Numerical Methods for Engineers (Indian Edition) 8th Edition By Steven Chapra, Raymond Canale (Solutions Manual)

1


CHAPTER 1 b



1.1 Webwillbillustratebtwobdifferentbmethodsbforbsolvingbthisbproblem:b(1)bseparationbofbvariables,bandb(2)
bLaplacebtransform.



dvb cb
=bgb−b vb
dt m

Separationbofbvariables:bSeparationbofbvariablesbgives



1
b c
b



bv

dvb=b dt
b
bgb−b


m

Thebintegralsbcanbbebevaluatedbas
b cb 
lnb gb−b vb
 mb 

−
b
=btb+bC
bcb/bm



wherebCb=babconstantbofbintegration,bwhichbcanbbebevaluatedbbybapplyingbthebinitialbconditionbtobyield
b cb 
lnb gb−b v(0)b
 m 
Cb = −  
b cb/bm

whichbcanbbebsubstitutedbbackbintobthebsolution
b cb  b cb 
lnb gb−b vb lnb gb−b v(0)b
 mb   m 
−   b
=btb−  
cb/bm cb/bm

Thisbresultbcanbbebrearrangedbalgebraicallybtobsolvebforbv,


vb=bv(0)e−(c/bm)tb +b
mgb
b

c
(
1−be−(cb/bm)tb )
wherebthebfirstbpartbisbthebgeneralbsolutionbandbthebsecondbpartbisbthebparticularbsolutionbforbthebconstantbfor
cingbfunctionbduebtobgravity.bForbthebcasebwhere,bv(0)b=b0,bthebsolutionbreducesbtobEq.b(1.10)


vb = b
mgb
b

c
(
1−be−(c/m)tb )
Laplacebtransformbsolution:bAnbalternativebsolutionbisbprovidedbbybapplyingbLaplacebtransformbtobthebdifferent
ialbequationbtobgive

g b cb
sVb(s)b−bv(0)b=b −b Vb(s)
s m

PROPRIETARYbMATERIAL.b ©bThebMcGraw-
HillbCompanies,bInc.b Allbrightsbreserved.b NobpartbofbthisbManualbmaybbebdisplayed,breproducedborbdistributedbinbanybf
ormborbbybanybmeans,bwithoutbthebpriorbwrittenbpermissionbofbthebpublisher,borbusedbbeyondbtheblimitedbdistributionbtob
teachersbandbeducatorsbpermittedbbybMcGraw-
Hillbforbtheirbindividualbcoursebpreparation.b IfbyoubarebabstudentbusingbthisbManual,byoubarebusingbitbwithoutbpermissio
n.

, 2


Solvebalgebraicallybforbthebtransformedbvelocity




PROPRIETARYbMATERIAL.b ©bThebMcGraw-
HillbCompanies,bInc.b Allbrightsbreserved.b NobpartbofbthisbManualbmaybbebdisplayed,breproducedborbdistributedbinbanybf
ormborbbybanybmeans,bwithoutbthebpriorbwrittenbpermissionbofbthebpublisher,borbusedbbeyondbtheblimitedbdistributionbtob
teachersbandbeducatorsbpermittedbbybMcGraw-
Hillbforbtheirbindividualbcoursebpreparation.b IfbyoubarebabstudentbusingbthisbManual,byoubarebusingbitbwithoutbpermissio
n.

, 3




Vb(s)b= v(0)
+
g (1)
sb+bcb/bm s(sb+bcb/bm)

Thebsecondbtermbonbthebrightbofbthebequalbsignbcanbbebexpandedbwithbpartialbfractions

g Ab B A(sb+bcb/bm)b+bBs
=b + =b (2)
s(sb+bcb/bm) s sb+bcb/bm s(sb+bcb/bm)

Bybequatingblikebtermsbinbthebnumerator,bthebfollowingbmustbhold

c
g b = b Ab 0b=b Asb+bBs
m

ThebfirstbequationbcanbbebsolvedbforbAb=bmg/c.bAccordingbtobthebsecondbequation,bBb=b–A,bsobBb=b–
mg/c.bSubstitutingbthesebbackbintob(2)bgives

g b=b
mgb/bcb b mgb/bc
−b
s(sb+bcb/bm) s sb+bcb/bm

ThisbcanbbebsubstitutedbintobEq.b1btobgive

v(0) mgb/bcb mgb/bc
Vb(s)b= +b −b
sb+bcb/bm s sb+bcb/bm

TakingbinversebLaplacebtransformsbyields

mgb mgb −(cb/bm)t
v(t)b=bv(0)e−(c/bm)tb +b +b e
c c

orbcollectingbterms


v(t)b=bv(0)e−(c/m)tb +b
mgb
b

c
(
1−be−(c/bm)tb )
1.2 Atbtb=b8bs,bthebanalyticalbsolutionbisb41.137b(Exampleb1.1).bThebrelativeberrorbcanbbebcalculatedbwith

analyticalb−bnumerical
absolutebrelativeberror = 100%
analytical

Thebnumericalbresultsbare:

absolute
step v(8)
b b b b b b b b b b b b b b b relativeb error
2 44.8700 9.074%
1 42.8931 4.268%
0.5 41.9901 2.073%

Theberrorbversusbstepbsizebcanbthenbbebplottedbas



PROPRIETARYbMATERIAL.b ©bThebMcGraw-
HillbCompanies,bInc.b Allbrightsbreserved.b NobpartbofbthisbManualbmaybbebdisplayed,breproducedborbdistributedbinbanybf
ormborbbybanybmeans,bwithoutbthebpriorbwrittenbpermissionbofbthebpublisher,borbusedbbeyondbtheblimitedbdistributionbtob
teachersbandbeducatorsbpermittedbbybMcGraw-
Hillbforbtheirbindividualbcoursebpreparation.b IfbyoubarebabstudentbusingbthisbManual,byoubarebusingbitbwithoutbpermissio
n.