1
CHAPTERj1
1.1 Wejwilljillustratejtwojdifferentjmethodsjforjsolvingjthisjproblem:j(1)jseparationjofjvariables,jandj(2)jL
aplacejtransform.
dvj cj
=jgj−j vj
dt m
Separationjofjvariables:jSeparationjofjvariablesjgives
1
j
j c
jv
dvj=j dt
j
jgj−j
m
Thejintegralsjcanjbejevaluatedjas
j cj
lnj gj−j vj
mj
−
j
=jtj+jCj
cj/jm
wherejCj=jajconstantjofjintegration,jwhichjcanjbejevaluatedjbyjapplyingjthejinitialjconditionjtojyield
j cj
lnj gj−j v(0)j
m
Cj = − j
cj/jm
whichjcanjbejsubstitutedjbackjintojthejsolution
j cj j cj
lnj gj−j vj lnj gj−j v(0)j
mj m
− j
=jtj−
cj/jm cj/jm
Thisjresultjcanjbejrearrangedjalgebraicallyjtojsolvejforjv,
vj=jv(0)e−(c/jm)tj +j
jmgj
c
(
1−je−(cj/jm)tj )
wherejthejfirstjpartjisjthejgeneraljsolutionjandjthejsecondjpartjisjthejparticularjsolutionjforjthejconstantjforcin
gjfunctionjduejtojgravity.jForjthejcasejwhere,jv(0)j=j0,jthejsolutionjreducesjtojEq.j(1.10)
vj = j
jmgj
c
(1− e j
−(c/m)tj
)
Laplacejtransformjsolution:jAnjalternativejsolutionjisjprovidedjbyjapplyingjLaplacejtransformjtojthejdiffe
rentialjequationjtojgive
g j cj
sVj(s)j−jv(0)j=j −j Vj(s)
s m
PROPRIETARYjMATERIAL.j ©jThejMcGraw-
HilljCompanies,jInc.j Alljrightsjreserved.j NojpartjofjthisjManualjmayjbejdisplayed,jreproducedjorjdistributedjinjanyjfor
mjorjbyjanyjmeans,jwithoutjthejpriorjwrittenjpermissionjofjthejpublisher,jorjusedjbeyondjthejlimitedjdistributionjtojteach
ersjandjeducatorsjpermittedjbyjMcGraw-
Hilljforjtheirjindividualjcoursejpreparation.j IfjyoujarejajstudentjusingjthisjManual,jyoujarejusingjitjwithoutjpermission.
, 2
Solvejalgebraicallyjforjthejtransformedjvelocity
PROPRIETARYjMATERIAL.j ©jThejMcGraw-
HilljCompanies,jInc.j Alljrightsjreserved.j NojpartjofjthisjManualjmayjbejdisplayed,jreproducedjorjdistributedjinjanyjfor
mjorjbyjanyjmeans,jwithoutjthejpriorjwrittenjpermissionjofjthejpublisher,jorjusedjbeyondjthejlimitedjdistributionjtojteach
ersjandjeducatorsjpermittedjbyjMcGraw-
Hilljforjtheirjindividualjcoursejpreparation.j IfjyoujarejajstudentjusingjthisjManual,jyoujarejusingjitjwithoutjpermission.
, 3
Vj(s)j = v(0) j
+
g (1)
sj+jcj/jm s(sj+jcj/jm)
Thejsecondjtermjonjthejrightjofjthejequaljsignjcanjbejexpandedjwithjpartialjfractions
g Aj B A(sj+jcj/jm)j+jBs
=j + =j (2)
s(sj+jcj/jm) s sj+jcj/jm s(sj+jcj/jm)
Byjequatingjlikejtermsjinjthejnumerator,jthejfollowingjmustjhold
c
g j = j Aj 0j=j Asj+jBs
m
ThejfirstjequationjcanjbejsolvedjforjAj=jmg/c.jAccordingjtojthejsecondjequation,jBj=j–A,jsojBj=j–
mg/c.jSubstitutingjthesejbackjintoj(2)jgives
g j =j
mgj/jcj j mgj/jc
−j
s(sj+jcj/jm) s sj+jcj/jm
ThisjcanjbejsubstitutedjintojEq.j1jtojgive
v(0) mgj/jcj mgj/jc
Vj(s)j = +j −j
sj+jcj/jm s sj+jcj/jm
TakingjinversejLaplacejtransformsjyields
mgj mgj −(cj/jm)t
v(t)j =jv(0)e−(c/jm)tj +j +j e
c c
orjcollectingjterms
v(t)j =jv(0)e−(c/m)tj +j
jmgj
c
(
1−je−(c/jm)tj )
1.2 Atjtj=j8js,jthejanalyticaljsolutionjisj41.137j(Examplej1.1).jThejrelativejerrorjcanjbejcalculatedjwith
analyticalj−jnumerical
absolutejrelativejerror = 100%
analytical
Thejnumericaljresultsjare:
absolute
j j jstep
j j j j j j j vj(8)
j j j j relativej error
2 44.8700 9.074%
1 42.8931 4.268%
0.5 41.9901 2.073%
Thejerrorjversusjstepjsizejcanjthenjbejplottedjas
PROPRIETARYjMATERIAL.j ©jThejMcGraw-
HilljCompanies,jInc.j Alljrightsjreserved.j NojpartjofjthisjManualjmayjbejdisplayed,jreproducedjorjdistributedjinjanyjfor
mjorjbyjanyjmeans,jwithoutjthejpriorjwrittenjpermissionjofjthejpublisher,jorjusedjbeyondjthejlimitedjdistributionjtojteach
ersjandjeducatorsjpermittedjbyjMcGraw-
Hilljforjtheirjindividualjcoursejpreparation.j IfjyoujarejajstudentjusingjthisjManual,jyoujarejusingjitjwithoutjpermission.
, 4
10%
8%
6%
4%
2%
0%
0 0.5 1 1.5 2 2.5
Thus,jhalvingjthejstepjsizejapproximatelyjhalvesjthejerror.
1.3 (a)jYoujarejgivenjthejfollowingjdifferentialjequationjwithjthejinitialjcondition,jv(tj=j0)j=j0,
dvj cj'j
=jgj−j v2j
dt m
Multiplyjbothjsidesjbyjm/c′jgives
mj dvj mj
=j gj−jv2
cj'j dt cj'
Definej aj= mgj/jcj'
mj dvj
=ja2j −jv2
cj'j dt
Integratejbyjseparationjofjvariables,
dv cj'
a j −jv
2 2
=
m
dt
Ajtablejofjintegralsjcanjbejconsultedjtojfindjthat
dxj a 1j xj
=j tanh−1j
j−jx
2 2 a a
Therefore,jthejintegrationjyields
1j vj cj'j
tanh−1j =j tj+jCja
a m
Ifjvj=j0jatjtj=j0,jthenjbecausejtanh–1(0)j=j0,jthejconstantjofjintegrationjCj=j0jandjthejsolutionjis
1j vj cj'j
tanh−1j =j tj
a a m
Thisjresultjcanjthenjbejrearrangedjtojyield
PROPRIETARYjMATERIAL.j ©jThejMcGraw-
HilljCompanies,jInc.j Alljrightsjreserved.j NojpartjofjthisjManualjmayjbejdisplayed,jreproducedjorjdistributedjinjanyjfor
mjorjbyjanyjmeans,jwithoutjthejpriorjwrittenjpermissionjofjthejpublisher,jorjusedjbeyondjthejlimitedjdistributionjtojteach
ersjandjeducatorsjpermittedjbyjMcGraw-
Hilljforjtheirjindividualjcoursejpreparation.j IfjyoujarejajstudentjusingjthisjManual,jyoujarejusingjitjwithoutjpermission.
CHAPTERj1
1.1 Wejwilljillustratejtwojdifferentjmethodsjforjsolvingjthisjproblem:j(1)jseparationjofjvariables,jandj(2)jL
aplacejtransform.
dvj cj
=jgj−j vj
dt m
Separationjofjvariables:jSeparationjofjvariablesjgives
1
j
j c
jv
dvj=j dt
j
jgj−j
m
Thejintegralsjcanjbejevaluatedjas
j cj
lnj gj−j vj
mj
−
j
=jtj+jCj
cj/jm
wherejCj=jajconstantjofjintegration,jwhichjcanjbejevaluatedjbyjapplyingjthejinitialjconditionjtojyield
j cj
lnj gj−j v(0)j
m
Cj = − j
cj/jm
whichjcanjbejsubstitutedjbackjintojthejsolution
j cj j cj
lnj gj−j vj lnj gj−j v(0)j
mj m
− j
=jtj−
cj/jm cj/jm
Thisjresultjcanjbejrearrangedjalgebraicallyjtojsolvejforjv,
vj=jv(0)e−(c/jm)tj +j
jmgj
c
(
1−je−(cj/jm)tj )
wherejthejfirstjpartjisjthejgeneraljsolutionjandjthejsecondjpartjisjthejparticularjsolutionjforjthejconstantjforcin
gjfunctionjduejtojgravity.jForjthejcasejwhere,jv(0)j=j0,jthejsolutionjreducesjtojEq.j(1.10)
vj = j
jmgj
c
(1− e j
−(c/m)tj
)
Laplacejtransformjsolution:jAnjalternativejsolutionjisjprovidedjbyjapplyingjLaplacejtransformjtojthejdiffe
rentialjequationjtojgive
g j cj
sVj(s)j−jv(0)j=j −j Vj(s)
s m
PROPRIETARYjMATERIAL.j ©jThejMcGraw-
HilljCompanies,jInc.j Alljrightsjreserved.j NojpartjofjthisjManualjmayjbejdisplayed,jreproducedjorjdistributedjinjanyjfor
mjorjbyjanyjmeans,jwithoutjthejpriorjwrittenjpermissionjofjthejpublisher,jorjusedjbeyondjthejlimitedjdistributionjtojteach
ersjandjeducatorsjpermittedjbyjMcGraw-
Hilljforjtheirjindividualjcoursejpreparation.j IfjyoujarejajstudentjusingjthisjManual,jyoujarejusingjitjwithoutjpermission.
, 2
Solvejalgebraicallyjforjthejtransformedjvelocity
PROPRIETARYjMATERIAL.j ©jThejMcGraw-
HilljCompanies,jInc.j Alljrightsjreserved.j NojpartjofjthisjManualjmayjbejdisplayed,jreproducedjorjdistributedjinjanyjfor
mjorjbyjanyjmeans,jwithoutjthejpriorjwrittenjpermissionjofjthejpublisher,jorjusedjbeyondjthejlimitedjdistributionjtojteach
ersjandjeducatorsjpermittedjbyjMcGraw-
Hilljforjtheirjindividualjcoursejpreparation.j IfjyoujarejajstudentjusingjthisjManual,jyoujarejusingjitjwithoutjpermission.
, 3
Vj(s)j = v(0) j
+
g (1)
sj+jcj/jm s(sj+jcj/jm)
Thejsecondjtermjonjthejrightjofjthejequaljsignjcanjbejexpandedjwithjpartialjfractions
g Aj B A(sj+jcj/jm)j+jBs
=j + =j (2)
s(sj+jcj/jm) s sj+jcj/jm s(sj+jcj/jm)
Byjequatingjlikejtermsjinjthejnumerator,jthejfollowingjmustjhold
c
g j = j Aj 0j=j Asj+jBs
m
ThejfirstjequationjcanjbejsolvedjforjAj=jmg/c.jAccordingjtojthejsecondjequation,jBj=j–A,jsojBj=j–
mg/c.jSubstitutingjthesejbackjintoj(2)jgives
g j =j
mgj/jcj j mgj/jc
−j
s(sj+jcj/jm) s sj+jcj/jm
ThisjcanjbejsubstitutedjintojEq.j1jtojgive
v(0) mgj/jcj mgj/jc
Vj(s)j = +j −j
sj+jcj/jm s sj+jcj/jm
TakingjinversejLaplacejtransformsjyields
mgj mgj −(cj/jm)t
v(t)j =jv(0)e−(c/jm)tj +j +j e
c c
orjcollectingjterms
v(t)j =jv(0)e−(c/m)tj +j
jmgj
c
(
1−je−(c/jm)tj )
1.2 Atjtj=j8js,jthejanalyticaljsolutionjisj41.137j(Examplej1.1).jThejrelativejerrorjcanjbejcalculatedjwith
analyticalj−jnumerical
absolutejrelativejerror = 100%
analytical
Thejnumericaljresultsjare:
absolute
j j jstep
j j j j j j j vj(8)
j j j j relativej error
2 44.8700 9.074%
1 42.8931 4.268%
0.5 41.9901 2.073%
Thejerrorjversusjstepjsizejcanjthenjbejplottedjas
PROPRIETARYjMATERIAL.j ©jThejMcGraw-
HilljCompanies,jInc.j Alljrightsjreserved.j NojpartjofjthisjManualjmayjbejdisplayed,jreproducedjorjdistributedjinjanyjfor
mjorjbyjanyjmeans,jwithoutjthejpriorjwrittenjpermissionjofjthejpublisher,jorjusedjbeyondjthejlimitedjdistributionjtojteach
ersjandjeducatorsjpermittedjbyjMcGraw-
Hilljforjtheirjindividualjcoursejpreparation.j IfjyoujarejajstudentjusingjthisjManual,jyoujarejusingjitjwithoutjpermission.
, 4
10%
8%
6%
4%
2%
0%
0 0.5 1 1.5 2 2.5
Thus,jhalvingjthejstepjsizejapproximatelyjhalvesjthejerror.
1.3 (a)jYoujarejgivenjthejfollowingjdifferentialjequationjwithjthejinitialjcondition,jv(tj=j0)j=j0,
dvj cj'j
=jgj−j v2j
dt m
Multiplyjbothjsidesjbyjm/c′jgives
mj dvj mj
=j gj−jv2
cj'j dt cj'
Definej aj= mgj/jcj'
mj dvj
=ja2j −jv2
cj'j dt
Integratejbyjseparationjofjvariables,
dv cj'
a j −jv
2 2
=
m
dt
Ajtablejofjintegralsjcanjbejconsultedjtojfindjthat
dxj a 1j xj
=j tanh−1j
j−jx
2 2 a a
Therefore,jthejintegrationjyields
1j vj cj'j
tanh−1j =j tj+jCja
a m
Ifjvj=j0jatjtj=j0,jthenjbecausejtanh–1(0)j=j0,jthejconstantjofjintegrationjCj=j0jandjthejsolutionjis
1j vj cj'j
tanh−1j =j tj
a a m
Thisjresultjcanjthenjbejrearrangedjtojyield
PROPRIETARYjMATERIAL.j ©jThejMcGraw-
HilljCompanies,jInc.j Alljrightsjreserved.j NojpartjofjthisjManualjmayjbejdisplayed,jreproducedjorjdistributedjinjanyjfor
mjorjbyjanyjmeans,jwithoutjthejpriorjwrittenjpermissionjofjthejpublisher,jorjusedjbeyondjthejlimitedjdistributionjtojteach
ersjandjeducatorsjpermittedjbyjMcGraw-
Hilljforjtheirjindividualjcoursejpreparation.j IfjyoujarejajstudentjusingjthisjManual,jyoujarejusingjitjwithoutjpermission.
