{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 2 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 301 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 305 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 306 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 307 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 308 "" 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 310 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 311 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 312 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 313 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 314 "" 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 315 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Hea ding 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 11 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 276 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 278 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 279 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 280 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 281 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 282 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 283 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 284 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 285 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 286 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 287 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 288 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 289 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 258 "" 0 "" {TEXT -1 33 "APPLICATION OF INTEGRATION - Wo rk" }}{PARA 4 "" 0 "" {TEXT -1 19 "Calculus II Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 256 11 "Objectives: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 77 "To apply integration \+ to problems involving work against the force of gravity." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 36 "To use Maple to p erform integration." }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 259 15 "Solved Example: " }}{PARA 289 "" 0 "" {TEXT -1 58 "(a) Find the work required to move \+ a rocket ship of mass " }{TEXT 280 1 "m" }{TEXT 306 1 " " }{TEXT -1 10 " from the " }}{PARA 272 "" 0 "" {TEXT -1 40 " surface of the \+ earth to the moon. " }}{PARA 259 "" 0 "" {TEXT 260 0 "" }}{PARA 260 " " 0 "" {TEXT -1 66 "(b) What initial velocity must be given to a cann onball of mass " }{TEXT 287 1 "m" }{TEXT 307 1 " " }{TEXT -1 7 " fire d " }}{PARA 273 "" 0 "" {TEXT -1 66 " from the surface of the ea rth if it is to land on the moon?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 10 " Solution:" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 261 18 "Needed information" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 0 "" }{TEXT 267 47 "G = 6.67*10^(-11), the gravitational constant; " }}{PARA 261 "" 0 "" {TEXT -1 44 "M = 5.98*1 0^24, the mass of the earth in kg;" }}{PARA 262 "" 0 "" {TEXT -1 50 "P = 6.38*10^6, the radius of the earth in meters;" }}{PARA 263 "" 0 " " {TEXT -1 43 "L = 7.35*10^22, the mass of the moon in kg;" }}{PARA 264 "" 0 "" {TEXT -1 61 "Q = 1.74*10^6, the radius of the moon in met ers; " }}{PARA 265 "" 0 "" {TEXT -1 89 "C = 3.84*10^8, the \+ distance from the center of the earth to the center of moon in meters. " }}{PARA 274 "" 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 77 "A n object moving from the surface of the earth to the surface of the mo on is " }}{PARA 271 "" 0 "" {TEXT -1 84 "under the influence of the gr avitational forces of both the earth and the moon. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 291 4 "If " }{TEXT 295 1 " v" }{TEXT 296 31 " is the initial velocity, the " }{TEXT -1 14 "kinet ic energy" }{TEXT 309 11 ", given by " }{TEXT -1 3 "k =" }{TEXT 312 1 " " }{XPPEDIT 18 0 "mv^2/2;" "6#*&%#mvG\"\"#F%!\"\"" }{TEXT 292 19 " , is equal to the " }}{PARA 0 "" 0 "" {TEXT 294 24 "force needed to lau nch a" }{TEXT -1 1 " " }{TEXT 293 36 "rocket ship or cannon ball of ma ss " }{TEXT -1 0 "" }{TEXT 298 1 "m" }{TEXT 297 1 "." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 262 52 " a) Find the gravitational force on the rocket ship " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 279 62 " and evaluate the integral for the work done by the force." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 0 "" }{TEXT -1 1 " " }{TEXT 264 4 " If " }{TEXT 308 2 "x " }{TEXT 310 66 " is the distance from the cente r of the earth to the rocket ship, " }}{PARA 0 "" 0 "" {TEXT 265 55 "t hen the force pulling toward the Earth is given by GM" }{TEXT 281 1 " m" }{TEXT 282 9 "/x^2. " }}{PARA 0 "" 0 "" {TEXT 266 52 "The force \+ pulling it toward the moon is given by GL" }{TEXT 283 1 "m" }{TEXT 284 12 "/(C - x)^2 ." }}{PARA 266 "" 0 "" {TEXT -1 71 "The net force o n the rocket is the difference between these two forces:" }}{PARA 267 "" 0 "" {TEXT -1 13 " force = GM" }{TEXT 285 1 "m" }{TEXT -1 11 "/x^ 2 - GL" }{TEXT 286 1 "m" }{TEXT -1 10 "/(C - x)^2" }}{PARA 268 "" 0 "" {TEXT -1 80 "Work done will be found by integrating the force from \+ the surface of the earth, " }{TEXT 311 1 "x" }{TEXT -1 9 " = P, to " } }{PARA 269 "" 0 "" {TEXT -1 25 "the surface of the moon, " }{TEXT 313 1 "x" }{TEXT -1 44 " = C - Q. This work is measured in joules. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "G := 6.67*10^(-11); M := 5. 98*10^24; P := 6.38*10^6; L := 7.35*10^22;Q := 1.74*10^6; C:=3.84*10 ^8;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG$\"++++qm!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG$\"++++!)f\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG$\"*+++Q'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"LG$\"++++]t\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG$\"*+++u \"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG$\"++++SQ!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "force := x -> G*M*m/x^2 - \+ G*L*m/(C - x)^2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&forceGf*6#%\" xG6\"6$%)operatorG%&arrowGF(,&**%\"GG\"\"\"%\"MGF/%\"mGF/9$!\"#F/**F.F /%\"LGF/F1F/,&%\"CGF/F2!\"\"F3F8F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "int(force(x), x = P.. (C - Q));\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&$\"+lA-ne!\"#\"\"\"%\"mGF(F(" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 269 79 "b) Set up an equation to find the force and evaluate the integral \+ for the work." }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 277 28 "Let the initial velocity be " }{TEXT 278 1 "v" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 270 25 " The cannonball of mass " }{TEXT 288 1 "m" }{TEXT 289 48 " m ust have sufficient kinetic energy, given by " }{XPPEDIT 18 0 "1/2;" " 6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 272 2 " " }{TEXT 290 1 "m" }{TEXT 275 1 " " }{XPPEDIT 18 0 "v^2;" "6#*$%\"vG\"\"#" }{TEXT 273 11 " , \+ " }}{PARA 0 "" 0 "" {TEXT 271 94 " to pass the point where the grav itational attraction of the moon and the Earth are equal. " }}{PARA 0 "" 0 "" {TEXT 274 61 " From that point it will fall to the surface of the moon. " }}{PARA 0 "" 0 "" {TEXT 276 86 " To find that point , we need to solve the equation Force = 0 for the variable Force." } {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve(forc e(x) = 0, x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+:D!)=V!\"\"$\"+ tnwcMF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "int (force(x), x = P..3.456766773*10^8);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$\"+ tq$\\7'!\"#\"\"\"%\"mGF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 299 26 "Find the initial veloc ity " }}{PARA 0 "" 0 "" {TEXT 300 73 "We get the required velocity by \+ setting the kinetic energy equal to this " }}{PARA 0 "" 0 "" {TEXT 301 23 "work and solving for v:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "solve(m*v^2/2 = .6124937076*10^8*m, v);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+'\\\"z16!\"&$!+'\\\"z16F%" }}}{PARA 275 "" 0 "" {TEXT -1 47 "The initial velocity required = 11067.91496 m/s" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 62 "________ ______________________________________________________" }}{PARA 276 " " 0 "" {TEXT -1 11 " ASSIGNMENT" }{TEXT 26 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 302 8 "P roblem:" }}{PARA 0 "" 0 "" {TEXT 304 80 "Taking into account the gravi tational pull of the sun, how much work is required" }}{PARA 0 "" 0 " " {TEXT 305 31 "to send a rocket ship of mass " }{TEXT 314 1 "m" } {TEXT 315 58 " from the surface of the earth to the surface of the su n?" }{TEXT -1 3 " " }}{PARA 277 "" 0 "" {TEXT -1 46 "Additional Data : Mass of sun = 1.97*" }{XPPEDIT 18 0 "10^30;" "6#*$\"#5\"#I " }{TEXT -1 4 " kg" }}{PARA 278 "" 0 "" {TEXT -1 58 " \+ Radius of sun = 6.95*" }{XPPEDIT 18 0 "10^8;" "6# *$\"#5\"\")" }{TEXT -1 3 " m" }}{PARA 279 "" 0 "" {TEXT -1 72 " \+ Distance from earth to sun = 1.49*" } {XPPEDIT 18 0 "10^11;" "6#*$\"#5\"#6" }{TEXT -1 3 " m" }}{PARA 284 " " 0 "" {TEXT -1 0 "" }}{PARA 280 "" 0 "" {TEXT -1 42 "Explain what the sign of the answer means." }}{PARA 281 "" 0 "" {TEXT -1 91 "Find the \+ point between the earth and the sun where the pull of the earth's grav ity matches " }}{PARA 287 "" 0 "" {TEXT -1 18 "that of the earth." }} {PARA 282 "" 0 "" {TEXT -1 92 "Calculate the initial velocity of a can nonball fired from the surface of the earth if it is " }}{PARA 288 "" 0 "" {TEXT -1 34 "to land on the surface of the sun." }}{PARA 283 "" 0 "" {TEXT -1 42 "Calculate the escape velocity for the sun." }}{PARA 0 "" 0 "" {TEXT 303 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 61 "_____________ ________________________________________________" }}{PARA 286 "" 0 "" {TEXT -1 1 " " }}{PARA 285 "" 0 "" {TEXT -1 96 "MSIP Grant #P120A80089 -98: \"Three Urban Calculus Reform Programs: Adopting the Best\" 1998 -2001 " }}}{MARK "7 5 10 4 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }