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1、10ConservationofMomentum10-1Newton'sThirdLawOnthebasisofNewton'ssecondlawofmotion,whichgivestherelation10-1Newton'sThirdLawbetweentheaccelerationofanybodyandtheforceactingonit,anyproblemin10-2Conservationofmomentummechanicscanbesolvedinpnnciple.Forexample,todeterminethemotionofafewparticle
2、s,onecanusethenumericalmethoddevelopedintheprecedingchapter.10-3Momentumisconserved!ButtherearegoodreasonstomakeafurtherstudyofNewton'slaws.First,there10--4Momentumandenergyarequitesimplecasesofmotionwhichcanbeanalyzednotonlybynumericalmethods,butalsobydirectmathematicalanalysis.Forexample
3、,althoughwe10-5Relativisticmomentum2knowthattheaccelerationofafallingbodyis32ft/sec,andfromthisfactcouldcalculatethemotionbynumericalmethods,itismucheasierandmoresatisfactory2toanalyzethemotionandfindthegeneralsolution,s=s0+v0t+16t•Inthesameway,althoughwecanworkoutthepositionsofaharmonicos
4、cillatorbynumericalmethods,itisalsopossibletoshowanalyticallythatthegeneralsolutionisasimplecosinefunctionoft,andsoitisunnecessarytogotoallthatarithmeticaltroublewhenthereisasimpleandmoreaccuratewaytogettheresult.Inthesamemanner,althoughthemotionofonebodyaroundthesun,determinedbygravitatio
5、n,canbecalculatedpointbypointbythenumericalmethodsofChapter9,whichshowthegeneralshapeoftheorbit,itisnicealsotogettheexactshape,whichanalysisrevealsasaperfectellipse.Unfortunately,therearereallyveryfewproblemswhichcanbesolvedexactlybyanalysis.Inthecaseoftheharmonicoscillator,forexample,ifth
6、espringforceisnotproportionaltothedisplacement,butissomethingmorecomplicated,onemustfallbackonthenumericalmethod.Oriftherearetwobodiesgoingaroundthesun,sothatthetotalnumberofbodiesisthree,thenanalysiscannotproduceasimpleformulaforthemotion,andinpracticetheproblemmustbedonenumeri•cally.That
7、isthefamousthree-bodyproblem,whichsolongchallengedhumanpowersofanalysis;itisveryinterestinghowlongittookpeopletoappreciatethefactthatperhapsthepowersofmathematicalanalysiswerelimitedanditmightbenecessarytousethenumericalmethods.Todayanenormousnumberofproblemst
