6#%\"sG/F5;\"\"!9$F.F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "freeFallTime := fsolve(y_freeFall(t ) = 0 , t , 0 .. infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-free FallTimeG$\"+0C?UD!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "v _freeFall(freeFallTime);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+f0!R\\# !\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 379 48 "According to our models, drag had little effect." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 42 " Exercise 4 Linear Dependence/Independence" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 18 "Backgroud Reading:" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "Tutor2R4.mws" 1 "Tutor2R4.mws" "" }{TEXT -1 44 " (The section on solving identitie s.) \n " }{HYPERLNK 17 "wronskian" 2 "linalg[wronskian]" "" }{TEXT -1 14 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Linear D ependence" }}{PARA 0 "" 0 "" {TEXT -1 56 "Find a dependence relations \+ for the elements of the set " }{TEXT 298 1 " " }{XPPEDIT 299 1 "\{2, c os(x)^2,cos(2*x)\}" "<%\"\"#*$-%$cosG6#%\"xGF#-F&6#*&F#\"\"\"F(F," } {TEXT -1 35 ". In other words, find constants " }{TEXT 303 1 "A" } {TEXT -1 2 ", " }{TEXT 304 2 " B" }{TEXT -1 7 " , and " }{TEXT 305 1 " C" }{TEXT -1 29 ", not all zero, such that " }{TEXT 301 2 " " } {XPPEDIT 300 1 "2*A+B*cos(x)^2+C*cos(2*x)=0" "/,(*&\"\"#\"\"\"%\"AGF&F &*&%\"BGF&*$-%$cosG6#%\"xGF%F&F&*&%\"CGF&-F,6#*&F%F&F.F&F&F&\"\"!" } {TEXT -1 12 " for all " }{TEXT 302 1 "x" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "trig_identity := identity( 2*A + B* cos(x)^2 + C*cos(2*x) = 0 , x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% .trig_identityG-%)identityG6$/,(%\"AG\"\"#*&%\"BG\"\"\"-%$cosG6#%\"xGF +F.*&%\"CGF.-F06#,$F2F+F.F.\"\"!F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solve(trig_identity , \{A,B,C\} );" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<%/%\"CGF%/%\"BG,$F%!\"#/%\"AG,$F%#\"\"\"\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "subs( \" , op(1, trig_identi ty));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%\"CG\"\"\"*&F%F&-%$cosG6# %\"xG\"\"#!\"#*&F%F&-F)6#,$F+F,F&F&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "testeq(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%true G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "Linear Independence" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Use the Wronskian \+ to prove that " }{XPPMATH 306 "6#<&*&%\"xG\"\"\"-%$expG6#*&%\"kGF&F%F &F&*&F%\"\"#F'F&*&F%\"\"$F'F&F'" }{TEXT -1 28 " are independent funct ions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "with(linalg): # Execute this command. The wronskian function is in the linalg package." }} {PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "v := vector( [ exp(k*x) , x *exp(k*x), x^2*exp(k*x), x^3*exp(k*x) ] ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7&-%$expG6#*&%\"kG\"\"\"%\"xGF.*&F/F. F)F.*&F/\"\"#F)F.*&F/\"\"$F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "wronskian( v , x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATR IXG6#7&7&-%$expG6#*&%\"kG\"\"\"%\"xGF-*&F.F-F(F-*&F.\"\"#F(F-*&F.\"\"$ F(F-7&*&F,F-F(F-,&F(F-*(F.F-F,F-F(F-F-,&F/F1*(F.F1F,F-F(F-F-,&F0F3*(F. F3F,F-F(F-F-7&*&F,F1F(F-,&F5F1*(F.F-F,F1F(F-F-,(F(F1F7\"\"%*(F.F1F,F1F (F-F-,(F/\"\"'F9FD*(F.F3F,F1F(F-F-7&*&F,F3F(F-,&F=F3*(F.F-F,F3F(F-F-,( F5FDF?FD*(F.F1F,F3F(F-F-,*F(FDF7\"#=FB\"\"**(F.F3F,F3F(F-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%$expG6#*&%\"kG\"\"\"%\"xGF*\"\"%\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 382 42 "Notice that the \+ Wronskian does not vanish." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Information" }} {EXCHG {PARA 261 "" 0 "" {TEXT -1 46 "02SF00R4.mws A MapleV Releas e 4 worksheet." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 42 "Author: Brian E. Blank (16 October 2000)" }}{PARA 264 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This document may \+ not be distributed by any medium," }}{PARA 0 "" 0 "" {TEXT -1 55 "incl uding print, disk, and electronic transfer, without" }}{PARA 0 "" 0 " " {TEXT -1 39 "prior written permission of the author." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 49 "For more informati on, please contact the author:" }}{PARA 266 "" 0 "" {TEXT -1 4 " \+ " }}{PARA 266 "" 0 "" {TEXT -1 32 " Department of Mathematics, " } }{PARA 0 "" 0 "" {TEXT -1 39 " Washington University in St. Louis " }}{PARA 0 "" 0 "" {TEXT -1 26 " St. Louis, MO 63130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Telepho ne: (314) 935-6763" }}{PARA 267 "" 0 "" {TEXT -1 44 " e -mail: brian@math.wustl.edu" }}{PARA 268 "" 0 "" {TEXT -1 0 "" }} {PARA 269 "" 0 "" {TEXT -1 56 "Copyright: \251 2000 Brian E. Blank, \+ All Rights Reserved." }}}}}{MARK "1 1 0" 0 }{VIEWOPTS 1 1 0 3 4 1802 }