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0 -1 0 } {PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 270 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 258 20 "Resistance to Motion" } }{PARA 258 "" 0 "" {TEXT 256 4 "HW 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 260 "Click on a [+] sign to expa nd a section. Click on a [-] sign to collapse a section. To do these e xercises you will have to insert execution groups. That can be done by clicking on the toolbar icon that looks like \"[>\". It can also be d one via the Insert menu." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 20 "Student Name and \+ ID:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 20 "S tudent Name and ID:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 0 {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 s olve first order differential equations that pertain to motion against resistance. The last exercise concerns the notions of linear depende nce and linear independence. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 7 " Reports" }}{PARA 0 "" 0 "" {TEXT -1 30 "Reports that you prepare with \+ " }{TEXT 267 5 "MAPLE" }{TEXT -1 112 " should be prepared with the s ame care that you would devote to laboratory reports in biology and ch emistry. " }{TEXT 269 46 "A report should not be a diary or history o f a" }{TEXT -1 1 " " }{TEXT 268 6 " MAPLE" }{TEXT -1 2 " " }{TEXT 270 1 " " }{TEXT 271 51 "session. Delete what is not needed for the r eport." }{TEXT -1 25 " All lines of the form " }{TEXT 259 6 "?topic " }{TEXT -1 82 " (that arise from help queries) should be erased. Al l errors should be erased. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "When you are printing a " }{TEXT 272 5 "MAPLE" }{TEXT -1 165 " report, think about the toner and paper resources tha t you are using. All commands must be terminated - either with the sta ndard terminator, the semicolon, or the " }{TEXT 276 17 "silent termin ator" }{TEXT -1 57 ", the colon. When you assign a variable, for examp le \n \"" }{TEXT 279 7 "x := 5;" }{TEXT -1 30 " \", there is no need to have " }{TEXT 273 5 "MAPLE" }{TEXT -1 12 " echo back " }{TEXT 260 6 "x := 5" }{TEXT -1 89 ". When this is printed, it simply wastes paper and ink. Choose the silent terminator \"" }{TEXT 261 7 "x := \+ 5:" }{TEXT -1 70 " \" instead. When you load a package (without the \+ silent terminator)," }{TEXT 262 1 " " }{TEXT -1 2 " " }{TEXT 274 5 "M APLE" }{TEXT -1 123 " will list the commands that become available wit h the package. This is fine - it will help you become familiar with wh at " }{TEXT 275 5 "MAPLE" }{TEXT -1 125 " makes available. However, t hese commands should not be part of a lab report. Reload the package w ith the silent terminator. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Much of the text in this worksheet should be de leted. For example, delete the " }{TEXT 263 12 "Introduction" }{TEXT -1 5 " and " }{TEXT 264 8 "Keywords" }{TEXT -1 34 " sections. Delete t his section on " }{TEXT 265 7 "Reports" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "Remember that your \+ worksheet should execute in the order that it has been written. In pa rticular, remember that the ditto refers to the result of the last exe cuted command - not the result of the command that physically precedes it. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 8 "Keywords" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "diff" 2 "diff" "" }{TEXT -1 4 ", \+ " }{HYPERLNK 17 "display" 2 "plots,display" "" }{TEXT -1 3 ", " } {HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 4 ", " }{HYPERLNK 17 " limit" 2 "limit" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "plot" 2 "plot" " " }{TEXT -1 4 ", " }{HYPERLNK 17 "plots" 2 "plots" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "plot,options" 2 "plot,options" "" }{TEXT -1 4 ", \+ " }{HYPERLNK 17 "restart" 2 "restart" "" }{TEXT -1 4 ", " } {HYPERLNK 17 "simplify" 2 "simplify" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "testeq" 2 "test eq" "" }{TEXT -1 3 ". " }{HYPERLNK 17 "unapply" 2 "unapply" "" }}} {SECT 0 {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 discussions that will h elp you do this homework. If they are in the same directory as this w orksheet, and if you have retained the filename under which they were \+ posted, then clicking on the hyperlink below will automatically open t hem. Use the Window menu to control the view when multiple files are o pened simultaneously. " }}{PARA 0 "" 0 "" {TEXT -1 4 " " } {HYPERLNK 17 "1.8epR4.mws" 1 "1.8epR4.mws" "" }{TEXT -1 81 " (Section s: Resistive Force Proportional to Velocity, Multicase Functions) \n \+ " }{HYPERLNK 17 "Tutor2R4.mws" 1 "Tutor2R4.mws" "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 40 "Exercise 1 Air Resistance - Linear Law" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 18 "Backgrou d Reading:" }{TEXT -1 7 " " }{HYPERLNK 17 "1.8epR4.mws" 1 "1.8ep R4.mws" "" }{TEXT -1 112 " (Sections: Resistive Force Proportional \+ to Velocity, Multicase Functions) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 95 " A Detailed Look at President Bush's Second Parachute Jump \+ " }}{PARA 0 "" 0 "" {TEXT -1 440 "As of this writing, former president George Bush has been the only American pres ident to have jumped from an airplane. He has done it three times. Dur ing World War II, the future president parachuted from his airplane wh en it was shot down. On March 25 1997, at the age of 72, the former pr esident made a recreational skydive/parachute jump. This problem conce rns that jump. (He made his third jump 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 president and his jumping gear was \+ given by " }{TEXT 365 2 " " }{XPPEDIT 366 1 "m=7.03*lb*s^2/ft" "/% \"mG**$\"$.(!\"#\"\"\"%#lbGF(%\"sG\"\"#%#ftG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 48 "and that gravitational acceleration is given by " }{TEXT 367 2 " " }{XPPEDIT 368 1 "g =32*ft/s" "/%\"gG*(\"#K\"\"\"%#ftGF&%\"s G!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "restart; m := 7.03; g := 32; # \+ Execute!" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Assume that the force of air re sistance was " }{TEXT 284 1 " " }{XPPEDIT 285 1 "-1.4*v(t)*lb*s/ft" ", $*,$\"#9!\"\"\"\"\"-%\"vG6#%\"tGF'%#lbGF'%\"sGF'%#ftGF&F&" }{TEXT -1 34 " during the skydive phase (where" }{TEXT 363 3 " " }{XPPEDIT 364 1 "v(t)" "-%\"vG6#%\"tG" }{TEXT -1 33 " is President Bush's veloc ity). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The differential equation of motion is therefore " }{TEXT 369 1 " " }{XPPEDIT 370 1 "m*diff(v(t),t) = - m*g- 1.4*v(t)" "/*&%\"mG\"\"\"-%%d iffG6$-%\"vG6#%\"tGF,F%,&*&F$F%%\"gGF%!\"\"*&$\"#9F0F%-F*6#F,F%F0" } {TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Enter this differential equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "? := m*diff( v(t),t) = ? ;" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 40 "The president jumped from a height of " }{TEXT 286 5 "1 2500" }{TEXT -1 57 " feet and did not open his parachute until he re ached " }{TEXT 287 4 "4500" }{TEXT -1 29 " feet. Compute the duratio n " }{TEXT 288 1 " " }{XPPEDIT 289 1 "tau" "I$tauG6\"" }{TEXT 290 1 " \+ " }{TEXT -1 17 "of this skydive. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dsolve( \{ ? , ? \}, v(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "v1 := unapply( ? , ? );" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "y1 := t -> ? + int(?, ? = ? .. ?);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "tau := solve( ? = ?, t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Obviously we wi ll use the positive solution!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tau := ?;" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 385 63 "What was the president's velocity when he opened his parachute?" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "? ;" }}}{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 Statue of Liberty at that consta nt speed then you crash into the ground in just under one second.)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 386 52 "Plot his velocity for the skydive phase of his jump." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot( ?, ? = ? .. tau);" }}}{PARA 0 "" 0 "" {TEXT -1 30 "\nAssign a name to \+ this plot:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "? := \":" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "As you c an see, the president pretty much achieved terminal velocity!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 387 52 "Plot his altitude for the skydive phase of his jump." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(?, ? = ? .. tau);" }}}{PARA 0 "" 0 "" {TEXT -1 30 "\nAssign a name to this plot:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "? := \":" }}}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "President Bush touched do wn on the ground " }{TEXT 291 3 "540" }{TEXT -1 113 " seconds after \+ he jumped from the plane. Continue to assume that air resistance prov ided 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\"k G6\"" }{TEXT -1 6 " . \n " }{TEXT 389 17 "Find the value of" }{TEXT -1 2 " " }{TEXT 296 1 " " }{XPPEDIT 297 1 "k" "I\"kG6\"" }{TEXT -1 120 " - it is different from the coefficient that was used in the skydive phase: that is the point of a parachute. \n\n\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "ode2 := m*diff(v(t),t) = -m*g - k*v (t); # Execute" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dsolve( \{ode2, ? = ?\}, v(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " v2 := unapply(?, ?);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y2 := t -> ? + int( ? , ? = ? .. ? \+ );" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k := fsolve( ? , k);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 388 82 "What was the presiden t's terminal velocity? With what velocity did he touch land?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "?; # Terminal Velocity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "?; #Velocity at touch down" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 390 66 "Plot the president's velocity for the pa rachute phase of the jump." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(v2(t), t = ? .. ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Assign a name t o this plot:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "? := \":" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT 391 66 "Plo t the president's altitude for the parachute phase of the jump." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(y2(t), t = ? .. ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "Assign a name to this plot:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "? := \":" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 392 29 "Combine the velocit y plots to" }{TEXT -1 2 " " }{HYPERLNK 17 "display" 2 "plots[display] " "" }{TEXT -1 3 " " }{TEXT 393 51 "a plot of his velocity for the e ntire 540 seconds. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 394 37 "Do the same for the height functions." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "?" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 259 "" 0 "" {TEXT -1 68 "Exercise 2 Linear Resistance - The Hill-Keller Theory of Sprinti ng" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 18 "Ba ckgroud Reading:" }{TEXT -1 7 " " }{HYPERLNK 17 "1.8epR4.mws" 1 "1.8epR4.mws" "" }{TEXT -1 55 " (Sections: Resistive Force Proporti onal to Velocity" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 285 "Since the 1920's, mathematical models have been develop ed to study running performance. These models rely on physical theory, empirical evidence, and statistical analysis. In 1973 J.B. Keller mod ified an existing mathematical theory of running to make it more appro priate 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 horizontal componen t of the propulsive force " }{TEXT 317 13 "per unit mass" }{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 m ass" }{TEXT -1 253 ". There are two components to the resistive force . There is an internal resistance to motion and there is also air res istance. We will neglect the air resistance in our analysis 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 35 " ). Obviously t his assumption on " }{TEXT 314 4 "p(t)" }{TEXT -1 123 " is not tena ble for races longer than a sprint. With the preceding assumptions, t he 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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 158 " In this exercise we will use the Hill-Keller \+ model to compare the performances of Ben Johnson and Carl Lewis at the 1987 World Championship meet in Rome." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "a) Maximum Velocity" }}{PARA 0 "" 0 "" {TEXT -1 78 "\nSol ve the velocity differential equation with appropriate initial conditi on:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(\{?,?\},?); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v := unapply(?,?);" }}} {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 " tends to infinity." }} {PARA 0 "" 0 "" {TEXT -1 126 "Although the model that has been develop ed is only applicable over a limited time range (and 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 the 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 the range of the mode l. " }}{PARA 0 "" 0 "" {TEXT -1 18 "The limit velocity" }{TEXT 331 2 " " }{TEXT 336 4 "vMax" }{TEXT -1 27 " is therefore called the " } {TEXT 332 16 "maximum velocity" }{TEXT -1 26 " of the runner. Comput e " }{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 16 "assume(tau > 0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vMax := limit(?, ?);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 "b) Displacemen t" }}{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 p roblem for " }{TEXT 348 4 "x(t)" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 18 "Show that " }{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#,$*&F&F,F/!\"\"F6F6F,F6F," }{TEXT -1 5 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{? , ?\}, x(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x := unapp ly( ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "For convenience, redefine:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "x := t -> vMax*(tau*exp(-1/t au*t)-tau+t); # Execute this line." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 "c) The 1987 World Championships" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 188 "At the 1987 World Championships in Rome , split times for the Men's 100 Meters competition show that the maxim um velocities of Ben Johnson (Canada) and Carl Lewis (United States) w ere both " }{TEXT 352 8 "11.8 m/s" }{TEXT -1 30 ". Johnson's winning t ime was " }{TEXT 353 6 "9.83 s" }{TEXT -1 35 " and Lewis's second pla ce time was " }{TEXT 354 6 "9.93 s" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "What were the approxima te values of " }{TEXT 356 1 "P" }{TEXT 355 1 " " }{TEXT -1 4 " ( " } {TEXT 380 2 "PJ" }{TEXT -1 8 " and " }{TEXT 382 2 "PL" }{TEXT -1 7 " ) and" }{TEXT 357 1 " " }{XPPEDIT 358 1 "tau" "I$tauG6\"" }{TEXT -1 5 " (" }{TEXT 383 4 "tauJ" }{TEXT -1 10 " and " }{TEXT 384 4 "tauL" }{TEXT -1 18 ") for these men?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "xJ := t -> ?;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "xL := t -> ?;" }}{PARA 11 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "tauJ := fsolve(xJ(?) = ?, tau); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "tauL := fsolve(? = ?, tau);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "PJ := ?;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " PL := ?;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 479 "According to the Hill-Keller mod el, can Johnson's victory be attributed to greater propulsive force, l esser resistance to motion, or to both factors? (One year later, the international governing body of track and field decided Johnson's vic tory could be attributed to the anabolic steroid stanozolol, for which he tested positive after winning a gold medal at the Seoul 1988 Olymp ic Games. Johnson did not receive the gold medal and his previous worl d records were disallowed.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {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 14 "xJ := t -> ?;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "xL := t -> ?;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot([?, ?], t = 0 ..?, color = [?, ?]); # Velocity \+ plot of the runners" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot([?, ?], t = 0 ..?, color = [?, ?]); \+ # Distance plot of the runners" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 37 " e) What was Johnson's Winning Margin?" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "? - ?;" }}}{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 0 {PARA 260 "" 0 "" {TEXT -1 41 "Exercise 3 Air Resistance - 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: Resistive 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 800 kg bomb was dropped from a Nakaji ma BN52 Kate bomber flying at an altitude of 3170 meters. The bomb st ruck the battleship USS Arizona, igniting its black powder magazine w hich in turn set off a series of catastrophic explosions. The ship san k in nine minutes with a death toll of 1177. An analysis of the attac k that was published in 1997 asserted that the flight of the bomb las ted 26 seconds. Let us accept that 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 0 {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 blo ck 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;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqn := dsolve( ode, v(t)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "with(student): \neqn2 \+ := isolate(eqn, v(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "_ C1 = solve(subs(t = 0, rhs(eqn2)), _C1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(\", eqn2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "v := unapply(rhs(\"), t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 "This gives you the downward ve locity function. Next comes the height function. 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 33 "y := t -> ? + Int(?, ? = ? .. t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Now make the definit ions:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "m := 800: g := \+ 9.81:" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }}{PARA 0 "" 0 "" {TEXT -1 72 "The height that the bomb reaches at 26 seconds depends on the v alue 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 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 376 1 "K" }{TEXT -1 14 " for which " }{TEXT 377 8 "f(K) > 0" }{TEXT -1 16 " . (Experiment!)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 102 "Then execute the following function. (Re ad 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 `Copyright: Brian E. Blank, 1996-200 0`;\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 fou r arguments`);\nelif not type(f, \{name,procedure\} ) then \n ERR OR(`bisection expects its first argument to be a name or procedure`); \nelif not type(a, realcons) or not type(args[3], realcons) then\n \+ ERROR(`bisection expects its second and third arguments to be real c onstants`);\nelif not type(f(a),realcons) or not type(f(b),realcons) t hen\n ERROR(`bisection expects its first argument to return a rea l number when evaluated at its second and third arguments.`);\nelif ev alf((args[1])(a)*(args[1])(b)) > 0 then\n ERROR(`No root found`); \nelif args[1](a) = 0 then RETURN(a);\nelif args[1](b) = 0 then RETURN (b);\nelse\n if nargs = 4 then epsilon := args[4];\n else ep silon := 1.0e-9;\n fi;\n while abs(a-b) > epsilon do\n if eva lf(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 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 64 "Fill in and execut e the next line to find the actual value of " }{TEXT 379 1 "K" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "K := bisection( ? , ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 58 "b) What was t he theoretical terminal velocity of the bomb." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 76 "c) What was the actual velocity of the bomb when it struck the USS Arizona. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 70 "d) Plot the heig ht and velocity of the bomb during its 26 second fall." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 97 "e) What would th e time of the fall have been had there been no air drag? The velocity \+ on impact?\n" }}{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 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 Readin g:" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 " Tutor2R4.mws" 1 "Tutor2R4.mws" "" }{TEXT -1 44 " (The section on sol ving identities.) \n " }{HYPERLNK 17 "wronskian" 2 "linalg[wronskian ]" "" }{TEXT -1 14 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Linear Dependence" }}{PARA 0 "" 0 "" {TEXT -1 56 "Find a depend ence relations for the elements of the set " }{TEXT 298 1 " " } {XPPEDIT 299 1 "\{2, cos(x)^2,cos(2*x)\}" "<%\"\"#*$-%$cosG6#%\"xGF#-F &6#*&F#\"\"\"F(F," }{TEXT -1 35 ". In other words, find constants " }{TEXT 303 1 "A" }{TEXT -1 2 ", " }{TEXT 304 2 " B" }{TEXT -1 7 " , an d " }{TEXT 305 1 "C" }{TEXT -1 29 ", not all zero, such that " } {TEXT 301 2 " " }{XPPEDIT 300 1 "2*A+B*cos(x)^2+C*cos(2*x)=0" "/,(*& \"\"#\"\"\"%\"AGF&F&*&%\"BGF&*$-%$cosG6#%\"xGF%F&F&*&%\"CGF&-F,6#*&F%F &F.F&F&F&\"\"!" }{TEXT -1 12 " for all " }{TEXT 302 1 "x" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "? := identity( ? , ? );" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 " Linear Independence" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 33 "Use the Wronskian to prove that " }{XPPMATH 306 "6#<&* &%\"xG\"\"\"-%$expG6#*&%\"kGF&F%F&F&*&F%\"\"#F'F&*&F%\"\"$F'F&F'" } {TEXT -1 28 " are independent functions." }}{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." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v := vector( ? ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "wronskian( ? , ? );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(?);" }}}{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 0 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and \+ Author Information" }}{EXCHG {PARA 261 "" 0 "" {TEXT -1 45 "02F00R4.mw s A MapleV Release 4 worksheet." }}{PARA 262 "" 0 "" {TEXT -1 0 " " }}{PARA 263 "" 0 "" {TEXT -1 44 "Author: Brian E. Blank (28 Septem ber 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 "including print, disk, and electronic transfer, w ithout" }}{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 information, please contact the author:" }}{PARA 266 "" 0 "" {TEXT -1 4 " " }}{PARA 266 "" 0 "" {TEXT -1 32 " Depart ment 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 " Telephone: (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 "10 1 2 0 " 28 }{VIEWOPTS 1 1 0 3 4 1802 }