heinemann baccalaureate physics sl chapter2d
DESCRIPTION
bookTRANSCRIPT
-
.FDIBOJDT
8IBUJTLJOFNBUJDT +INEMATICSISTHESTUDYOFMOTION!SWEHAVEALREADYSEENDISTANCEISASCALARANDDISPLACEMENTISAVECTORQUANTITY$ISTANCETRAVELLEDISASIMPLEMEASUREOFLENGTHWHILEDISPLACEMENTISDElNEDASTHEDISTANCETRAVELLEDINAPARTICULARDIRECTION
7HENWEWANTTODESCRIBEHOWFASTSOMETHINGISMOVINGWESAYTHAT
SPEEDDISTANCE???????TIME
VELOCITYDISPLACEMENT???????????TIME
4HESESIMPLEEQUATIONSCANBEUSEDFORTWOSITUATIONSlRSTLYWHENTHEBODYISMOVINGATACONSTANTSPEEDANDSECONDLYINORDERTOlNDTHEAVERAGESPEED)NTHESECONDCASEWENEEDTOKNOWTHETOTALDISTANCEANDTHETOTALTIME
8PSLFEFYBNQMF
#ALCULATEYOURAVERAGESPEEDIFYOURJOURNEYTOSCHOOLTAKESHALFANHOURANDTHEDISTANCEISKM
4PMVUJPOSPEEDKM??????H
KMH
,JOFNBUJDT
ii>ii iwi`>Vii]iV]ii`]>`>VVii> >i`vviiViLiii>>i>`>i>}i>iv
ii`]iV>`>VVii> "iiV``iVii>vv
>VVii>i`>Li>i`{ `ivi>VVii>v>L`v>}>>Vi>i>
v>Vii>VVii>}vviiv>x -iLi}ii>vv>VVii>i`
iVLiiivviVv>i>Viv>}LiV >>`>>i`>Viqi}>]`>Viiqi}>]
iVqi}>>`>VVii>qi}>n >V>i>`iiiiv`>Viiqi}>>`
iVqi}>]>`i>i>`iiVqi}>>`>VVii>qi}>
iiii>iiVi>``i
&YBNJOFSTIJOU 4QFFEEJTUBODF@@@@@@@@UJNF
-
.FDIBOJDT
4HISWORKSEVENTHOUGHYOURCARORBUSHASTOSLOWDOWNSPEEDUPANDSOMETIMESSTOPACCORDINGTOTHETRAFlCCONDITIONS
)FYOURVEHICLEISCHANGINGSPEEDORDIRECTIONITMUSTBECHANGINGVELOCITYWHICHMEANSTHATITISACCELERATING!CCELERATIONISDElNEDASTHECHANGEINVELOCITYPERUNITTIME!CCELERATIONACANBEWRITTENAS
A$V???$T
WHERE$VISCHANGEINVELOCITYAND$TISCHANGEINTIME
8PSLFEFYBNQMF
!CARSTARTSFROMRESTANDACCELERATESSTEADILYTOASPEEDOFMSINATIMEOFS&INDTHEMAGNITUDEOFTHEACCELERATIONOFTHECAR
4PMVUJPO
AMS???????S
MS
4HEUNITSOFVELOCITYANDACCELERATIONAREDIFFERENTANDYOUMUSTALWAYSBECAREFULTOMEASURESPEEDANDVELOCITYINMSBUTACCELERATIONINMS
)FABALLISROLLINGDOWNASLOPEITSSPEEDWILLBEINCREASING7ECANlNDTHEAVERAGESPEEDOFTHEBALLBYDIVIDINGTHETOTALDISTANCEBYTHETOTALTIMEASSEENEARLIER!NOTHERWAYTODESCRIBETHEMOTIONISBYUSINGTHEINSTANTANEOUSSPEEDOFTHEBALLTHESPEEDATANYINSTANTINTIME9OUCANlNDTHISBYDRAWINGADISTANCEnTIMEGRAPHOFTHEMOTIONANDMEASURINGTHETANGENTATAPARTICULARINSTANTnWEWILLSEEHOWTHISWORKSLATERONINTHISCHAPTER
5IFFRVBUJPOTGPSVOJGPSNMZBDDFMFSBUFENPUJPO)FSPEEDORVELOCITYISCHANGINGATASTEADYRATETHENTHEACCELERATIONISCONSTANT)NTHESESITUATIONSWECANUSETHEEQUATIONSOFUNIFORMLYACCELERATEDMOTIONWHICHWEWILLCALLTHESUVATEQUATIONS4HESEAREASETOFEQUATIONSWITHlVEVARIABLESASFOLLOWSS DISTANCE MU INITIALVELOCITY MSV lNALVELOCITY MSA ACCELERATION MST TIME S
*UJTFTTFOUJBMUIBUZPVSFNFNCFSUIBUATJOUIFTFFRVBUJPOTTUBOETGPSEJTUBODFUSBWFMMFEBOEOPUGPSTQFFE
-
4HEREAREFOURSUVATEQUATIONSANDTHEIRPOWERISTHATIFWEKNOWANYTHREEOFTHElVEVARIABLESBYUSINGTHEEQUATIONSCAREFULLYWECANSOLVEFORTHEOTHERTWOUNKNOWNQUANTITIES
4HESEARETHEEQUATIONS VUAT
SUV????? T
SUT?AT
VUAS
8PSLFEFYBNQMFT
!BALLISTHROWNVERTICALLYUPWARDSWITHANINITIALVELOCITYOFMS4AKINGTHEACCELERATIONDUETOGRAVITYASMSANDNEGLECTINGAIRRESISTANCElNDA THETIMETAKENTORETURNTOTHETHROWERSHAND
B THEMAXIMUMHEIGHTREACHED
4PMVUJPOA S UMS
V AMS
T VUAT
T T TS
4IMETORETURNTOTHETHROWERSHANDWILLBETWICETHISS
B S UMS
V AMS
TS
SUV????? T
S??????
SM
'JHVSF 5IFNBYJNVNTQFFEXJMMCFSFBDIFEKVTUBGUFSUIFCBMMMFBWFTUIFHJSMTIBOE
&YBNJOFSTIJOU 'JSTUMJTUUIFTVWBUWBSJBCMFT
5IFODIPPTFUIFCFTUFRVBUJPOUIFPOFXJUIUIFSFMFWBOUWBSJBCMFT
*OUIJTDBTFUIFmOBMWFMPDJUZXIFOUIFCBMMSFBDIFTUIFNBYJNVNIFJHIUXJMMCF[FSP
5IFBDDFMFSBUJPOXJMMCFOFHBUJWFJGXFDPOTJEFSUIFCBMMUSBWFMMJOHVQXBSETCFDBVTFJUXJMMCFTMPXJOHEPXO
&YBNJOFSTIJOU "MXBZTXSJUFEPXOUIFFRVBUJPOZPVBSFVTJOHTPNFUJNFTZPVDBOHFUBNBSLGPSKVTUXSJUJOHEPXOUIFDPSSFDUFRVBUJPO
&YBNJOFSTIJOU /FYUQMVHJOUIFOVNCFSTBOETPMWFGPSUIFVOLOPXO
5IJTUZQFPGQSPCMFNJOWPMWJOHHSBWJUZJTBMXBZTTZNNFUSJDBMUIFEJTUBODFUJNFBOETQFFEHPJOHVQBSFUIFTBNFBTUIPTFDPNJOHCBDLEPXO
&YBNJOFSTIJOU 5PmOEUIFNBYJNVNIFJHIUXFBHBJOMJTUUIFWBSJBCMFTGPSUIFNPUJPOVQXBSET
&YBNJOFSTIJOU $IPPTFBOFRVBUJPOBOEXSJUFJUEPXO
-
.FDIBOJDT
&YBNJOFSTIJOU 5IJTUJNFUIFJOJUJBMTQFFEJT[FSPBOEUIFBDDFMFSBUJPOJTQPTJUJWF"HBJOXFMJTUUIFWBSJBCMFT
&YBNJOFSTIJOU 5PmOEUIFUJNFJOUIJTDBTFXFDPVMEVTFBOZPGUIFPUIFSUISFFFRVBUJPOT
!STONEISDROPPEDINTOTHESEAFROMACLIFFMHIGH4AKINGTHEACCELERATIONDUETOGRAVITYASMSANDNEGLECTINGAIRRESISTANCElNDA THESPEEDOFTHESTONEWHENITHITSTHESEAB THETIMETAKENTOREACHTHESEA
A SM U V AMS
T
VUAS V VMS
B SM U VMS
AMS
T SUT?AT
T T
T TS
4RYTHISLASTPROBLEMFORYOURSELFUSINGTHEOTHERTWOEQUATIONSTOMAKESUREYOUGETTHESAMEVALUEFORTHETIMETAKEN
)NTHETWOEXAMPLESABOVETHEACCELERATIONOFTHEBODIESWASGTHEACCELERATIONDUETOGRAVITY4HISVARIESFROMPLACETOPLACEONTHE%ARTHSSURFACEBUTISOFTENTAKENASMSORAPPROXIMATEDTOMS
4HESUVATEQUATIONSWORKFORANYSITUATIONWHERETHEACCELERATIONOFTHEBODYISCONSTANTANDYOUMUSTLEARNTOSPOTTHETYPEOFPROBLEMWHERETHESUVATEQUATIONSWILLPROVIDETHESOLUTION
'JHVSF "DDFMFSBUJPOJTQPTJUJWFJOUIJTDBTF5IFTUPOFBDDFMFSBUFTVOUJMJUIJUTUIFXBUFS
*OBOFYQFSJNFOUBTNBMMTUFFMCBMMJTmMNFEBOEUJNFEBTJUGBMMTGSPNSFTUOFYUUPBNFUSFSVMFS$BMDVMBUFUIFUJNFUBLFOGPSUIFCBMMUPGBMMB GSPNUIF[FSPNBSLUPUIFDNNBSLPOUIFSVMFSCGSPNUIFDNNBSLUPUIFFOEPGUIFSVMFS/FHMFDUBJSSFTJTUBODFBOEUBLFHNT
"DBSJTUSBWFMMJOHBUNTXIFOBEPHSVOTPVUJOUPUIFSPBE*GUIFESJWFSTSFBDUJPOUJNFJTTBOEUIFDBSEFDFMFSBUFTTUFBEJMZBUBSBUFPGNTDBMDVMBUFUIFNJOJNVNEJTUBODFUSBWFMMFECFGPSFUIFDBSDBODPNFUPSFTU
&YFSDJTFT
5PWJFXBOBOJNBUJPOPGNPUJPOJO%WJTJUIFJOFNBOODPVLIPUMJOLTFOUFSUIFFYQSFTTDPEF4BOEDMJDLPOUIF8FCMJOL
-
)NTHEREALWORLDBODIESDONOTFALLINAVACUUMTHEYEXPERIENCETHEEFFECTSOFAIRRESISTANCE!SLONGASTHEBODYISSMALLANDHARDANDTHEHEIGHTISNOTTOOGREATTHENWECANIGNORETHESEEFFECTS/THERWISEAIRRESISTANCEORDRAGBECOMESIMPORTANTOBVIOUSLYTHEMOTIONOFAFALLINGBODYSUCHASAFEATHERORALEAFWILLBESIGNIlCANTLYCHANGEDBYTHEAIRTHROUGHWHICHITFALLS)FAFEATHERORALEAFWERETOFALLINAVACUUMTHEYWOULDDROPJUSTLIKEASTONE
)NITIALLYTHESKYDIVERSSPEEDISLOWSOTHEAIRRESISTANCEISMUCHSMALLERTHANTHEWEIGHTANDTHESKYDIVERACCELERATES!STHESPEEDINCREASESTHEAIRRESISTANCEINCREASESUNTILAPOINTISREACHEDWHENTHEDRAGFORCEISEQUALANDOPPOSITETOTHEWEIGHT4HENTHESKYDIVERWILLFALLATACONSTANTMAXIMUMSPEEDCALLEDTHETERMINALVELOCITY7HENTHEPARACHUTEOPENSTHEAIRRESISTANCEINCREASESSIGNIlCANTLYANDANEWSLOWERTERMINALVELOCITYISREACHEDENABLINGASAFELANDING
&ROM/TO!THEVELOCITYISLOWANDACCELERATIONFAIRLYCONSTANT&ROM!TO"AIRRESISTANCEBECOMESGREATERASTHEVELOCITYINCREASESSOTHERATEOFACCELERATIONDECREASES&ROM"TO#ACCELERATIONISZEROTHESKYDIVERISFALLINGATCONSTANTTERMINALVELOCITY!T#THEPARACHUTEOPENSANDVELOCITYRAPIDLYDECREASES&ROM$TO%ANEWLOWERTERMINALVELOCITYISREACHED!T%THESKYDIVERLANDSONTHEGROUND
5IFBDDFMFSBUJPOEVFUPHSBWJUZJTTMJHIUMZMPXFSPUIFDPBTUPG*OEJBBOETMJHIUMZIJHIFSJOUIF4PVUI1BDJmDPXOJOHUPUIFOBUVSFPGUIFSPDLTJOUIF&BSUITDSVTU
'PSBQFSTPOJOGSFFGBMMUFSNJOBMWFMPDJUZJTBCPVULNIPSNTTJNJMBSUPBTQFFEJOHDBS
'JHVSF "WFMPDJUZUJNFHSBQIPGBQBSBDIVUFKVNQ
&YBNJOFSTIJOU 8IFOZPVBSFBTLFEUPTLFUDIBHSBQIZPVTIPVMEESBXBQBJSPGMBCFMMFEBYFTXJUIVOJUTBOEBMJOFTIPXJOHUIFUSFOE:PVEPOPUIBWFUPTIPXOVNCFSTPOBTLFUDIHSBQICVUZPVTIPVMEESBXJUDMFBSMZXJUIBSVMFSBOEQFODJM
*NBHJOFXIBUIBQQFOTUPUIFTQFFEPGBTLZEJWFSXIPKVNQTPVUPGBQMBOFGBMMTGSFFGPSBXIJMFBOEmOBMMZPQFOTBQBSBDIVUF
-
.FDIBOJDT
5PWJFXBOBOJNBUJPOPGUIFMVOBSMBOEFSWJTJU)FJOFNBOODPVLIPUMJOLTFOUFSUIFFYQSFTTDPEF4BOEDMJDLPO8FCMJOL
(SBQIJOHNPUJPO7EHAVEALREADYSEENSOMEEXAMPLESOFMOTIONGRAPHSBUTNOWWEWILLLOOKATTHEMINMOREDETAIL.OTETHATTIMEALWAYSGOESONTHEXAXIS
%JTUBODFPSEJTQMBDFNFOUWTUJNF
4HENEXTGRAPHSHOWSDISPLACEMENTAGAINSTTIMEREMEMBERTHATDISPLACEMENTISAVECTORQUANTITY4HEWAYWESHOWTHEDIFFERENCEBETWEENSCALARSANDVECTORSONAGRAPHISTODRAWTHESCALARABOVETHEXAXISONLYANDTHEVECTORBOTHABOVEANDBELOW
4HESLOPEORGRADIENTOFADISTANCEnTIMEGRAPHGIVESTHESPEEDWHILEFORADISPLACEMENTnTIMEGRAPHTHEGRADIENTGIVESTHEVELOCITY)N&IGUREABOVETHECARISTRAVELLINGATCONSTANTVELOCITYFROM/TO!BECAUSETHEGRADIENTISCONSTANT&ROM!TO"THEVELOCITYISDECREASINGBECAUSETHEGRADIENTISDECREASINGINOTHERWORDSTHECARISDECELERATING
*OUIFQMBOFJOXIJDI/JDIPMBT"MLFNBEFXBTnZJOHXBTTFUPOmSFBOEIJTQBSBDIVUFEFTUSPZFE/JDIPMBTDIPTFUPKVNQXJUIPVUBQBSBDIVUFSBUIFSUIBOTUBZJOUIFCVSOJOHQMBOF"MUIPVHIIFGFMMOFBSMZNIFXBTFYUSFNFMZMVDLZCFDBVTFIFCMBDLFEPVUMBOEJOHSFMBYFEUISPVHIQJOFUSFFTBOEJOUPTOPX)FTVFSFEPOMZNJOPSTQSBJOTDVUTBOECSVJTFTBOETVSWJWFEVOUJM
'JHVSF 'PVSEJTUBODFUJNFHSBQIT5IFDBSTIPXOJOUIFTFHSBQITJTBUSFTUOPUNPWJOH NPWJOHBUBDPOTUBOUTQFFE BDDFMFSBUJOH EFDFMFSBUJOH
'JHVSF %JTQMBDFNFOUUJNFHSBQI0o" DPOTUBOUWFMPDJUZGPSXBSET"o# EFDFMFSBUJOH#o$ TUPQQFE$o% BDDFMFSBUJOHBOEDIBOHJOH
EJSFDUJPO%o& DPOTUBOUWFMPDJUZCBDLUPTUBSUJOH
QPJOU&o' DPOTUBOUWFMPDJUZCBDLXBSET'o( EFDFMFSBUJOH(o) TUPQQFE)o* BDDFMFSBUJOHBOEBHBJODIBOHJOH
EJSFDUJPO*o+ DPOTUBOUWFMPDJUZCBDLUPTUBSUJOH
QPJOU
-
7FMPDJUZWTUJNF)TISVITALLYIMPORTANTTHATYOUDONOTMIXUPTHEVARIOUSDIFFERENTTYPESOFMOTIONGRAPH4HENEXTONESWEWILLCONSIDERARESPEEDANDVELOCITYAGAINSTTIME4RYANDDESCRIBEWHATISHAPPENINGTOACARMOVINGASFOLLOWS
4HECARSTARTSFROMRESTWHENTHESPEEDISZEROANDTHENFROM/TO!ITSSPEEDINCREASESATASTEADYRATE4HISISKNOWNASCONSTANTACCELERATION4HENFROM!TO"THECARISNOTSTOPPEDBUTISMOVINGATCONSTANTSPEED&ROM"TO#WEHAVECONSTANTDECELERATIONANDlNALLYTHECARAGAINCOMESTOREST!SLONGASTHECARISTRAVELLINGINASTRAIGHTLINEINTHESAMEDIRECTIONTHISGRAPHWOULDBEEXACTLYTHESAMEWITHEITHERSPEEDORVELOCITYONTHEYAXIS
4HEGRADIENTOFAVELOCITYnTIMEGRAPHGIVESTHEACCELERATION3OIN&IGUREABOVEFROM/TO!THEACCELERATIONISCONSTANTBECAUSETHEGRADIENTISCONSTANTTHEGRAPHISASTRAIGHTLINE&ROM!TO"THEGRADIENTISZEROBECAUSETHERISEOR$YTERMISZEROSOTHEACCELERATIONISZERO&ROM"TO#THEGRADIENTISSTEEPERTHANFROM/TO!ANDTHISTELLSUSTHATTHERATEOFDECELERATIONWASMORERAPIDTHANTHERATEOFACCELERATION
.OWWEWILLANALYSETHESAMEGRAPHUSINGNUMBERS
4HEAREAUNDERAVELOCITYnTIMEGRAPHGIVESTHEDISPLACEMENT)NTHISCASETHEAREAISTHESUMOFATRIANGLEARECTANGLEANDASMALLERTRIANGLE2EMEMBERTHATTHEAREAOFATRIANGLEISHALFTHEBASETIMESTHEHEIGHT
'JHVSF 5IFJOTUBOUBOFPVTWFMPDJUZJTGPVOECZCZNFBTVSJOHUIFTMPQFPGUIFUBOHFOUUPUIFDVSWF" 5IFUBOHFOUMJOFJTTUFFQBUUIJTQPJOU
JOEJDBUJOHBIJHIJOTUBOUBOFPVTWFMPDJUZ
# #FDBVTFUIFDVSWFIBTnBUUFOFEPVUBUUIJTQPJOUUIFUBOHFOUJTMFTTTUFFQBOEUIFJOTUBOUBOFPVTWFMPDJUZJTMPXFS
'JHVSF 7FMPDJUZoUJNFHSBQI
5IFTMPQFPSHSBEJFOUPGBHSBQI
SJTF@@@SVOZZ@@@@@@YY
$Z@@@$Y
'JHVSF 'JOEJOHEJTUBODFUSBWFMMFEGSPNBWFMPDJUZUJNFHSBQI0o" DPOTUBOUBDDFMFSBUJPO
NT@@@@@@@T NT
"o# DPOTUBOUNBYJNVNWFMPDJUZPGNTGPST
#o$ DPOTUBOUEFDFMFSBUJPO
NT@@@@@@@T NT
-
.FDIBOJDT
$ISPLACEMENT?SMSSMS?SMS
MMM M
9OUNEEDTOBEABLETOSKETCHANDLABELTHESETYPESOFGRAPHFORVARIOUSSITUATIONS
"DDFMFSBUJPOWTUJNF
4HELASTTYPEOFMOTIONGRAPHWEAREGOINGTOLOOKATISACCELERATIONAGAINSTTIME)FTHEACCELERATIONISUNIFORMORCONSTANTTHENTHESEGRAPHSWILLBESIMPLYVERTICALANDHORIZONTALSTRAIGHTLINES!NACCELERATIONnTIMEGRAPHFORTHECARIN&IGUREWOULDLOOKLIKETHIS
4HEAREAUNDERANACCELERATIONnTIMEGRAPHGIVESTHECHANGEINVELOCITY
3OFROM/TO!THECHANGEINVELOCITYTHEAREAOFTHERECTANGLE SMS MS
&ROM!TO"THEAREAUNDERTHEGRAPHISZEROSOTHECHANGEINVELOCITYISZERO
9OUCANCHECKFORYOURSELFTHATTHEAREAFROM"TO#ISMSINTHEOPPOSITEDIRECTIONANDTHATTHISALLlTSWITHTHEDATAIN&IGURE
)FTHEACCELERATIONISNOTCONSTANTWECANNOTUSETHESUVATEQUATIONSTOSOLVETHEPROBLEMS4HEAREAUNDERTHEGRAPHWILLSTILLTELLUSTHECHANGEINVELOCITYHOWEVERASSHOWNINTHENEXTEXAMPLE
"HJSMUISPXTBCBMMVQJOUIFBJSBOEUIFODBUDIFTJU4LFUDIBTQFFEoUJNFHSBQIBOEUIFOBWFMPDJUZoUJNFHSBQIGPSUIFNPUJPOPGUIFCBMM
"CBMMJTESPQQFEPOUPUIFnPPSBOECPVODFTUXJDF4LFUDIBWFMPDJUZoUJNFHSBQIBOEUIFOBTQFFEoUJNFHSBQIGPSUIFNPUJPOPGUIFCBMM
&YFSDJTFT
'JHVSF"DDFMFSBUJPOoUJNFHSBQI0o" DPOTUBOUBDDFMFSBUJPOPGNT
GPST"o# [FSPBDDFMFSBUJPOGPST#o$ DPOTUBOUOFHBUJWFBDDFMFSBUJPOPS
EFDFMFSBUJPOPGNTGPST
-
8PSLFEFYBNQMF
&INDTHECHANGEINVELOCITYOFABODYTHATACCELERATESASSHOWNINTHEGRAPHBELOW
4PMVUJPO#HANGEINVELOCITY?SMS
SMS MSMS
MS
3FMBUJWFWFMPDJUZ)MAGINEYOUARERIDINGINATRAINTRAVELLINGATKMH!NOTHERTRAINOVERTAKESYOUATASPEEDOFKMH)TLOOKSLIKETHEOTHERTRAINISMOVINGQUITESLOWLY4HISISBECAUSERELATIVETOYOUTHESPEEDOFTHEOTHERTRAINISONLYKMH
4HINKABOUTTHEFOLLOWINGSCENARIO$IEGOISSTANDINGONTHEPLATFORMAND-ARIAONATRAINPASSESTHROUGH7ECANTALKABOUTTHERELATIVEMOTIONINTWOWAYSTHEMOTIONOF-ARIARELATIVETO$IEGOORTHEMOTIONOF$IEGORELATIVETO-ARIA4HEMAGNITUDEORSPEEDWILLBETHESAMEBUTTHEDIRECTIONWILLCLEARLYBEDIFFERENT
'JHVSF "DDFMFSBUJPOUJNFHSBQI5IFDIBOHFJOWFMPDJUZJTUIFBSFBPGUSJBOHMF"#$QMVTBSFBPGSFDUBOHMF0"$%
5PWJFX5IFNPWJOHNBOTJNVMBUJPOWJTJU)FJOFNBOODPVLIPUMJOLTFOUFSUIFFYQSFTTDPEF4BOEDMJDLPOUIF8FCMJOL
'JHVSF 3FMBUJWFWFMPDJUZ5IFWFMPDJUZPG.BSJBSFMBUJWFUP%JFHPJTNTUPUIFSJHIU5IFWFMPDJUZPG%JFHPSFMBUJWFUP.BSJBJTNTUPUIFMFGU
-
.FDIBOJDT
3OFARWEHAVEBEENLOOKINGATRELATIVEMOTIONINONEDIMENSION4OUNDERSTANDHOWITWORKSINTWODIMENSIONSIMAGINEYOUAREINACARDRIVINGDOWNALONGSTRAIGHTROAD!BIRDISmYINGATRIGHTANGLESTOTHEROADANDYOUCANSEEITOUTOFTHEWINDOW
1SBDUJDBMIJOUTPOESBXJOHHSBQIT7HENYOUAREASKEDTOSKETCHORPLOTAMOTIONGRAPHTHINKABOUTTHEFOLLOWINGPOINTS )STHISAGRAPHOFDISTANCEDISPLACEMENTSPEEDVELOCITYORACCELERATION4HEYAREALLUSUALLYSKETCHEDAGAINSTTIMEBUTTHEYAREALLDIFFERENTGRAPHS
)STHISASCALARORAVECTORGRAPH)NOTHERWORDSDOYOUNEEDTODRAWITJUSTABOVETHELINEORBOTHABOVEANDBELOW
2EMEMBERTOLABELTHEAXESWITHBOTHTHENAMEANDTHEUNITSOMETIMESYOUWILLGETAMARKFORJUSTDOINGTHATALONE
$OESTHEGRAPHSTARTFROMTHEORIGINTHEZEROZEROPOINT )STHEGRAPHGOINGTOBEASTRAIGHTLINEORASMOOTHCURVE -AKESUREYOUPLOTTHEGRAPHWITHASENSIBLESCALEUSENUMBERSLIKEANDNOTNUMBERSLIKEAND
!IMFORYOURGRAPHTOlLLATLEASTHALFTHEAVAILABLESPACE -EASURETHEGRADIENTORTANGENTOVERATLEASTHALFTHESPACE ,INKWHATYOUNEEDTOlNDTOTHEGRADIENTINTERCEPTORAREAUNDERTHEGRAPH !LWAYSREMEMBERTHEEQUATIONOFASTRAIGHTLINEISOFTHEFORMYMXCANDLINKITWHENEVERPOSSIBLETOANYGRAPHYOUCOMEACROSS
-UCHOFTHISAPPLIESTOOTHERTYPESOFPHYSICSGRAPHSNOTJUSTMOTIONGRAPHS
'JHVSF5IFNPUJPOPGUIFCJSESFMBUJWFUPUIFDBSJTEJFSFOUGSPNUIFNPUJPOPGUIFDBSSFMBUJWFUPUIFCJSE
:PVXJMMMFBSOBMPUNPSFBCPVUSFMBUJWFNPUJPOJGZPVTUVEZ0QUJPO%XIJDIJODMVEFT&JOTUFJOTUIFPSZPG4QFDJBM3FMBUJWJUZ
)FBEPODBSDPMMJTJPOTBSFMJLFMZUPIBWFTFSJPVTDPOTFRVFODFT0OFSFBTPOJTUIBUUIFSFMBUJWFTQFFETXJMMBEEVQBOETPUIFSFTVMUJOHEBNBHFXJMMBMTPJODSFBTF
-
'PSDFXFJHIUBOENBTT)NSIMPLETERMSAFORCEISAPULLORAPUSH&ORCEISAVECTORQUANTITYANDISMEASUREDINNEWTONS4HEREAREMANYDIFFERENTTYPESOFFORCEINCLUDING FRICTION NORMALORSUPPORTINGORCONTACTFORCE TENSIONANDCOMPRESSION AIRRESISTANCEORDRAG UPTHRUSTORBUOYANCY LIFT THRUST WEIGHT
7EIGHTISTHEPULLOFGRAVITYONABODY!SSUMINGTHATTHEBODYISONTHESURFACEOFTHEEARTHITISDIRECTEDTOWARDSTHECENTREOFTHEEARTH5SUALLYWECALLTHISDIRECTION@DOWN
3TRICTLYSPEAKINGWEIGHTANDMASSARENOTTHESAMETHING7EIGHTISAVECTORMEASUREDINNEWTONSANDMASSISASCALARMEASUREDINKILOGRAMS-ASSCANBEDESCRIBEDASTHEAMOUNTOFMATTERINABODY
)NEVERYDAYSPEECHWEIGHTANDMASSARETAKENTOMEANTHESAMEBUTINPHYSICSWENEEDTOREMEMBERTHATTHEREISADIFFERENCE
'PSDFTBOEEZOBNJDT
ii>ii >V>iii}v>L`}iii7} `ivivVi>V}>LiV>``>viiL``>}>
iii}ivVi>V} iiiii>vVi`vvii>{ ->i iw>vx iVLii>iv iw> ->iiV`v>>>iL -iLi}>>>iLn ->i iiV`>v -iLi} iiV`> iwii>i>`i iiiii`i>i>}vViLii}>
vViqi}> ->ii>vVi>vi>i -iLi}i>`i{ ->i i`>vx Vi>iv i`>
'JHVSF 'PSDFDBOCFBQVMMPSBQVTI
*GZPVXFOUUPUIFNPPOZPVSNBTTXPVMEOPUDIBOHFCVUZPVSXFJHIUXPVMEEFDSFBTFCFDBVTFUIFQVMMPGHSBWJUZJTMFTTPOUIFNPPO
-
.FDIBOJDT
8PSLFEFYBNQMF
7HATISTHEWEIGHTOFAKGMASSONTHESURFACEOFTHEEARTHWHEREGMS
4PMVUJPO 7MG 7KGMS
.
#BMBODFEGPSDFT(ERETRANSLATINGMEANSMOVINGFROMONEPLACETOANOTHERINASTRAIGHTLINE)FABODYISINTRANSLATIONALEQUILIBRIUMTHENITWILLNOTBEACCELERATING4HISISTHEBASICIDEADESCRIBEDIN.EWTONSlRSTLAW
7EHAVEALREADYSEENSOMEFREEBODYDIAGRAMSIN#HAPTERBUTHEREARESOMEMOREEXAMPLES
4HEHELIUMBALLOONISATRESTSOMUSTBEINTRANSLATIONALEQUILIBRIUM
UPTHRUSTWEIGHTTENSION
4HEWEIGHTOFTHEBALLOONWILLBESMALLBUTCANNOTBEZERO)FTHESTRINGBREAKSTHENTHETENSIONBECOMESZEROANDTHEBALLOONWILLACCELERATEUPWARDS
"TUSPOBVUTPOUIFNPPOXFSFBCMFUPUBLFHJBOUTUFQTCFDBVTFUIFJSXFJHIUXBTMFTTBMUIPVHIUIFJSNBTTIBEOUDIBOHFE
8FJHIUJTSFMBUFEUPNBTTJOBTJNQMFXBZ 8NH8 JTUIFXFJHIUNFBTVSFEJO/N JTUIFNBTTNFBTVSFEJOLHH JTUIFBDDFMFSBUJPOEVFUP
HSBWJUZNFBTVSFEJONT
*GBCPEZJTJOUSBOTMBUJPOBMFRVJMJCSJVNJUNFBOTUIBUUIFGPSDFTBDUJOHPOUIFCPEZJOBMMUISFFEJNFOTJPOTBSFCBMBODFE
'JHVSF /PUJDFUIBUUIFEJSFDUJPOPGUIFUFOTJPOJOUIFTUSJOHNVTUCFEPXOXBSETUPCBMBODFUIFGPSDFT
-
4HEFREEBODYDIAGRAMMUSTSHOWONLYTHETHREEREALFORCESACTINGONTHEBLOCK3INCETHEBLOCKISACCELERATINGTHEFORCESARENOTBALANCEDDOWNTHEINCLINE
X&FRY&.
WHERE&FRISTHEFRICTIONALFORCEAND&.ISTHENORMALORPERPENDICULARFORCE
5SINGGEOMETRYYOUCANSEETHATTHEANGLEOFTHEINCLINEISEQUALTOTHEANGLEBETWEENTHEWEIGHTVECTORANDTHEYCOMPONENT4HISISBECAUSETHEREAREINATRIANGLEANDTHENORMALFORCEISATTOTHEINCLINE
5SINGTRIGONOMETRYWECANTHENDEDUCETHAT
SINU X???MG
XMGSINU
COSUY???MG
YMGCOSU
/FXUPOTMBXTPGNPUJPO
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
'JHVSF *GUIFQFSTPOQVMMTIBSEFSBOEKVTUTVDDFFETJOMJGUJOHUIFCMPDLUIFOUIFDPOUBDUGPSDFCFDPNFT[FSPBOEUIFQVMMJOHGPSDFJTFRVBMUPUIFXFJHIU
*TBBD/FXUPO
'JHVSF *ODMJOFEQMBOFUIFXFJHIUJTSFTPMWFEJOUPYBOEZDPNQPOFOUTQBSBMMFMBOEQFSQFOEJDVMBSUPUIFJODMJOF5IFUISFFGPSDFTBDUJOHPOUIFCMPDLBSFNHUIFXFJHIU
'/BOG'GS
-
.FDIBOJDT
.EWTONSlRSTLAWSTATESTHATABODYWILLEITHERREMAINATRESTORMOVEWITHCONSTANTVELOCITYUNLESSACTEDONBYANUNBALANCEDFORCE
7EHAVEALREADYSEENHOWTHISLAWWORKSANDITISSIMPLYCOMMONSENSETHATSOMETHINGDOESNOTSTARTTOMOVEUNLESSTHEREISRESULTANTFORCEACTINGONIT-OREDIFlCULTTOUNDERSTANDISTHEIDEATHATABODYCANBEINEQUILIBRIUMEVENTHOUGHITISMOVING)TCANBEDIFlCULTTOACCEPTTHATABODYCANMOVEWITHOUTAFORCEBEINGEXERTEDASTHISGOESAGAINSTBOTHCOMMONSENSEANDTHETEACHINGOF!RISTOTLE
.EWTONSSECONDLAWTELLSUSWHATHAPPENSWHENANUNBALANCEDFORCEACTSANDISEXPRESSEDINTHEMOSTIMPORTANTEQUATIONOFBASICMECHANICS
&MA& ISTHERESULTANTORUNBALANCEDFORCEMEASUREDIN.M ISTHEMASSMEASUREDINKGA ISTHEACCELERATIONMEASUREDINMS
.OTETHATTHEEARLIERDElNITIONOFWEIGHTISASPECIALCASEOFTHISGENERALLAW
&ORACONSTANTMASSTHISMEANSTHATFORCEISDIRECTLYPROPORTIONALTOACCELERATION
)FWETAKEACONSTANTOUTOFASIMPLEEQUATIONTHENWEGETARELATIONSHIPTHATISDIRECTLYPROPORTIONAL&OREXAMPLEIF
YMXANDMISACONSTANTTHEN Y^X#ONVERSELYIFTWOTHINGSAREDIRECTLYPROPORTIONALTOEACHOTHERANDWEPUTINACONSTANTTHENWEGETANEQUALITYOREQUATION&OREXAMPLEIF
&^XANDKISACONSTANT
THEN &KX
8PSLFEFYBNQMF
&INDTHEACCELERATIONOFABLOCKOFMASSGPULLEDACROSSATABLEWITHAFORCEOF.IFTHEREISAFRICTIONALFORCEOF.
4PMVUJPO &MA
A&??M
A .?????KG
AMS TOTHERIGHT
1SBDUJDBMIJOUTPODBMDVMBUJPOT4HEBASICMETHODFORNUMERICALPHYSICSPROBLEMSISNOTCOMPLICATED #HOOSETHECORRECTEQUATION 5SUALLYYOUCANlNDTHEEQUATIONSINTHE$ATA"OOKLETBUTSOMETIMESYOUSIMPLYHAVETOMEMORISETHEM
'JHVSF 5IFSFTVMUBOUGPSDF/UPUIFSJHIU
*GUXPWBSJBCMFTBSFEJSFDUMZQSPQPSUJPOBMUPFBDIPUIFSJUNFBOTUIBUJGPOFJODSFBTFTUIFPUIFSXJMMJODSFBTFJOQSPQPSUJPOPSBUUIFTBNFSBUF"HSBQIPGUIFUXPWBSJBCMFTXJMMCFBTUSBJHIUMJOFUISPVHIUIFPSJHJO
"SJTUPUMF#$XBTPOFPGUIFNPTUJNQPSUBOUPGUIFBODJFOU(SFFLQIJMPTPQIFSTBOEIFUVUPSFE"MFYBOEFSUIF(SFBU"SJTUPUMFUIPVHIUUIBUBIFBWZPCKFDUXPVMEGBMMNPSFRVJDLMZUIBOBMJHIUPOFBOEUIBUBGPSDFXBTOFDFTTBSZGPSBOPCKFDUUPNPWF
(BMJMFPXBTCPSOJO1JTB*UBMZBOEJONBOZXBZTXBTUIFGPVOEFSPGNPEFSOQIZTJDT(BMJMFPUIPVHIUUIBUBIFBWZPCKFDUBOEBMJHIUPCKFDUTIPVMEGBMMBUUIFTBNFSBUFBOEUIBUBCPEZDPVMENPWFGPSFWFSJGOPGPSDFXFSFBDUJOH
8IPXBTSJHIU"SJTUPUMFPS(BMJMFP
5IFGPSDFJO/FXUPOTTFDPOEMBXJTBMXBZTUIFSFTVMUBOUPSVOCBMBODFEGPSDF
$POTUBOUWFMPDJUZNFBOTUXPUIJOHTDPOTUBOUTQFFEBOEDPOTUBOUEJSFDUJPOJOPUIFSXPSETDPOTUBOUTQFFEJOBTUSBJHIUMJOF
-
-AKESUREYOUKNOWWHATTHESYMBOLSLETTERSSTANDFOR 7HENSOLVINGTHEPROBLEMALWAYSWRITEDOWNTHEEQUATIONYOUAREUSINGSOMETIMESYOUWILLGETAMARKFORJUSTWRITINGTHEEQUATION
0LUGINTHENUMBERS -AKESURETHEUNITSARECONSISTENTIFNOTCHANGETHEMALLTO3)UNITS 'IVEYOURlNALANSWERTOTHECORRECTNUMBEROFSIGNIlCANTlGURESANDMAKESUREYOUHAVEINCLUDEDTHECORRECTUNIT
"EFOREYOUMOVEONASKYOURSELFWHETHERTHEANSWERYOUHAVEGIVENISSENSIBLEANDREALISTIC
.PNFOUVNBOEJNQVMTF4HEMOMENTUMOFABODYISAPROPERTYTHATDEPENDSONITSMASSANDITSVELOCITY,INEARMOMENTUMISDElNEDASTHEPRODUCTOFMASSANDVELOCITY,INEARSIMPLYMEANSINASTRAIGHTLINE
.OTETHATTHEDElNITIONREFERSTOVELOCITYNOTSPEEDMOMENTUMMASSVELOCITY
PMV
P ISTHEMOMENTUMMEASUREDINKGMS
M ISTHEMASSMEASUREDINKGV ISTHEVELOCITYMEASUREDINMS
'IVENTHATMASSUSUALLYREMAINSCONSTANTTHISISOFTENWRITTENPM$V
8PSLFEFYBNQMF
&INDTHEMOMENTUMOFACAROFMASSKGTRAVELLINGATACONSTANTVELOCITYOFKMH
4PMVUJPOKMH?????????MS
MS
PMV PKGMS
PKGMS
5PWJFXBTJNVMBUJPOPGGPSDFTJOPOFEJNFOTJPOo5IFSBNQWJTJU)FJOFNBOODPVLIPUMJOLTFOUFSUIFFYQSFTTDPEF4BOEDMJDLPOUIF8FCMJOL "USBJOPGNBTTLHJTUSBWFMMJOHBUNTXIFOUIFCSBLFTBSFBQQMJFEBOEJU
EFDFMFSBUFTTUFBEJMZ5IFUSBJOUSBWFMTBEJTUBODFPGNCFGPSFDPNJOHUPBIBMUB $BMDVMBUFUIFEFDFMFSBUJPOPGUIFUSBJOC 'JOEUIFBWFSBHFCSBLJOHGPSDF
"MBSHFIFMJVNCBMMPPOJTBUUBDIFEUPUIFHSPVOECZUXPmYJOHSPQFT&BDISPQFNBLFTBOBOHMFPGXJUIUIFHSPVOE5IFSFJTBGPSDF'WFSUJDBMMZVQXBSETPG/5IFUPUBMNBTTPGUIFCBMMPPOBOEJUTCBTLFUJTLHB 4UBUFUIFNBHOJUVEFPGUIFSFTVMUBOUGPSDFXIFOJUJTBUUBDIFEUPUIFHSPVOEC $BMDVMBUFUIFUFOTJPOJOFJUIFSPGUIFmYJOHSPQFTD 5IFmYJOHSPQFTBSFSFMFBTFEBOEUIFCBMMPPOBDDFMFSBUFTVQXBSET$BMDVMBUFUIFNBHOJUVEF
PGUIJTJOJUJBMBDDFMFSBUJPOE 5IFCBMMPPOSFBDIFTBUFSNJOBMWFMPDJUZTBGUFSUBLFP5IFVQXBSEGPSDF'SFNBJOT
DPOTUBOU%FTDSJCFIPXUIFNBHOJUVEFPGUIFBJSSFTJTUBODFPOUIFCBMMPPOWBSJFTEVSJOHUIFmSTUTPGJUTnJHIU
&YFSDJTFT
&YBNJOFSTIJOU 5IFWFMPDJUZJTOPUHJWFOJO4*VOJUTTPXFNVTUDPOWFSU
.PNFOUVNJTBWFDUPSRVBOUJUZ
-
.FDIBOJDT
$POTFSWBUJPOPGNPNFOUVN
-OMENTUMISIMPORTANTFORSOLVINGPROBLEMSINVOLVINGCOLLISIONSANDEXPLOSIONS7ESAYTHATMOMENTUMISCONSERVEDINTHESEINTERACTIONS4HISMEANSTHATTHETOTALMOMENTUMBEFOREANEVENTISEQUALTOTHETOTALMOMENTUMAFTERTHEEVENT
8PSLFEFYBNQMF
!BALLOFMASSKGISDROPPEDANDHITSTHEmOORWITHAVELOCITYOFMS)TBOUNCESBACKWITHAVELOCITYOFMS&INDTHECHANGEOFMOMENTUM
4PMVUJPO4HECHANGEINMOMENTUMlNALMOMENTUMINITIALMOMENTUM
$PMVMU
$PKGMSKGMS
$PKGMSKGMS
$PKGMS
4HISIDEACANBEEXPRESSEDASAPROBLEMSOLVINGEQUATIONINDIFFERENTWAYS
MVMV
MVMVMV
MVMVMV
4HEVERSIONWECHOOSEDEPENDSONTHEPROBLEMWHETHERITISACOLLISIONOREXPLOSIONANDWHETHERTHETWOBODIESSTICKTOGETHERAFTERTHECOLLISION
'JHVSF
'JHVSF 7FDUPSRVBESBOU
5IFMBXPGDPOTFSWBUJPOPGMJOFBSNPNFOUVNTUBUFTUIBUUIFUPUBMNPNFOUVNSFNBJOTDPOTUBOUJOBOZJOUFSBDUJPOQSPWJEJOHUIFSFJTOPFYUFSOBMGPSDF
'PSBWFDUPSQSPCMFNMJLFUIJTXFNVTUVTFBDPOWFOUJPOGPSEJSFDUJPO6TVBMMZXFUBLFUIFEJSFDUJPOTVQPSUPUIFSJHIUBTQPTJUJWFBOEEPXOPSUPUIFMFGUBTOFHBUJWF
-
8PSLFEFYBNQMF
!CAROFMASSKGTRAVELLINGATMSCOLLIDESHEADONWITHATRUCKOFMASSKG)FBOTHVEHICLESCOMETORESTIMMEDIATELYAFTERTHECRASHWHATWASTHEVELOCITYOFTHETRUCK
4PMVUJPOMVMVMV
KGMSKGVKGMS KGMSV V
?????? VMS
*NQVMTF4HECHANGEINMOMENTUMOFABODYISALSOKNOWNASTHEIMPULSE)MPULSEISDElNEDASTHEPRODUCTOFAFORCEANDTHETIMEDURINGWHICHITACTS
)MPULSE&$TM$V
4HEUNITFORIMPULSEIS.SANDTHISCANALSOBEUSEDASAUNITOFMOMENTUM
)MAGINEWHATHAPPENSIFAGIRLKICKSAFOOTBALLANDABOYKICKSABRICKOFTHESAMEMASSSAYG"OTHTHEBALLANDTHEBRICKAREACCELERATEDFROMRESTTOAVELOCITYOFMS
4HEDIFFERENCEISTHATTHEBRICKISHARDERTHANTHEBALLSOTHEGIRLSFOOTISINCONTACTWITHTHEBALLFORALONGERTIMEINTERVAL)FTHEGIRLSFOOTISINCONTACTWITHTHEBALLFORSTHENWECANCALCULATETHEFORCEEXERTEDLIKETHIS &$TM$V
&SKGMS
& ???
&.
)FTHEBOYSFOOTISINCONTACTWITHTHEBRICKFORSTHENTHEFORCEWOULDBEMUCHGREATER &$TM$V
&SKGMS
& ????
&.
&YBNJOFSTIJOU 4JODFUIFCPEJFTJOWPMWFETUJDLUPHFUIFSBGUFSUIFDPMMJTJPOVTFUIJTFRVBUJPO
&YBNJOFSTIJOU 5IFmOBMWFMPDJUZWJT[FSPBOEXFXJMMUBLFUIFEJSFDUJPOPGNPUJPOPGUIFDBSBTQPTJUJWF
'JHVSF *UJTOPUBHPPEJEFBUPLJDLBCSJDL'PSUIFTBNFDIBOHFJONPNFOUVNJGUIFUJNFPGDPOUBDUJTTNBMMFSUIFGPSDFXJMMCFHSFBUFS8FTBZUIBUUIFUJNFJTJOWFSTFMZQSPQPSUJPOBMUPUIFGPSDF
-
.FDIBOJDT
2EMEMBERTHAT.EWTONSSECONDLAWIS &MA!SWESAWEARLIERINTHECHAPTER A$V???$T
3UBSTITUTINGFORAINTHESECONDLAWGIVES
&M$V????$T
+NOWINGTHAT M$VP7ECANREWRITE.EWTONSSECONDLAWAS
&$P???$T
.OTICETHISCANBEREARRANGEDTOGIVETHEIMPULSEEQUATION &$T$PM$V)NWORDSWESAYTHATIMPULSEISTHECHANGEINMOMENTUMWHILEFORCEISTHERATEOFCHANGEOFMOMENTUM
/FXUPOTUIJSEMBXPGNPUJPO.EWTONSTHIRDLAWSTATESTHATIFBODY!EXERTSAFORCEONBODY"THENBODY"EXERTSANEQUALANDOPPOSITEFORCEONBODY!
#LEARLYTHEREMUSTALWAYSBETWODIFFERENTBODIESINVOLVEDINTHETHIRDLAW!NEASYWAYTODEMONSTRATETHISISWITHTWOSPRINGBALANCES
4HEREAREEXAMPLESOFTHIRDLAWPAIRSOFFORCESALLAROUNDUSHEREAREJUSTAFEWEXAMPLES 7HENYOUKICKAFOOTBALLORABRICKYOURFOOTEXERTSAFORCEONTHEBALLANDTHEBALLEXERTSANEQUALANDOPPOSITEFORCEONYOURFOOT4HEBALLMOVESHOPEFULLYTOWARDSTHEGOALANDYOUCANFEELTHEFORCEONYOURFOOT
7HENACANNONISlREDTHECANNONBALLMOVESFORWARDBUTTHEBALLEXERTSANEQUALANDOPPOSITEFORCEONTHECANNONSOITRECOILS
)FYOUPLAYTENNISYOURRACKETEXERTSAFORCEONTHEBALLANDTHEBALLEXERTSANEQUALANDOPPOSITEFORCEONYOURRACKET
4HE%ARTHEXERTSAFORCEONTHE-OONANDTHE-OONEXERTSANEQUALANDOPPOSITEFORCEONTHE%ARTH7EKNOWTHE%ARTHPULLSTHE-OONBECAUSEITREMAINSINORBITAROUNDUSANDWEKNOWTHE-OONPULLSONTHE%ARTHBECAUSEITINmUENCESTHETIDESINTHEOCEANS
5IFQIPUPTIPXTUIFSBDLFUTUSJOHTCFJOHTUSFUDIFEBOEEFGPSNFE
'JHVSF 5XPTQSJOHCBMBODFT5IFSFBEJOHPO"XJMMBMXBZTCFUIFTBNFBTUIFSFBEJOHPO#
&EVDBUJPOJTOPUUIFTBNFJOFWFSZDPVOUSZBOEEFQFOETPOEJFSFODFTJOMBOHVBHFQPMJUJDTDVMUVSFBOESFMJHJPOBNPOHNBOZPUIFSUIJOHT*OBMMFEVDBUJPOBMTZTUFNTUIFSFJTIPXFWFSBHSFFNFOUJOUIFWBMJEJUZPGUIFJEFBTCFIJOE/FXUPOTUISFFMBXTPGNPUJPO
-
1IZTJDBMXPSL3OMETIMESINSCIENCEWEUSEWORDSINADIFFERENTSENSETOTHEWAYTHEYARENORMALLYUSEDINTHE%NGLISHLANGUAGE!LTHOUGHAWORDOFTENLOOKSSOUNDSANDISSPELTTHESAMEITSOMETIMESHASADIFFERENTMEANINGINPHYSICS/NEEXAMPLEISTHEDElNITIONOFWORKINTHESTUDYOFPHYSICS
7ORKISDElNEDASTHEFORCETIMESTHEDISTANCEMOVEDINTHEDIRECTIONOFTHEFORCE
4HISMEANSTHATIFTHEREISNOMOVEMENTINTHEDIRECTIONOFTHEFORCETHENNOWORKISDONE)FYOURCARISSTUCKINTHEMUDORTHESANDYOUCANPUSHITASHARDASYOULIKEANDGETCOMPLETELYEXHAUSTEDBUTIFYOUDONOTMOVEITTHENYOUHAVEDONENOWORKONTHECAR
7ORKISASCALARQUANTITYANDISMEASUREDINJOULES)TISINDEPENDENTOFTHEMASSOFTHEBODYBEINGMOVEDANDTHEPATHTAKEN)TONLYDEPENDSONTHEMAGNITUDEOFTHEFORCEANDTHEDISTANCEMOVEDINTHEDIRECTIONOFTHEFORCE
4HEEQUATIONFORWORKIS7&SCOSU
7 ISTHEWORKMEASUREDIN*& ISTHEFORCEMEASUREDIN.S ISTHEDISTANCEMOVEDMEASUREDINMU ISTHEANGLEBETWEENTHEAPPLIEDFORCEANDTHEDIRECTIONOFMOTION
8PSLFEFYBNQMF
!FORCEOF.ACTINGATTOTHEHORIZONTALPULLSABLOCKOFMASSKGADISTANCEOFMACROSSASMOOTHSURFACE(OWMUCHWORKISDONE
8PSLFOFSHZBOEQPXFS
ii>ii "i>i>L iiii`iL>V>vViLii}>
vVi`>Vii}> -iLi}i`iL>vVi{ "i>i>LiVii}x "i>i>LV>}i}>>>i>ii} ->iiVivVi>vii} `vviivvii}>``iVLii>ivi
>v>vii}viv>in }Liiii>V>`i>VV iwii iwi>`>iVVivivwViV -iLi}]ii}>`i
'JHVSF 8PSLEPOFQVMMJOHBCMPDL
-
.FDIBOJDT
4PMVUJPO4HESIZEOFTHEMASSMAKESNODIFFERENCETOTHEWORKDONE)FTHESURFACEISSMOOTHTHEREISNOFRICTIONSOTHERESULTANTFORCEINTHEXDIRECTIONISTHEXCOMPONENTOFTHE.FORCE3INCETHEXCOMPONENTISADJACENTTOANGLE COSX??? XCOS. 7&SCOSU 7COS.M 7*
7HENASERIESOFWEIGHTSAREHUNGONASPRINGCAUSINGITTOSTRETCHWORKISDONE4HEFORCEISTHEWEIGHTANDTHEDISTANCEMOVEDISTHEEXTENSIONOFTHESPRING!SLONGASTHESPRINGISNOTSTRETCHEDBEYONDACERTAINPOINTCALLEDTHEELASTICLIMITTHENTHEFORCEISDIRECTLYPROPORTIONALTOTHEEXTENSION!GRAPHOFFORCEAGAINSTEXTENSIONISASTRAIGHTLINETHROUGHTHEORIGIN
4HEWORKDONEBYTHISNONCONSTANTFORCEISFOUNDFROMTHEAREAUNDERTHEGRAPH
WORKDONE?X& 7?XKX 7?KX
)NPHYSICSENERGYANDWORKAREVERYCLOSELYLINKEDINSOMESENSESTHEYARETHESAMETHING"OTHARESCALARQUANTITIESANDBOTHAREMEASUREDINJOULES)FYOUHAVEENERGYYOUCANDOWORK
)NGENERALWESAYTHATWORKDONEENERGYCONVERTED
7$%7HENWORKISDONEENERGYISCONVERTEDORCHANGEDTODIFFERENTTYPES)NTHECASEOFTHESPRINGBEINGSTRETCHEDTHEWORKDONEINCREASESTHEELASTICPOTENTIALENERGYOFTHESPRING
ELASTICPOTENTIALENERGY?KX
,JOFUJDFOFSHZ+INETICENERGY%KISTHEENERGYABODYHASBECAUSEITISMOVING4HEWORDKINETICREFERSTOMOTION)MAGINEABLOCKOFMASSMACCELERATEDFROMRESTBYARESULTANTFORCE&!FTERTRAVELLINGADISTANCESTHEMASSHASANACCELERATIONA 4HEWORKDONE7&S BUTFROM.EWTONSSECONDLAW&MA SOTHAT7MASSUBSTITUTINGFOR& &ROMTHESUVATEQUATIONSVUAS SINCEUTHENVAS AS?V
3INCE7MAS$% ANDTHEENERGYCONVERTEDISKINETICENERGY KINETICENERGYMASM?V
%K?MV
'JHVSF "HSBQIPGGPSDFBHBJOTUFYUFOTJPOGPSBTUSFUDIFETQSJOH
5IFHSBEJFOUL'@Y
'LYLJTUIFTQSJOHDPOTUBOU
&OFSHZJTEFmOFEBTUIFBCJMJUZUPEPXPSL
&YBNJOFSTIJOU 4PNFUJNFTJOBOFYBNZPVXJMMCFBTLFEUPEFSJWFBOFRVBUJPOTVDIBTUIFLJOFUJDFOFSHZEFSJWBUJPOIFSF.BOZTUVEFOUTmOEUIJTEJDVMUBOEUIFSFJTSFBMMZPOMZPOFXBZUPJNQSPWFZPVSTLJMMT:PVTJNQMZIBWFUPQSBDUJTFBOESFQFBUUIFNXJUIUIFCPPLDMPTFEBOEUIFOQSBDUJTFTPNFNPSF
&YBNJOFSTIJOU ,JOFUJDFOFSHZJTPGUFOBCCSFWJBUFEUP,&PS&L
-
8PSLFEFYBNQMF
&INDTHEKINETICENERGYOFATRUCKOFMASSKGTRAVELLINGATASPEEDOFMS
4PMVUJPO %K
?MV
%K?KGMS
%K*
1PUFOUJBMFOFSHZ'RAVITATIONALPOTENTIALENERGY%PISTHEENERGYABODYHASBECAUSEOFITSPOSITIONINPARTICULARITSVERTICALHEIGHTRELATIVETOAGIVENPOINT
.OWIMAGINEAMASSMLIFTEDAVERTICALHEIGHTH(EREWORKISBEINGDONEAGAINSTTHEPULLOFGRAVITY
THEWORKDONE7&S
HERETHEFORCEISTHEWEIGHTMG
ANDTHEDISTANCEISTHEHEIGHTH
SOTHEWORK7MGH
4HEWORKISINCREASINGTHEGRAVITATIONALPOTENTIALENERGYOFTHEMASS
7$%P $%PMG$H
8PSLFEFYBNQMF
4HE,EANING4OWEROF0ISAISMHIGHA&INDTHEGRAVITATIONALPOTENTIALENERGYOFAKGCANNONBALLATTHETOPOFTHETOWER
B)FTHECANNONBALLISDROPPEDFROMTHETOPOFTHETOWERWHATWILLBEITSSPEEDWHENITHITSTHEGROUND.EGLECTAIRRESISTANCEANDTAKEGMS
4PMVUJPOA $%PMG$H
$%PKGMSM
$%P*
B $%P%K
*?MV
*?KGV
V????
VMS
&YBNJOFSTIJOU 8FDPVMETPMWFUIJTVTJOHUIFTVWBUFRVBUJPOTCVUJGBJSSFTJTUBODFJTOFHMFDUFEBMMUIFHSBWJUBUJPOBMQPUFOUJBMFOFSHZXJMMCFDIBOHFEUPLJOFUJDFOFSHZ
5IFSFJTBGBNPVTTUPSZBCPVU(BMJMFPESPQQJOHDBOOPOCBMMTPGEJFSFOUXFJHIUTGSPNUIFUPQPGUIF-FBOJOH5PXFSPG1JTBUPEFNPOTUSBUFUIBUIFBWZPCKFDUTBOEMJHIUFSPCKFDUTGBMMBUUIFTBNFSBUF5IJTTUPSZNBZPSNBZOPUCFUSVF
5PWJFXBNPSFSFDFOUWFSTJPOPG(BMJMFPTFYQFSJNFOUWJTJU)FJOFNBOODPVLIPUMJOLTFOUFSUIFFYQSFTTDPEF4BOEDMJDLPO8FCMJOL
&YBNJOFSTIJOU 1PUFOUJBMFOFSHZJTPGUFOBCCSFWJBUFEUP1&PS&Q
-
.FDIBOJDT
$POTFSWBUJPOPGFOFSHZ4HEIDEAOFONEFORMOFENERGYCHANGINGINTOANOTHERISSUMMARISEDINAVERYIMPORTANTPRINCIPLEORLAWOFPHYSICSTHEPRINCIPLECONSERVATIONOFENERGY
7EUSUALLYCLASSIFYENERGYASTHEFOLLOWINGFORMS
+INETICENERGY
'RAVITATIONALPOTENTIALENERGY
%LASTICPOTENTIALENERGY
4HERMALORHEATENERGY
,IGHTENERGY
3OUNDENERGY
#HEMICALENERGY
%LECTRICALENERGY
-AGNETICENERGY
.UCLEARENERGY
)NANYGIVENTRANSFORMATIONSOMEOFTHEENERGYISALMOSTINEVITABLYCHANGEDTOHEAT
4HEREAREENERGYCHANGESHAPPENINGALLAROUNDUSALLOFTHETIME(EREAREAFEWEXAMPLESTOTHINKABOUT
!BOUNCINGBALL4HEGRAVITATIONALPOTENTIALENERGYISCHANGEDTOKINETICENERGYANDBACKAGAINWITHEACHBOUNCE7HILETHEBALLISINCONTACTWITHTHEGROUNDSOMEENERGYISSTOREDASELASTICPOTENTIALENERGY%VENTUALLYTHEBALLCOMESTORESTANDALLTHEENERGYHASBECOMELOWGRADEHEAT
!NAEROPLANETAKINGOFF!SFUELISBURNEDINTHEENGINESCHEMICALENERGYISCONVERTEDTOHEATLIGHTANDSOUND4HEPLANEACCELERATESDOWNTHERUNWAYANDITSKINETICENERGYINCREASES4HEREWILLBEHEATENERGYDUETOFRICTIONBETWEENTHETYRESANDTHERUNWAY!STHEPLANETAKESOFFANDCLIMBSINTOTHESKYITSGRAVITATIONALPOTENTIALENERGYINCREASES
!NUCLEARPOWERSTATION.UCLEARENERGYFROMTHEURANIUMFUELCHANGESTOTHERMALENERGYTHATISUSEDTOBOILWATER4HEKINETICENERGYOFTHESTEAMMOLECULESDRIVESTURBINESANDASTHEKINETICENERGYOFTHETURBINESINCREASESITINTERACTSWITHMAGNETICENERGYTOGIVEELECTRICALENERGY
!LAPTOPCOMPUTER4HEELECTRICALENERGYFORMTHEMAINSORCHEMICALENERGYFROMTHEBATTERYISCHANGEDTOLIGHTONTHESCREENANDSOUNDTHROUGHTHESPEAKERS4HEREISKINETICENERGYINTHEFANMAGNETICENERGYINTHEMOTORSANDPLENTYOFHEATISGENERATED
!HUMANBODY-OSTOFTHECHEMICALENERGYFROMOURFOODISCHANGEDTOHEATTOKEEPUSALIVE3OMEISCHANGEDTOKINETICENERGYASWEMOVEORGRAVITATIONALPOTENTIALENERGYIFWECLIMBSTAIRS4HEREISALSOELASTICPOTENTIALENERGYINOURMUSCLESSOUNDENERGYWHENWETALKANDELECTRICALENERGYINOURNERVESANDBRAIN
5IFQSJODJQMFPGDPOTFSWBUJPOPGFOFSHZTUBUFTUIBUFOFSHZDBOOPUCFDSFBUFEPSEFTUSPZFEPOMZDIBOHFEGSPNPOFGPSNUPBOPUIFS
"MMPGUIFFOFSHZPO&BSUIIBTDPNFPSJHJOBMMZGSPNUIF4VO*TUIJTTUBUFNFOUUSVF )PXEPZPVLOPX
&YBNJOFSTIJOU *OBOFYBNEPOPUTBZUIBUFOFSHZJTAMPTUCVUJOTUFBETBZJUIBTCFFODIBOHFEPSDPOWFSUFEUPPUIFSGPSNT
-
4HEREISAUSEFULRELATIONSHIPBETWEENKINETICENERGYANDMOMENTUMWHICHWILLHELPYOUTOSOLVENUMERICALPROBLEMS %K
?MVMV
????
-ULTIPLYINGTOPANDBOTTOMBYMGIVES %KMV?????M
-OMENTUM PMV
3QUARINGGIVES PMV
&INALLYSUBSTITUTINGINTHEABOVEEQUATION %KP???M
$PMMJTJPOT7HENWELOOKEDATMOMENTUMEARLIERWEMENTIONEDTHATITWASIMPORTANTFORCOLLISIONS.OWWEMUSTDISTINGUISHBETWEENTWODIFFERENTTYPESOFCOLLISIONS7HENELASTICCOLLISIONSOCCURMOMENTUMISCONSERVEDANDKINETICENERGYISALSOCONSERVED4HISMEANSTHATNOENERGYISCHANGEDTOHEATINANELASTICCOLLISION)NFACTTHEREISREALLYNOSUCHTHINGASAPERFECTLYELASTICCOLLISIONALTHOUGHOFCOURSESOMECOLLISIONSAREMOREELASTICTHANOTHERS!FTERANELASTICCOLLISIONTHEBODIESALWAYSMOVESEPARATELY
$URINGINELASTICCOLLISIONSMOMENTUMISCONSERVEDBUTKINETICENERGYISNOTCONSERVED)NOTHERWORDSSOMEOFTHEKINETICENERGYISCONVERTEDTOHEATANDOTHERFORMSOFENERGY!FTERANINELASTICCOLLISIONTHEBODIESMAYSTICKTOGETHER
+INETICENERGYCANSOMETIMESINCREASEFOREXAMPLEIFTHEREISANINPUTOFCHEMICALENERGYFROMAFUELOREXPLOSIVEMATERIAL
1PXFS0OWERISAMEASUREOFHOWFASTWORKISDONEORHOWQUICKLYENERGYISCONVERTED0OWERISASCALARQUANTITY
POWERENERGYORWORK?????????????TIME
)FTHEENERGYORWORKISMEASUREDINJOULESANDTHETIMEINSECONDSTHENTHEPOWERWILLBEMEASUREDINWATTS7
7*S
!7LIGHTBULBCONVERTSELECTRICALENERGYTOHEATANDLIGHTATTHERATEOF*EVERYSECOND
3OMETIMESYOUWILLSEEPOWERGIVENINK7OREVEN-7
1BSUJDMFQIZTJDJTUTCFMJFWFTPNFDPMMJTJPOTCFUXFFOTVCBUPNJDQBSUJDMFTBSFQFSGFDUMZFMBTUJD
&YBNJOFSTIJOU *OFYBNRVFTUJPOTBCPVUDPMMJTJPOTBOEFYQMPTJPOTFTQFDJBMMZNVMUJQMFDIPJDFBTTVNFUIBUNPNFOUVNJTBMXBZTDPOTFSWFE
#JMMJBSECBMMDPMMJTJPOTBSFIJHIMZFMBTUJDCFDBVTFUIFCBMMTBSFTPIBSE*OBQFSGFDUMZFMBTUJDDPMMJTJPOPOFCBMMDPVMETUPQBOEUIFPUIFSNPWFPXJUIUIFTBNFTQFFEBTUIFmSTU*OQSBDUJDFUIJTDBOOPUIBQQFOCFDBVTFTPNFPGUIFLJOFUJDFOFSHZDIBOHFTUPIFBUBOETPVOE
1PXFSJTEFmOFEBTUIFSBUFPGEPJOHXPSL
-
.FDIBOJDT
8PSLFEFYBNQMF
#ALCULATETHEPOWEROFAWORKERINASUPERMARKETWHOSTACKSSHELVESMHIGHWITHCARTONSOFORANGEJUICEEACHOFMASSKGATTHERATEOFCARTONSPERMINUTE
4PMVUJPO
POWERWORK?????TIME
POWER&S??T
POWER.M?????????????????S
POWER7
4HEREISANOTHERWAYTOCALCULATEPOWER
POWER&S??T &S?T&V
)NOTHERWORDSPOWERISEQUALTOFORCETIMESVELOCITYPROVIDINGTHEVELOCITYISCONSTANT
&GmDJFODZ%FlCIENCYISARATIOOFHOWMUCHWORKENERGYORPOWERWEGETOUTOFASYSTEMCOMPAREDTOHOWMUCHISPUTIN
EFlCIENCYUSEFULOUTPUT???????????TOTALINPUT
3INCEITISARATIOTHEREISNOUNITANDWECANEXPRESSEFlCIENCYASAPERCENTAGEBYMULTIPLYINGBY.OREALMACHINEORSYSTEMCANEVERBEEFlCIENTBECAUSETHEREWILLALWAYSBESOMEENERGYCHANGEDTOHEATDUETOFRICTIONAIRRESISTANCEOROTHERCAUSES
+BNFT8BUUXBTB4DPUTNBOXIPTFXPSLPOUIFTUFBNFOHJOFXBTBNBKPSJOnVFODFPOUIF*OEVTUSJBM3FWPMVUJPO
5IJTFOHSBWJOHTIPXTBTUFBNFOHJOFEFTJHOFECZ+BNFT8BUU
'SPNUIFEBZTPG"SJTUPUMFUISPVHIUIFUJNFPG(BMJMFP/FXUPO8BUUBOE+PVMFSJHIUVQVOUJM&JOTUFJOJOUIFFBSMZUIDFOUVSZTDJFOUJTUTXPSLFENPTUMZBMPOF/PXBEBZTTDJFOUJTUTPGUFOXPSLJONVMUJOBUJPOBMUFBNT5IFTFUFBNTNBZNBLFJNQPSUBOUEJTDPWFSJFTJOmFMETTVDIBTQBSUJDMFQIZTJDTPSXPSLPOTPMWJOHUIFHSFBUHMPCBMQSPCMFNTMJLF)*7"JETBOEHMPCBMXBSNJOH
0OFPGUIFNPTUFDJFOUEFWJDFTJTUIFFMFDUSJDUSBOTGPSNFS5IFZDBOCFPWFSFDJFOUCFDBVTFTJODFUIFSFBSFOPNPWJOHQBSUTUIFSFBSFOPFOFSHZMPTTFTEVFUPGSJDUJPOPSBJSSFTJTUBODF
-
&YBNJOFSTIJOU 5IFFDJFODZJTXIJDIFRVBMT
8PSLFEFYBNQMF
!CARENGINEHASANEFlCIENCYOFANDPRODUCESANAVERAGEOFK*OFUSEFULWORKPERSECOND(OWMUCHENERGYISCONVERTEDTOHEATPERSECOND
4PMVUJPO EFlCIENCY
USEFULOUTPUT???????????TOTALINPUT
*?????????TOTALINPUT
TOTALINPUT?????? *
)FTHETOTALINPUTISK*ANDK*OFUSEFULWORKAREOBTAINEDTHENTHEHEATPRODUCEDMUSTBE K*K*K*
5IJTRVFTUJPOJTBCPVUESJWJOHBNFUBMCBSJOUPUIFHSPVOE-BSHFNFUBMCBSTDBOCFESJWFOJOUPUIFHSPVOEVTJOHBIFBWZGBMMJOHPCKFDU
*OUIFTJUVBUJPOTIPXOUIFPCKFDUIBTBNBTTLHBOEUIFNFUBMCBSIBTBNBTTPGLH5IFPCKFDUTUSJLFTUIFCBSBUBTQFFEPGNT*UDPNFTUPSFTUPOUIFCBSXJUIPVUCPVODJOH"TBSFTVMUPGUIFDPMMJTJPOUIFCBSJTESJWFOJOUPUIFHSPVOEUPBEFQUIPGNB %FUFSNJOFUIFTQFFEPGUIFCBSJNNFEJBUFMZBGUFSUIFPCKFDUTUSJLFTJUC %FUFSNJOFUIFBWFSBHFGSJDUJPOBMGPSDFFYFSUFECZUIFHSPVOEPOUIFCBS
5IJTRVFTUJPOJTBCPVUFTUJNBUJOHFOFSHZDIBOHFTGPSBOFTDBMBUPSNPWJOHTUBJSDBTF5IFEJBHSBNCFMPXSFQSFTFOUTBOFTDBMBUPS1FPQMFTUFQPOUPJUBUQPJOU"BOETUFQPBUQPJOU#B 5IFFTDBMBUPSJTNMPOHBOENBLFTBOBOHMF
PGXJUIUIFIPSJ[POUBM"UGVMMDBQBDJUZQFPQMFTUFQPOBUQPJOU"BOETUFQPBUQPJOU#FWFSZNJOVUF
J $BMDVMBUFUIFQPUFOUJBMFOFSHZHBJOFE CZBQFSTPOPGXFJHIU/JO NPWJOHGSPN"UP# JJ &TUJNBUFUIFFOFSHZTVQQMJFECZ UIFFTDBMBUPSNPUPSUPUIFQFPQMFFWFSZNJOVUFXIFOUIFFTDBMBUPSJTXPSLJOHBUGVMM DBQBDJUZ JJJ 4UBUFPOFBTTVNQUJPOUIBUZPVIBWFNBEFUPPCUBJOZPVSBOTXFSUPJJ
5IFFTDBMBUPSJTESJWFOCZBOFMFDUSJDNPUPSUIBUIBTBOFDJFODZPGC 6TJOHZPVSBOTXFSUPBJJ
DBMDVMBUFUIFNJOJNVNJOQVUQPXFSSFRVJSFECZUIFNPUPSUP
ESJWFUIFFTDBMBUPS
&YFSDJTFT
&YBNJOFSTIJOU L++
-
.FDIBOJDT
.PUJPOJOBDJSDMF)FABODYISTRAVELLINGINACIRCULARPATHTHENITISCONSTANTLYCHANGINGDIRECTIONATNOPOINTISITTRAVELLINGINASTRAIGHTLINE%VENIFITISTRAVELLINGATCONSTANTORUNIFORMSPEEDABODYTHATISMOVINGROUNDACIRCLEMUSTBECHANGINGVELOCITY4HISISBECAUSEVELOCITYISAVECTORANDACHANGEINVELOCITYCANBEACHANGEOFSPEEDORDIRECTION
)FTHEVELOCITYOFTHEBODYISCHANGINGTHENBYDElNITIONTHEBODYMUSTBEACCELERATING4HISTYPEOFACCELERATIONISCALLEDCENTRIPETALACCELERATION4HEWORDCENTRIPETALMEANSCENTRESEEKINGANDTHEDIRECTIONOFTHEACCELERATIONISALWAYSTOWARDSTHECENTREOFTHECIRCLE
4HEEQUATIONFORCENTRIPETALACCELERATIONISAV
??R
A CENTRIPETALACCELERATION MS
V SPEED MS
R RADIUSOFTHECIRCLE M
3INCETHEBODYMOVESWITHCONSTANTSPEEDWECANUSETHESIMPLEEQUATION
SPEEDDISTANCE???????TIME
)FTHEBODYGOESROUNDTHEENTIRECIRCLEONCETHENTHISCANBEWRITTEN
VPR????4
PRISTHECIRCUMFERENCEOFACIRCLE4ISTHEPERIODORTIMEPERIODANDISTHETIMETAKENFORONECOMPLETEREVOLUTION
)FWESQUARETHISEQUATIONITBECOMES
VPR?????4
6OJGPSNDJSDVMBSNPUJPO{
ii>ii{ >>iV`>}>>i>i>VVii>v>>Vi
}V>ii`>VVi`iVi`>`iViiviVVi
{ iiivVii>>VVii>{ `ivivVi`V}VV>>>{{ -iLi}VV>
'JHVSF "QBSUJDMFNPWJOHJOBDJSDMF5IFJOTUBOUBOFPVTWFMPDJUZWJTUIFUBOHFOUUPUIFDJSDMF5IFEJSFDUJPOPGUIFDFOUSJQFUBMGPSDFXJMMCFJOUIFTBNFEJSFDUJPOBTUIFDFOUSJQFUBMBDDFMFSBUJPO
-
3UBSTITUTINGFORVINTHECENTRIPETALACCELERATIONANDCANCELLINGONEOFTHERTERMSGIVES
APR????4
)FABODYISACCELERATINGWEKNOWFROM.EWTONSSECONDLAWTHATITMUSTBEBEINGACTEDONBYANUNBALANCEDFORCE4HEFORCECAUSINGTHECENTRIPETALACCELERATIONISCALLEDTHECENTRIPETALFORCE4HEDIRECTIONOFTHECENTRIPETALFORCEISALSOTOWARDSTHECENTREOFTHECIRCLETHISISLOGICALBECAUSEBODIESACCELERATEINTHEDIRECTIONOFTHERESULTANTFORCE
4OUNDERSTANDTHISTHINKABOUTACARDRIVINGROUNDACIRCULARTRACK)FTHEFRICTIONISTOOLOWFOREXAMPLEIFTHECARSKIDSONAPATCHOFICETHENTHECARWILLmYOFFATATANGENT
4HEFRICTIONISTHEFORCETHATTURNSTHECARINTOTHEBENDSANDITISDIRECTEDTOWARDSTHECENTEROFTHECIRCLE4HEFRICTIONHEREISTHECENTRIPETALFORCE
)NTHESAMEWAYWHENTHE-OONORBITSTHE%ARTHTHEREISACENTRIPETALFORCEFROMTHE-OONTOWARDSTHECENTREOFTHE%ARTH)NTHISCASETHEFORCEISGRAVITATIONAL)FYOUWHIRLARUBBERBUNGORASTONETIEDTOAPIECEOFSTRINGAROUNDYOURHEADTHENTHECENTRIPETALFORCETHATKEEPSTHESTONETURNINGISTHETENSIONINTHESTRING)FTHESTRINGBREAKSANDTHETENSIONBECOMESZEROTHEBUNGORSTONEmIESOFFINASTRAIGHTLINE
)FTHEEQUATIONFORCENTRIPETALACCELERATIONIS
AV??R
ANDWEKNOWTHAT &MA
WECANSUBSTITUTEFORATOGIVE
&MV????R
4HISISTHEEQUATIONFORCENTRIPETALFORCE
8PSLFEFYBNQMF
!BALLOFMASSGTIEDTOTHEENDOFAPIECEOFSTRINGISWHIRLEDBYASTUDENTINAHORIZONTALCIRCLEOFRADIUSMATCONSTANTSPEED4HEBALLMAKESREVOLUTIONSPERSECONDA &INDTHEACCELERATIONOFTHEBALLB &INDTHEFORCETHATTHESTUDENTMUSTEXERTONTHESTRING
5IFDFOUSJQFUBMGPSDFJTTJNQMZBOFYUSBOBNFXFHJWFUPBGPSDFUIBUJTBMSFBEZUIFSFJUJTOPUBOFXPSEJFSFOUGPSDF
'JHVSF 5IFSFNVTUCFGSJDUJPOCFUXFFOUIFDBSUZSFTBOEUIFSPBEUPLFFQUIFDBSHPJOHSPVOE
&YBNJOFSTIJOU *GZPVBSFBTLFEUPESBXUIFGPSDFPOBCPEZNPWJOHJOBDJSDMFBMXBZTESBXJUQPJOUJOHUPXBSETUIFDFOUSFPGUIFDJSDMF%POPUCFUFNQUFEUPBEEBOPUIFSBSSPXQPJOUJOHPVUXBSETUPCBMBODFUIFGPSDFTUIFZBSFOPUCBMBODFECFDBVTFUIFCPEZJTBDDFMFSBUJOH
-
.FDIBOJDT
4PMVUJPOA SPEEDDISTANCE???????TIME
VPM??????????????S
VMS
AV??R
AMS??????????M
MS
B &MV????R
KGMS????????????????????M
.
&YBNJOFSTIJOU 4JODFUIFCBMMJTNPWJOHJOBDJSDMFUIFBDDFMFSBUJPOXJMMCFDFOUSJQFUBM
"OBJSDSBGUBDDFMFSBUFTGSPNSFTUBMPOHBIPSJ[POUBMTUSBJHIUSVOXBZBOEUIFOUBLFTP5IFNBTTPGUIFBJSDSBGUJTLHB 5IFBWFSBHFSFTVMUBOUGPSDFPOUIFBJSDSBGUXIJMFUSBWFMMJOHBMPOHUIFSVOXBZJTL/5IF
TQFFEPGUIFBJSDSBGUKVTUBTJUMJGUTPJTNT&TUJNBUFUIFEJTUBODFUSBWFMMFEBMPOHUIFSVOXBZ
C 5IFBJSDSBGUDMJNCTUPBIFJHIUPGN$BMDVMBUFUIFQPUFOUJBMFOFSHZHBJOFEEVSJOHUIFDMJNC
D 8IFOBQQSPBDIJOHJUTEFTUJOBUJPOUIFQJMPUQVUTUIFBJSDSBGUJOUPBIPMEJOHQBUUFSO5IJTNFBOTUIFBJSDSBGUnJFTBUBDPOTUBOUTQFFEPGNTJOBIPSJ[POUBMDJSDMFBOEUIFSBEJVTPGUIFDJSDMFJTN'PSUIFBJSDSBGUJOUIFIPMEJOHQBUUFSO
J DBMDVMBUFUIFNBHOJUVEFPGUIFSFTVMUBOUGPSDFPOUIFBJSDSBGU JJ TUBUFUIFEJSFDUJPOPGUIFSFTVMUBOUGPSDF
"HFPTUBUJPOBSZTBUFMMJUFGPSFYBNQMFBDPNNVOJDBUJPOTTBUFMMJUFPSCJUTUIF&BSUIXJUIBUJNFQFSJPEPGIPVST*GUIFEJTUBODFGSPNUIFTBUFMMJUFUPUIFDFOUSFPGUIF&BSUIJTNDBMDVMBUFUIFBDDFMFSBUJPOPGUIFTBUFMMJUF
&YFSDJTFT
!BALLINITIALLYATRESTTAKESTIMETTOFALLTHROUGHAVERTICALDISTANCEH)FAIRRESISTANCEISIGNOREDTHETIMETAKENFORTHEBALLTOFALLFROMRESTTHROUGHAVERTICALDISTANCEHIS! T" T# T$ T
!RAINDROPFALLINGTHROUGHAIRREACHESATERMINALVELOCITYBEFOREHITTINGTHEGROUND!TTERMINALVELOCITYTHEFRICTIONALFORCEONTHERAINDROPIS! ZERO" LESSTHANTHEWEIGHTOFTHERAINDROP# GREATERTHANTHEWEIGHTOFTHERAINDROP$ EQUALTOTHEWEIGHTOFTHERAINDROP
1SBDUJDFRVFTUJPOT
-
!NATHLETERUNSROUNDACIRCULARTRACKATCONSTANTSPEED7HICHONEOFTHEFOLLOWINGGRAPHSBESTREPRESENTSTHEVARIATIONWITHTIMETOFTHEMAGNITUDEDOFTHEDISPLACEMENTOFTHEATHLETEFROMTHESTARTINGPOSITIONDURINGONELAPOFTHETRACK
! "
# $
!BALLISDROPPEDFROMRESTATTIMETONTOAHORIZONTALSURFACEFROMWHICHITREBOUNDS4HEGRAPHSHOWSTHEVARIATIONOFTIMETWITHSPEEDVOFTHEBALL
7HICHONEOFTHEFOLLOWINGBESTREPRESENTSTHEPOINTATWHICHTHEBALLJUSTLOSESCONTACTWITHTHESURFACEAFTERTHElRSTBOUNCE!"#$
*UANISSTANDINGONTHEPLATFORMATARAILWAYSTATION!TRAINPASSESTHROUGHTHESTATIONWITHSPEEDMSINTHEDIRECTIONSHOWNMEASUREDRELATIVETOTHEPLATFORM#ARMENISWALKINGALONGONEOFTHECARRIAGESOFTHETRAINWITHASPEEDOFMSMEASUREDRELATIVETOTHECARRIAGEINTHEDIRECTIONSHOWN6ELOCITYISMEASUREDASPOSITIVEINTHEDIRECTIONSHOWNONTHEDIAGRAM
4HEVELOCITYOF#ARMENRELATIVETO*UANIS! MS
" MS
# MS
$ MS
-
.FDIBOJDT
4HEGRAPHBELOWSHOWSTHEVARIATIONWITHTIMEOFTHEDISTANCEMOVEDBYACARALONGASTRAIGHTROAD$URINGWHICHTIMEINTERVALDOESTHECARHAVEITSGREATESTACCELERATION
4HEVARIATIONWITHTIMETOFTHESPEEDVOFACARMOVINGALONGASTRAIGHTROADISSHOWNBELOW
7HICHAREA33OR3ORCOMBINATIONOFAREASREPRESENTSTHETOTALDISTANCEMOVEDBYTHECARDURINGTHETIMETHATITSSPEEDISREDUCING! 3" 3# 33$ 333
7HENABODYISACCELERATINGTHERESULTANTFORCEACTINGONITISEQUALTOITS! CHANGEOFMOMENTUM" RATEOFCHANGEOFMOMENTUM# ACCELERATIONPERUNITOFMASS$ RATEOFCHANGEOFKINETICENERGY
!BALLOFMASSMTRAVELLINGINADIRECTIONATRIGHTANGLESTOAVERTICALWALLSTRIKESTHEWALLWITHASPEEDV)TREBOUNDSATRIGHTANGLESTOTHEWALLWITHASPEEDV4HEBALLISINCONTACTWITHTHEWALLFORATIME$T4HEMAGNITUDEOFTHEFORCETHATTHEBALLEXERTSONTHEWALLIS
! MVV?????????$T
$ MVV$T
" MVV?????????$T
# MVV$T
-
!SPHEREOFMASSMSTRIKESAVERTICALWALLANDBOUNCESOFFITASSHOWNBELOW
4HEMAGNITUDEOFTHEMOMENTUMOFTHESPHEREJUSTBEFOREIMPACTISP"ANDJUSTAFTERIMPACTISP!4HESPHEREISINCONTACTWITHTHEWALLFORTIMET4HEMAGNITUDEOFTHEAVERAGEFORCEEXERTEDBYTHEWALLONTHESPHEREIS
! P"P!????????T
" P"P!????????T
# P"P!????????MT
$ P"P!????????MT
!TRUCKCOLLIDESHEADONWITHALESSMASSIVECARMOVINGINTHEOPPOSITEDIRECTIONTOTHETRUCK$URINGTHECOLLISIONTHEAVERAGEFORCEEXERTEDBYTHETRUCKONTHECARIS&4ANDTHEAVERAGEFORCEEXERTEDBYTHECARONTHETRUCKIS7HICHONEOFTHEFOLLOWINGSTATEMENTSISCORRECT! &4WILLALWAYSBEGREATERINMAGNITUDETHAN" &4WILLALWAYSBEEQUALINMAGNITUDETO# &4WILLBEGREATERINMAGNITUDETHANONLYWHENTHESPEEDOFTHECARISLESSTHANTHESPEEDOFTHETRUCK
$ &4WILLBEEQUALINMAGNITUDETOONLYWHENTHESPEEDOFTHETRUCKISEQUALTOTHESPEEDOFTHECAR
!SMALLBOATINSTILLWATERISGIVENANINITIALHORIZONTALPUSHTOGETITMOVING4HEBOATGRADUALLYSLOWSDOWN7HICHOFTHEFOLLOWINGSTATEMENTSISTRUEFORTHEFORCESACTINGONTHEBOATASITSLOWSDOWN! 4HEREISAFORWARDFORCETHATDIMINISHESWITHTIME" 4HEREISABACKWARDFORCETHATDIMINISHESWITHTIME# 4HEREISAFORWARDFORCEANDABACKWARDFORCEBOTHOFWHICHDIMINISHWITHTIME$ 4HEREISAFORWARDFORCEANDABACKWARDFORCETHATAREALWAYSEQUALANDOPPOSITE
4HEGRAPHBELOWSHOWSTHEVARIATIONWITHDISPLACEMENTDOFTHEFORCE&APPLIEDBYASPRINGONACART4HEWORKDONEBYTHEFORCEINMOVINGTHECARTTHROUGHADISTANCEOFCMIS! *" *# *$ *
-
.FDIBOJDT
7HICHONEOFTHEFOLLOWINGISATRUESTATEMENTABOUTENERGY! %NERGYISDESTROYEDDUETOFRICTIONALFORCES" %NERGYISAMEASUREOFTHEABILITYTODOWORK# -OREENERGYISAVAILABLEWHENTHEREISALARGERPOWER$ %NERGYANDPOWERBOTHMEASURETHESAMEQUANTITY
!NELECTRICTRAINDEVELOPSAPOWEROF-7WHENTRAVELLINGATACONSTANTSPEEDOFMS4HENETRESISTIVEFORCEACTINGONTHETRAINIS! -." K.# K.$ .
7HICHOFTHEFOLLOWINGQUANTITIESISARECONSERVEDINANINELASTICCOLLISIONINANISOLATEDSYSTEMOFTWOOBJECTS
,INEARMOMENTUMOFSYSTEM +INETICENERGYOFSYSTEM
! 9ES 9ES
" 9ES .O
# .O 9ES
$ .O .O
4HEDIAGRAMBELOWREPRESENTSENERGYTRANSFERSINANENGINE
4HEEFlCIENCYOFTHEENGINEISGIVENBYTHEEXPRESSION
! %7???%).
" %7????%/54
# %/54????%).
$ %/54????%7
!MACHINELIFTSANOBJECTOFWEIGHT.TOAHEIGHTOFM4HEMACHINEHASANOVERALLEFlCIENCYOF4HEWORKDONEBYTHEMACHINEINRAISINGTHEOBJECTIS! *" *# *$ *
-
!NELECTRICMOTORWITHANINPUTPOWEROF7PRODUCES7OFMECHANICALPOWER4HEEFlCIENCYOFTHEMOTORIS! " # $
!STONEOFMASSMISATTACHEDTOASTRINGANDMOVESROUNDINAHORIZONTALCIRCLEOFRADIUS2ATCONSTANTSPEED64HEWORKDONEBYTHEPULLOFTHESTRINGONTHESTONEINONECOMPLETEREVOLUTIONIS! ZERO" PM6
# PM6??????2
$ PM6??????2
4WOOBJECTS8AND9AREMOVINGAWAYFROMTHEPOINT04HEDIAGRAMBELOWSHOWSTHEVELOCITYVECTORSOFTHETWOOBJECTS
7HICHOFTHEFOLLOWINGVELOCITYVECTORSBESTREPRESENTSTHEVELOCITYOFOBJECT8RELATIVETOOBJECT9
4HISQUESTIONISABOUTTHROWING
ASTONEFROMACLIFF!NTONIASTANDSATTHEEDGEOFAVERTICALCLIFFANDTHROWSASTONEVERTICALLYUPWARDS
4HESTONELEAVES!NTONIASHANDWITHSPEEDVMS4HEACCELERATIONOFFREEFALLGISMSANDALLDISTANCEMEASUREMENTSARETAKENFROMTHEPOINTWHERETHESTONELEAVES!NTONIASHANDA )GNORINGAIRRESISTANCECALCULATE
I THEMAXIMUMHEIGHTREACHEDBYTHESTONE II THETIMETAKENBYTHESTONETOREACHITSMAXIMUMHEIGHT4HETIMEBETWEENTHESTONELEAVING!NTONIASHANDANDHITTINGTHESEAISSB $ETERMINETHEHEIGHTOFTHECLIFF
-
.FDIBOJDT
4HISQUESTIONISABOUTTHECOLLISIONBETWEENTWORAILWAYTRUCKSCARTS
A $ElNELINEARMOMENTUM)NTHEDIAGRAMONTHERIGHT
TRUCK!ISMOVINGALONGAHORIZONTALTRACK)TCOLLIDESWITHASTATIONARYTRUCK"ANDONCOLLISIONTHETWOJOINTOGETHER)MMEDIATELYBEFORETHECOLLISIONTRUCK!ISMOVINGWITHSPEEDMS)MMEDIATELYAFTERCOLLISIONTHESPEEDOFTHETRUCKSISV
4HEMASSOFTRUCK!ISKGANDTHEMASSOFTRUCK"ISKG
B I #ALCULATETHESPEEDVIMMEDIATELYAFTERTHECOLLISION
II #ALCULATETHETOTALKINETICENERGYLOSTDURINGTHECOLLISION
!NELEVATORLIFTSTARTSFROMRESTONTHEGROUNDmOORANDCOMESTORESTATAHIGHERmOOR)TSMOTIONISCONTROLLEDBYANELECTRICMOTOR!SIMPLIlEDGRAPHOFTHEVARIATIONOFTHEELEVATORSVELOCITYWITHTIMEISSHOWNBELOW
4HEMASSOFTHEELEVATORISKG5SETHISINFORMATIONTOCALCULATE
A THEACCELERATIONOFTHEELEVATORDURINGTHElRSTS
B THETOTALDISTANCETRAVELLEDBYTHEELEVATOR
C THEMINIMUMWORKREQUIREDTORAISETHEELEVATORTOTHEHIGHERmOOR
D THEMINIMUMAVERAGEPOWERREQUIREDTORAISETHEELEVATORTOTHEHIGHERmOOR
E THEEFlCIENCYOFTHEELECTRICMOTORTHATLIFTSTHEELEVATORGIVENTHATTHEINPUTPOWERTOTHEMOTORISK7
-
4HISQUESTIONISABOUTMOMENTUMANDTHEKINEMATICSOFAPROPOSEDJOURNEYTO*UPITERA 3TATETHELAWOFCONSERVATIONOFMOMENTUM!SOLARPROPULSIONENGINEUSESSOLARPOWERTOIONIZEATOMSOFXENONANDTOACCELERATETHEM!SARESULTOFTHEACCELERATIONPROCESSTHEIONSAREEJECTEDFROMTHESPACESHIPWITHASPEEDOFMS
B 4HEMASSNUCLEONNUMBEROFTHEXENONUSEDIS$EDUCETHATTHEMASSOFONEIONOFXENONISKG
C 4HEORIGINALMASSOFTHEFUELISKG$EDUCETHATIFTHEENGINEEJECTSXENONIONSEVERYSECONDTHEFUELWILLLASTFORYEARSYEARS
D 4HEMASSOFTHESPACESHIPISKG$EDUCETHATTHEINITIALACCELERATIONOFTHESPACESHIPISMS
4HEGRAPHBELOWSHOWSTHEVARIATIONWITHTIMETOFTHEACCELERATIONAOFTHESPACESHIP4HESOLARPROPULSIONENGINEISSWITCHEDONATTIMETWHENTHESPEEDOFTHESPACESHIPISMS
E %XPLAINWHYTHEACCELERATIONOFTHESPACESHIPISINCREASINGWITHTIMEF 5SINGDATAFROMTHEGRAPHCALCULATETHESPEEDOFTHESPACESHIPATTHETIMEWHENTHEXENONFUELHASALLBEENUSED
G 4HEDISTANCEOFTHESPACESHIPFROM%ARTHWHENTHESOLARPROPULSIONENGINEISSWITCHEDONISVERYSMALLCOMPAREDTOTHEDISTANCEFROM%ARTHTO*UPITER4HEFUELRUNSOUTWHENTHESPACESHIPISADISTANCEOFMFROM*UPITER%STIMATETHETOTALTIMETHATITWOULDTAKETHESPACESHIPTOTRAVELFROM%ARTHTO*UPITER