Page 1

M ATTER & I NT E RA C TIONS

FOURTH

EDITION

C H A B AY • S H E R W O O D


$)"15&3



F1

2F1

s 2s L0

5IF .PNFOUVN 1SJODJQMF 0#+&$5*7&4 "GUFS TUVEZJOH UIJT DIBQUFS ZPV TIPVME CF BCMF UP 6TF CPUI JUFSBUJWF BOE BOBMZUJDBM UFDIOJRVFT UP QSFEJDU UIF GVUVSF NPUJPO PG B TZTUFN UIBU JT TVCKFDUFE UP B DPOTUBOU OFU GPSDF 6TF BO JUFSBUJWF BQQSPBDI UP QSFEJDU UIF GVUVSF NPNFOUVN PG BO PCKFDU UIBU JT TVCKFDUFE UP B WBSZJOH OFU GPSDF %SBX BOE JOUFSQSFU HSBQIT PG UIF DPNQPOFOUT PG QPTJUJPO PS WFMPDJUZ WT UJNF $BMDVMBUF UIF GPSDF FYFSUFE CZ B TQSJOH PO BO PCKFDU JO DPOUBDU XJUI JU $BMDVMBUF UIF BQQSPYJNBUF HSBWJUBUJPOBM GPSDF PO BO PCKFDU OFBS UIF &BSUIT TVSGBDF

 5)& .0.&/56. 13*/$*1-& *O NBOZ SFBM TJUVBUJPOT UIF NPNFOUVN BOE UIFSFGPSF UIF WFMPDJUZ PG B NPWJOH PCKFDU BSF DPOUJOVBMMZ DIBOHJOH EVF UP JOUFSBDUJPOT PG UIF PCKFDU XJUI JUT TVSSPVOEJOHT *O PSEFS UP QSFEJDU UIF NPUJPO PG TVDI B TZTUFN XIFUIFS JU JT B NFUFPS B UFOOJT CBMM PS BO FMFDUSPO XF OFFE UP CF BCMF UP FYQSFTT NBUIFNBUJDBMMZ UIF SFMBUJPOTIJQ CFUXFFO JOUFSBDUJPO BOE NPNFOUVN DIBOHF 5IF .PNFOUVN 1SJODJQMF NBLFT B RVBOUJUBUJWF DPOOFDUJPO CFUXFFO BNPVOU PG JOUFSBDUJPO BOE DIBOHF PG NPNFOUVN 5IF .PNFOUVN 1SJODJQMF JT UIF Ã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

4ZTUFN BOE 4VSSPVOEJOHT 5IF .PNFOUVN 1SJODJQMF SFMBUFT BO JOUFSBDUJPO RVBOUJUBUJWFMZ UP B DIBOHF JO NPNFOUVN 5P BOBMZ[F B DIBOHF JO NPNFOUVN XF OFFE UP TQFDJGZ DMFBSMZ UIF PCKFDUT XIPTF NPNFOUVN DIBOHF XF XJTI UP LOPX 0OF PS NPSF PCKFDUT DBO CF DPOTJEFSFE UP CF B iTZTUFNu &WFSZUIJOH UIBU JT OPU JODMVEFE JO UIF TZTUFN 




$IBQUFS  5IF .PNFOUVN 1SJODJQMF JT QBSU PG UIF iTVSSPVOEJOHTu 'PS FYBNQMF UIF SFE EBTIFE MJOF JO 'JHVSF  JOEJDBUFT UIBU XF IBWF DIPTFO UIF TLBUFS BT UIF TZTUFN BOE UIBU UIFSFGPSF BMM PUIFS PCKFDUT JODMVEJOH UIF XBMM UIF JDF UIF BJS BOE UIF &BSUI BSF QBSU PG UIF TVSSPVOEJOHT 5IF .PNFOUVN 1SJODJQMF SFMBUFT UIF DIBOHF JO NPNFOUVN PG B TZTUFN UP UIF BNPVOU PG JOUFSBDUJPO XJUI JUT TVSSPVOEJOHT /P NBUUFS XIBU TZTUFN XF DIPPTF UIF .PNFOUVN 1SJODJQMF XJMM DPSSFDUMZ QSFEJDU UIF CFIBWJPS PG UIF TZTUFN *O UIJT DIBQUFS XF XJMM BMXBZT DPOTJEFS B TJOHMF PCKFDU UP CF UIF TZTUFN BOE FWFSZUIJOH FMTF JO UIF 6OJWFSTF UP CF UIF TVSSPVOEJOHT )PXFWFS JO MBUFS DIBQUFST XF XJMM TFF UIBU JU JT QPTTJCMF BOE PGUFO VTFGVM UP DIPPTF B TZTUFN DPNQSJTJOH TFWFSBM PCKFDUT " GVOEBNFOUBM QSJODJQMF TVDI BT UIF .PNFOUVN 1SJODJQMF BQQMJFT UP BOZ TZTUFN OP NBUUFS IPX DPNQMFY

$BVTF BOE &GGFDU 'JHVSF  " TLBUFS QVTIFT PGG GSPN B XBMM *G XF DIPPTF UIF TLBUFS BT UIF TZTUFN JOEJDBUFE CZ UIF EBTIFE MJOF UIF XBMM UIF JDF UIF BJS BOE UIF &BSUI BSF JODMVEFE JO UIF TVSSPVOEJOHT

5IF .PNFOUVN 1SJODJQMF JT B GVOEBNFOUBM QSJODJQMF UIBU JT BMTP LOPXO BT /FXUPOT TFDPOE MBX *U SFTUBUFT BOE FYUFOET /FXUPOT à STU MBX PG NPUJPO JO B RVBOUJUBUJWF DBVTBM GPSN UIBU DBO CF VTFE UP QSFEJDU UIF CFIBWJPS PG PCKFDUT 5IF WBMJEJUZ PG UIF .PNFOUVN 1SJODJQMF IBT CFFO WFSJàFE UISPVHI B WFSZ XJEF WBSJFUZ PG PCTFSWBUJPOT BOE FYQFSJNFOUT JOWPMWJOH MBSHF BOE TNBMM PCKFDUT NPWJOH TMPXMZ PS BU TQFFET OFBS UIF TQFFE PG MJHIU *U JT B TVNNBSZ PG UIF XBZ JOUFSBDUJPOT BGGFDU NPUJPO JO UIF SFBM XPSME 5)& .0.&/56. 13*/$*1-&  OFU ΔU ΔQ = ' 5IF DIBOHF PG NPNFOUVN PG B TZTUFN UIF FGGFDU JT FRVBM UP UIF OFU GPSDF BDUJOH PO UIF TZTUFN UJNFT UIF EVSBUJPO PG UIF JOUFSBDUJPO UIF DBVTF  5IF UJNF JOUFSWBM ΔU NVTU CF TNBMM FOPVHI UIBU UIF OFU GPSDF JT OFBSMZ DPOTUBOU EVSJOH UIJT UJNF JOUFSWBM 5IF .PNFOUVN 1SJODJQMF JT B GVOEBNFOUBM QSJODJQMF CFDBVTF JU BQQMJFT UP FWFSZ QPTTJCMF TZTUFN OP NBUUFS IPX MBSHF PS TNBMM GSPN DMVTUFST PG HBMBYJFT UP TVCBUPNJD QBSUJDMFT BOE OP NBUUFS IPX GBTU JU JT NPWJOH JU JT USVF GPS FWFSZ LJOE PG JOUFSBDUJPO FMFDUSJD HSBWJUBUJPOBM FUD  JU SFMBUFT BO FGGFDU DIBOHF JO NPNFOUVN UP B DBVTF BO JOUFSBDUJPO  5P CFHJO UP VOEFSTUBOE UIJT FRVBUJPO XF XJMM DPOTJEFS FBDI RVBOUJUZ JOWPMWFE

 OFU ΔU ΔQ = '

$IBOHF PG .PNFOUVN ΔQ

 OFU ΔU ΔQ = '

 'PSDF '

"T XF TBX JO UIF QSFWJPVT DIBQUFS UIF DIBOHF JO NPNFOUVN ΔQ = QG −QJ PG B TZTUFN DBO JOWPMWF B DIBOHF JO UIF NBHOJUVEF PG NPNFOUVN UIF EJSFDUJPO PG NPNFOUVN PS CPUI NBHOJUVEF BOE EJSFDUJPO

4DJFOUJTUT BOE FOHJOFFST FNQMPZ UIF DPODFQU PG iGPSDFu UP RVBOUJGZ JOUFSBDUJPOT CFUXFFO UXP PCKFDUT 'PSDF JT B WFDUPS RVBOUJUZ CFDBVTF B GPSDF IBT B NBHOJUVEF BOE JT FYFSUFE JO B QBSUJDVMBS EJSFDUJPO &YBNQMFT PG GPSDFT JODMVEF UIF GPMMPXJOH 5IF SFQVMTJWF FMFDUSJD GPSDF B QSPUPO FYFSUT PO BOPUIFS QSPUPO 5IF BUUSBDUJWF HSBWJUBUJPOBM GPSDF UIF &BSUI FYFSUT PO ZPV 5IF GPSDF UIBU B DPNQSFTTFE TQSJOH FYFSUT PO ZPVS IBOE 5IF GPSDF PO B TQBDFDSBGU CZ FYQBOEJOH HBTFT JO B SPDLFU FOHJOF 5IF GPSDF PG UIF TVSSPVOEJOH BJS PO UIF QSPQFMMFS PG BO BJSQMBOF


 5IF .PNFOUVN 1SJODJQMF



.FBTVSJOH B 'PSDF

s 2s

" TJNQMF XBZ UP NFBTVSF GPSDF JT UP VTF UIF TUSFUDI PS DPNQSFTTJPO PG B TQSJOH *O 'JHVSF  XF IBOH B CMPDL GSPN B TQSJOH BOE OPUF UIBU UIF TQSJOH JT TUSFUDIFE B EJTUBODF T 5IFO XF IBOH UXP TVDI CMPDLT GSPN UIF TQSJOH BOE XF TFF UIBU UIF TQSJOH JT TUSFUDIFE UXJDF BT NVDI #Z FYQFSJNFOUBUJPO XF à OE UIBU BOZ TQSJOH NBEF PG UIF TBNF NBUFSJBM BOE QSPEVDFE UP UIF TBNF TQFDJàDBUJPOT CFIBWFT JO UIF TBNF XBZ 4JNJMBSMZ XF DBO PCTFSWF IPX NVDI UIF TQSJOH DPNQSFTTFT XIFO UIF TBNF CMPDLT BSF TVQQPSUFE CZ JU 'JHVSF   8F àOE UIBU POF CMPDL DPNQSFTTFT UIF TQSJOH CZ UIF TBNF EJTUBODF |T| BOE UXP CMPDLT DPNQSFTT JU CZ |T| $PNQSFTTJPO DBO CF DPOTJEFSFE OFHBUJWF TUSFUDI CFDBVTF UIF MFOHUI PG UIF TQSJOH EFDSFBTFT

6OJU PG 'PSDF 'JHVSF  4USFUDIJOH PG B TQSJOH JT B NFBTVSF PG GPSDF

s 2s

8F DBO VTF B TQSJOH UP NBLF B TDBMF GPS NFBTVSJOH GPSDFT DBMJCSBUJOH JU JO UFSNT PG IPX NVDI GPSDF JT SFRVJSFE UP QSPEVDF B HJWFO TUSFUDI 5IF 4* VOJU PG GPSDF JT UIF OFXUPO BCCSFWJBUFE BT / 0OF OFXUPO JT B SBUIFS TNBMM GPSDF " OFXUPO JT BQQSPYJNBUFMZ UIF EPXOXBSE HSBWJUBUJPOBM GPSDF PG UIF &BSUI PO B TNBMM BQQMF PS BCPVU B RVBSUFS PG B QPVOE *G ZPV IPME B TNBMM BQQMF BU SFTU JO ZPVS IBOE ZPV BQQMZ BO VQXBSE GPSDF PG BCPVU POF OFXUPO DPNQFOTBUJOH GPS UIF EPXOXBSE QVMM PG UIF &BSUI

5IF .FBOJOH PG OFU

'JHVSF  $PNQSFTTJPO PG B TQSJOH JT BMTP B NFBTVSF PG GPSDF

*O QIZTJDT UIF XPSE OFU IBT B QSFDJTF NFBOJOH JU JT UIF TVN PG BMM DPOUSJCVUJPOT UP B RVBOUJUZ CPUI QMVTFT BOE NJOVTFT #FDBVTF B GPSDF IBT B NBHOJUVEF BOE B EJSFDUJPO JU JT B WFDUPS TP UIF OFU GPSDF PO B TZTUFN JT UIF WFDUPS TVN PG BMM UIF GPSDFT FYFSUFE PO UIF TZTUFN CZ  OFU PO B TZTUFN BDUJOH BMM UIF PCKFDUT JO UIF TVSSPVOEJOHT *U JT UIF OFU GPSDF ' GPS TPNF UJNF ΔU UIBU DBVTFT B DIBOHF PG NPNFOUVN %&'*/*5*0/ 0' /&5 '03$&

 OFU ΔU ΔQ = '

 + '  + · · ·  OFU ≡ ' ' 5IF iOFUu GPSDF BDUJOH PO B TZTUFN BU BO JOTUBOU JT UIF WFDUPS TVN PG BMM UIF GPSDFT FYFSUFE PO UIF TZTUFN CZ BMM UIF PCKFDUT JO UIF TVSSPVOEJOHT BU UIBU JOTUBOU

F1 System F3

Fnet = F1+ F2+F3 F2

'JHVSF  5IF OFU GPSDF PO UIF TZTUFN JT UIF WFDUPS TVN PG BMM UIF JOEJWJEVBM GPSDFT BDUJOH PO UIF TZTUFN

5IF OFU GPSDF BDUJOH PO B TZTUFN BU BO JOTUBOU JT UIF WFDUPS TVN PG BMM PG UIF GPSDFT FYFSUFE PO UIF TZTUFN CZ BMM UIF PCKFDUT JO UIF TVSSPVOEJOHT XIJDI BSF DBMMFE iFYUFSOBMu GPSDFT 'JHVSF   5IFSF NBZ CF GPSDFT JOUFSOBM UP UIF TZTUFN FYFSUFE CZ POF PCKFDU JO UIF TZTUFN PO BOPUIFS PCKFDU JO UIF TZTUFN CVU TVDI JOUFSOBM GPSDFT DBOOPU DIBOHF UIF NPNFOUVN PG UIF TZTUFN 8F XJMM TFF JO EFUBJM XIZ UIJT JT TP JO $IBQUFS  CVU UIF CBTJD SFBTPO JT UIBU JOUFSOBM GPSDFT DBODFM FBDI PUIFST FGGFDUT " GPSDF UIBU PCKFDU  JO B TZTUFN FYFSUT PO PCKFDU  JO UIF TZTUFN EPFT DIBOHF UIF NPNFOUVN PG PCKFDU  CVU PCKFDU  FYFSUT B GPSDF JO UIF PQQPTJUF EJSFDUJPO PO PCKFDU  UIBU DIBOHFT UIF NPNFOUVN PG PCKFDU  JO UIF PQQPTJUF XBZ TP UIBU UIF DIBOHFT JO NPNFOUB PG UIF UXP PCKFDUT BEE VQ UP [FSP 26&45*0/ " CBMM GBMMJOH UPXBSE UIF &BSUI DPOTJTUT PG NBOZ BUPNT &BDI BUPN JO UIF CBMM FYFSUT GPSDFT PO OFJHICPSJOH BUPNT JO UIF CBMM UIF &BSUI FYFSUT GPSDFT PO FBDI BUPN JO UIF CBMM BOE UIF BJS FYFSUT GPSDFT PO BUPNT BU UIF TVSGBDF PG UIF CBMM 5BLJOH UIF CBMM BT UIF TZTUFN BOE UIF &BSUI BOE BJS BT UIF TVSSPVOEJOHT XIJDI PG UIFTF GPSDFT BSF FYUFSOBM GPSDFT BOE XIJDI BSF JOUFSOBM GPSDFT




$IBQUFS  5IF .PNFOUVN 1SJODJQMF

F

'JHVSF  " CMPDL NPVOUFE PO B OFBSMZ GSJDUJPOMFTT BJS USBDL QVMMFE CZ B TQSJOH

 OFU ΔU ΔQ = '

5IF &BSUI BOE UIF BJS BSF QBSU PG UIF TVSSPVOEJOHT TP UIF GPSDF FYFSUFE CZ UIF &BSUI BOE UIF GPSDF FYFSUFE CZ UIF BJS BSF CPUI FYUFSOBM GPSDFT 5IFZ BSF GPSDFT FYFSUFE PO UIF DIPTFO TZTUFN UIF CBMM CZ PCKFDUT JO UIF TVSSPVOEJOHT UIF &BSUI BOE UIF BJS  'PSDFT UIBU BUPNT JO UIF CBMM FYFSU PO OFJHICPSJOH BUPNT JO UIF CBMM BSF JOUFSOBM GPSDFT 5IF JOUFSOBM GPSDFT EPOU DPOUSJCVUF UP UIF OFU  OFU UIBU DIBOHFT UIF CBMMT NPNFOUVN BT QSFEJDUFE CZ UIF .PNFOUVN GPSDF '  OFU ΔU 1SJODJQMF ΔQ = ' 8F à OE FYQFSJNFOUBMMZ UIBU UIF NBHOJUVEF PG UIF OFU GPSDF BDUJOH PO BO PCKFDU BGGFDUT UIF NBHOJUVEF PG UIF DIBOHF JO JUT NPNFOUVN .BOZ JOUSPEVDUPSZ QIZTJDT MBCPSBUPSJFT IBWF BJS USBDLT MJLF UIF POF JMMVTUSBUFE JO 'JHVSF  5IF MPOH USJBOHVMBS CBTF IBT NBOZ TNBMM IPMFT JO JU BOE BJS VOEFS QSFTTVSF JT CMPXO PVU UISPVHI UIFTF IPMFT 5IF BJS GPSNT B DVTIJPO VOEFS UIF HMJEFS BMMPXJOH JU UP DPBTU TNPPUIMZ XJUI WFSZ MJUUMF GSJDUJPO 4VQQPTF UIBU XF QMBDF B CMPDL PO B HMJEFS PO B MPOH BJS USBDL BOE BUUBDI B DBMJCSBUFE TQSJOH UP JU 'JHVSF   8F QVMM PO UIF TQSJOH TP UIBU JU JT TUSFUDIFE B EJTUBODF T TP UIBU JU FYFSUT B GPSDF ' PO UIF CMPDL BOE XF QVMM GPS B TIPSU UJNF $IPPTF UIF CMPDL BOE HMJEFS BT UIF TZTUFN TP UIBU UIF OFU GPSDF PO UIF TZTUFN JT KVTU UIF TQSJOH TJODF GSJDUJPO XJUI UIF USBDL JT WFSZ TNBMM EVF UP UIF BJS DVTIJPO 8F PCTFSWF UIBU UIF NPNFOUVN PG UIF TZTUFN PG CMPDL BOE HMJEFS JODSFBTFT GSPN [FSP UP BO BNPVOU NW *G JOTUFBE XF QVMM PO UIF TQSJOH TP UIBU JU TUSFUDIFT B EJTUBODF T UIF TQSJOH FYFSUT B GPSDF ' PO UIF TZTUFN BOE XF PCTFSWF UIBU UIF TZTUFNT NPNFOUVN JODSFBTFT UP NW JO UIF TBNF BNPVOU PG UJNF "QQBSFOUMZ UIF NBHOJUVEF PG UIF DIBOHF JO NPNFOUVN JT QSPQPSUJPOBM UP UIF NBHOJUVEF PG UIF OFU GPSDF BQQMJFE UP UIF PCKFDU

%VSBUJPO PG *OUFSBDUJPO ΔU "OPUIFS FYQFSJNFOU XJMM TIPX VT UIBU OPU POMZ UIF NBHOJUVEF PG UIF GPSDF CVU BMTP UIF MFOHUI PG UJNF EVSJOH XIJDI JU BDUT PO BO PCKFDU BGGFDUT UIF DIBOHF PG NPNFOUVN PG UIF PCKFDU 4VQQPTF UIBU XF BHBJO QMBDF B CMPDL PO BO BJS USBDL BOE BUUBDI B TQSJOH UP JU 8F QVMM PO UIF TQSJOH GPS B UJNF ΔU XJUI B GPSDF ' BOE XF PCTFSWF UIBU UIF NPNFOUVN PG UIF TZTUFN PG CMPDL QMVT HMJEFS JODSFBTFT GSPN [FSP UP BO BNPVOU NW )PXFWFS JG XF SFQFBU UIF FYQFSJNFOU CVU QVMM GPS B UJNF UXJDF BT MPOH XJUI UIF TBNF GPSDF ' UIF TZTUFNT NPNFOUVN JODSFBTFT UP NW UIF DIBOHF JO JUT NPNFOUVN JT UXJDF BT HSFBU 8F PCTFSWF UIBU UIF NBHOJUVEF PG UIF DIBOHF JO NPNFOUVN JT EJSFDUMZ QSPQPSUJPOBM UP UIF MFOHUI PG UJNF ΔU = U − U EVSJOH XIJDI UIF GPSDF BDUT PO UIF PCKFDU

 OFU ΔU ΔQ = '

 OFU ΔU *NQVMTF ' 5IF BNPVOU PG JOUFSBDUJPO BGGFDUJOH BO PCKFDU JODMVEFT CPUI UIF TUSFOHUI PG  OFU BOE UIF EVSBUJPO ΔU PG UIF UIF JOUFSBDUJPO FYQSFTTFE BT UIF OFU GPSDF ' JOUFSBDUJPO &JUIFS B CJHHFS GPSDF PS BQQMZJOH UIF GPSDF GPS B MPOHFS UJNF XJMM DBVTF NPSF DIBOHF PG NPNFOUVN 5IF QSPEVDU PG B GPSDF BOE B UJNF JOUFSWBM JT DBMMFE iJNQVMTFu %&'*/*5*0/ 0' *.16-4&  *NQVMTF = 'ΔU *NQVMTF IBT VOJUT PG / · T OFXUPOTFDPOET

5IF UJNF JOUFSWBM ΔU NVTU CF TNBMM FOPVHI UIBU UIF OFU GPSDF JT OFBSMZ DPOTUBOU EVSJOH UIJT UJNF JOUFSWBM /PX XF DBO TUBUF UIF .PNFOUVN 1SJODJQMF UIJT XBZ 5IF DIBOHF PG NPNFOUVN PG B TZTUFN JT FRVBM UP UIF OFU JNQVMTF BQQMJFE UP UIF TZTUFN


 5IF .PNFOUVN 1SJODJQMF



26&45*0/ "SF UIF VOJUT PG JNQVMTF BOE NPNFOUVN UIF TBNF *G OPU IPX BSF UIFZ SFMBUFE

5IF VOJUT PG JNQVMTF BSF / · T B GPSDF PG  / BDUJOH EVSJOH B UJNF PG  TFDPOE HJWFT BO JNQVMTF PG  / · T  5IF VOJUT PG NPNFOUVN BSF LH · NT TP FWJEFOUMZ  / · T =  LH · NT BOE UIFSFGPSF  / =  LH · NT  &9".1-&

*NQVMTF "QQMJFE UP B #BTFCBMM +VTU CFGPSF JU UPVDIFT UIF CBU UIF NPNFOUVN PG B CBTFCBMM JT    LH · NT "GUFS UIF CBUUFS IBT IJU UIF CBMM BOE UIF CBMM JT OP MPOHFS JO DPOUBDU XJUI UIF CBU UIF NPNFOUVN PG UIF CBTFCBMM JT −  − LH · NT 8IBU XBT UIF JNQVMTF BQQMJFE UP UIF CBMM EVSJOH JUT DPOUBDU XJUI UIF CBU

4PMVUJPO

 OFU ΔU ΔQ = '  OFU ΔU −  − LH · NT −    LH · NT = '  OFU ΔU −  − LH · NT = '  OFU PS ΔU TFQBSBUFMZ 8JUIPVU NPSF JOGPSNBUJPO XF DBOOPU à OE UIF WBMVFT PG '

&9".1-&

pi pf 'JHVSF  " CBMM IJUT B XBMM BOE SFCPVOET

4PMVUJPO

$PMMJTJPO XJUI B 8BMM " UFOOJT CBMM USBWFMJOH JO UIF −Y EJSFDUJPO XJUI NBHOJUVEF PG NPNFOUVN  LH · NT IJUT B XBMM BOE SFCPVOET 'JHVSF   "GUFS UIF DPMMJTJPO UIF NBHOJUVEF PG UIF CBMMT NPNFOUVN JT OFBSMZ UIF TBNF CVU UIF CBMM JT USBWFMJOH JO UIF +Y EJSFDUJPO 8IBU XBT UIF JNQVMTF BQQMJFE UP UIF CBMM CZ UIF XBMM

"MUIPVHI UIF NBHOJUVEF PG UIF CBMMT NPNFOUVN EJE OPU DIBOHF TJHOJà DBOUMZ JUT EJSFDUJPO EJE DIBOHF TP UIF JNQVMTF JT OPU [FSP  OFU ΔU ΔQ = '  OFU ΔU    LH · NT − −   LH · NT = '  OFU ΔU    LH · NT = ' /PUF UIBU JO UIJT DPMMJTJPO UIF NBHOJUVEF PG UIF DIBOHF JO UIF CBMMT NPNFOUVN XIJDI JT FRVBM UP UIF NBHOJUVEF PG UIF JNQVMTF XBT BCPVU UXJDF BT MBSHF BT UIF CBMMT PSJHJOBM NPNFOUVN

&9".1-&

p1

p2

'JHVSF  *OJUJBM BOE Ã OBM NPNFOUB PG B IPDLFZ QVDL #PUI NPNFOUB MJF JO UIF Y[ QMBOF UIJT JT B UPQ WJFX MPPLJOH EPXO PO UIF JDF GSPN BCPWF

4PMVUJPO

'JOEJOH %JSFDUJPO PG *NQVMTF (SBQIJDBMMZ " IPDLFZ QVDL JT TMJEJOH BDSPTT UIF JDF XJUI NPNFOUVN Q TIPXO JO 'JHVSF  XIFO B IPDLFZ QMBZFS IJUT JU TIBSQMZ XJUI B IPDLFZ TUJDL "GUFSXBSE UIF QVDL IBT NPNFOUVN Q  #PUI Q BOE Q MJF JO UIF Y[ QMBOF 8IBU XBT UIF EJSFDUJPO PG UIF JNQVMTF BQQMJFE UP UIF QVDL CZ UIF TUJDL 8IBU XBT UIF EJSFDUJPO PG UIF OFU GPSDF FYFSUFE PO UIF QVDL EVSJOH JUT TIPSU DPOUBDU XJUI UIF IPDLFZ TUJDL

8F DBO à OE UIF EJSFDUJPO PG ΔQ HSBQIJDBMMZ BT TIPXO JO 'JHVSF  'SPN UIF WFDUPS FRVBUJPO  OFU ΔU ΔQ = '




$IBQUFS  5IF .PNFOUVN 1SJODJQMF

p2 F

p2 – p1

 OFU ΔU NVTU IBWF UIF TBNF EJSFDUJPOT 4JODF ΔU JT B QPTJUJWF XF TFF UIBU ΔQ BOE ' TDBMBS UIF EJSFDUJPO PG UIF OFU GPSDF JT UIF TBNF BT UIF EJSFDUJPO PG UIF JNQVMTF  OFU CZ UIF QPTJUJWF TDBMBS ΔU DBOU DIBOHF UIF CFDBVTF NVMUJQMZJOH UIF WFDUPS ' EJSFDUJPO PG UIF WFDUPS

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 "/"-:5*$"- 13&%*$5*0/ $0/45"/5 /&5 '03$& *O DPOUSBTU UP BO JUFSBUJWF TPMVUJPO BO BOBMZUJDBM TPMVUJPO UP B QSPCMFN ZJFMET B NBUIFNBUJDBM GVODUJPO UIBU HJWFT UIF WBMVF PG B RVBOUJUZ PG JOUFSFTU BT B GVODUJPO PG UJNF *G TVDI B GVODUJPO FYJTUT POF DBO VTF JU UP Ã OE UIF QPTJUJPO PS WFMPDJUZ PG B TZTUFN BU B HJWFO UJNF JO POF TUFQ JOTUFBE PG UBLJOH NBOZ TNBMM TUFQT UP HFU UIJT JOGPSNBUJPO *U JT OPU BMXBZT QPTTJCMF UP HFU BO BOBMZUJDBM TPMVUJPO TP JUFSBUJWF NFUIPET BSF PG NPSF HFOFSBM BQQMJDBCJMJUZ 0OF DBTF JO XIJDI BO JUFSBUJWF TPMVUJPO JT QPTTJCMF JT UIF DBTF PG DPOTUBOU OFU GPSDF 8F XJMM EFSJWF UIJT BOBMZUJDBM TPMVUJPO UP JMMVTUSBUF UIF BQQSPBDI BOE UIF OBUVSF PG UIF SFTVMUT

"QQSPYJNBUJPOT GPS "WFSBHF 7FMPDJUZ *O UIF QSFWJPVT JUFSBUJWF TPMVUJPO GPS UIF NPUJPO PG B UFOOJT CBMM XF VTFE UIF BQQSPYJNBUJPO QGVUVSF ≈ WGVUVSF ≈ WBWH N GPS UIF BWFSBHF WFMPDJUZ PWFS FBDI UJNF JOUFSWBM ΔU 26&45*0/ 8IZ EJEOU XF DBMDVMBUF UIF BWFSBHF WFMPDJUZ EJSFDUMZ GSPN UIF EFà OJUJPO

8F DBOU DBMDVMBUF UIF BWFSBHF WFMPDJUZ EJSFDUMZ GSPN UIF EFÃ OJUJPO

vx

WBWH ≡

vfx (vix + vfx) 2 vix ti

tf t

'JHVSF  WY JT DIBOHJOH BU B DPOTUBOU SBUF MJOFBSMZ XJUI UJNF TP UIF BSJUINFUJD BWFSBHF JT FRVBM UP WBWH Y

ΔS SG −SJ = ΔU ΔU

CFDBVTF UIF à OBM MPDBUJPO SG PG UIF NPWJOH CBMM JT VOLOPXO 4JODF XF EP OPU IBWF BOZ PUIFS JOGPSNBUJPO UIBU XPVME BMMPX VT UP EFUFSNJOF UIF BWFSBHF WFMPDJUZ XF OFFE UP àOE BO BQQSPYJNBUF WBMVF GPS UIF BWFSBHF WFMPDJUZ JO UIJT TJUVBUJPO 5IFSF BSF UXP QPTTJCMF BQQSPBDIFT  *G UIF DIBOHF JO WFMPDJUZ JT TNBMM PWFS UIF UJNF JOUFSWBM PG JOUFSFTU XF DBO VTF UIF àOBM WFMPDJUZ PS UIF JOJUJBM WFMPDJUZ BT BO BQQSPYJNBUJPO UP UIF BWFSBHF WFMPDJUZ  "MUFSOBUJWFMZ XF DBO BWFSBHF UXP WBMVFT PG WFMPDJUZ UP BQQSPYJNBUF UIF BWFSBHF WFMPDJUZ 8IBU NJHIU CF B CFUUFS BQQSPYJNBUJPO GPS WBWH JG UIF OFU GPSDF PO B TZTUFN JT DPOTUBOU

26&45*0/

-FUT DPOTJEFS VTJOH UIF BSJUINFUJD BWFSBHF WBWH ≈ vx

WGVUVSF +WOPX 

5IF BSJUINFUJD BWFSBHF MJFT CFUXFFO UIF UXP WBMVFT 'PS FYBNQMF UIF BSJUINFUJD BWFSBHF PG  BOE  JT  +  / = / =  IBMGXBZ CFUXFFO  BOE  5IF BSJUINFUJD BWFSBHF JT PGUFO B HPPE BQQSPYJNBUJPO CVU JU JT OPU OFDFTTBSJMZ FRVBM UP UIF USVF BWFSBHF WBWH = ΔS/ΔU 5IF BSJUINFUJD BWFSBHF EPFT OPU HJWF UIF USVF BWFSBHF VOMFTT FBDI DPNQPOFOU PG W JT DIBOHJOH BU B DPOTUBOU SBUF BT JO 'JHVSF  XIJDI JT UIF DBTF POMZ JG UIF OFU GPSDF JT DPOTUBOU 5IF EFUBJMFE QSPPG UIBU WBWH = WJ +WG / XIFO W DIBOHFT BU B DPOTUBOU SBUF JT NPSF DPNQMJDBUFE UIBO POF NJHIU FYQFDU BOE JT HJWFO JO PQUJPOBM 4FDUJPO 

vfx (vix + vfx) 2 vix vavg, x

ti

tf t

'JHVSF  8IFO WY EPFT OPU DIBOHF MJOFBSMZ XJUI UJNF UIF BSJUINFUJD BWFSBHF NBZ CF TJHOJÃ DBOUMZ EJGGFSFOU GSPN UIF USVF BWFSBHF WBMVF PG WBWH Y

26&45*0/

GPS WBWH

8IFO JT UIF BSJUINFUJD BWFSBHF B QPPS BQQSPYJNBUJPO

5IF BSJUINFUJD BWFSBHF JT B QPPS BQQSPYJNBUJPO GPS WBWH JG UIF SBUF PG DIBOHF PG WFMPDJUZ UIF BDDFMFSBUJPO B = EW/EU JT OPU DPOTUBOU 'JHVSF  TIPXT


 "OBMZUJDBM 1SFEJDUJPO $POTUBOU /FU 'PSDF



BO FYBNQMF PG TVDI B TJUVBUJPO "U MPX TQFFET W  D UIF BDDFMFSBUJPO JT EJSFDUMZ QSPQPSUJPOBM UP UIF SBUF PG DIBOHF PG NPNFOUVN CFDBVTF  OFU = ΔQ/ΔU ≈ N EW/EU TP XF DBO DPODMVEF UIBU EW/EU XJMM CF DPOTUBOU ' JG UIF OFU GPSDF PO UIF TZTUFN JT DPOTUBOU &9".1-&

5IF "SJUINFUJD "WFSBHF $BO #F B 1PPS "QQSPYJNBUJPO *G ZPV ESJWF  NJ/I JO UIF +Y EJSFDUJPO GPS GPVS IPVST BOE UIFO  NJ/I JO UIF +Y EJSFDUJPO GPS BO IPVS IPX GBS IBWF ZPV USBWFMFE 'JHVSF 

4PMVUJPO

vx vix vavg, x

vfx

*G XF VTF WY BU UIF CFHJOOJOH BOE BU UIF FOE PG UIJT  I UJNF JOUFSWBM UP EFUFSNJOF UIF BSJUINFUJD BWFSBHF WFMPDJUZ ZPV XPVME DBMDVMBUF UIBU ZPV IBWF USBWFMFE  NJ WBWH Y ≈

(vix + vfx) 2

 +  NJI =  NJI 

YG − YJ ≈  NJI  I =  NJ )PXFWFS ZPV IBWF BDUVBMMZ USBWFMFE TJHOJà DBOUMZ GBSUIFS t

'JHVSF  #FDBVTF WY PG UIF DBS EPFT OPU DIBOHF MJOFBSMZ XJUI UJNF WBWH Y JT TJHOJÃ DBOUMZ EJGGFSFOU GSPN UIF BSJUINFUJD BWFSBHF PG WJ Y + WG Y 

YG − YJ =  NJI  I +  NJI  I =  NJ 5IF Y DPNQPOFOU PG ZPVS BDUVBM BWFSBHF WFMPDJUZ XBT WBWH Y =

 NJ =  NJI I

"11309*."5& "7&3"(& 7&-0$*5: WBWH ≈

WGVUVSF +WOPX 

5IJT JT FYBDUMZ DPSSFDU JG BMM DPNQPOFOUT PG W DIBOHF MJOFBSMZ XJUI UJNF BDDFMFSBUJPO JT DPOTUBOU  )PXFWFS JU NBZ CF B QPPS BQQSPYJNBUJPO JG UIF OFU GPSDF JT OPU DPOTUBOU

"OBMZUJDBM 4PMVUJPO GPS $POTUBOU /FU 'PSDF 5IF UFSN BOBMZUJDBM EFTDSJCFT B TPMVUJPO UIBU DBO CF XSJUUFO TZNCPMJDBMMZ BMHFCSBJDBMMZ  *O NBOZ DBTFT JU JT OPU QPTTJCMF UP à OE BO BOBMZUJDBM TPMVUJPO GPS UIF NPUJPO PG B TZTUFN )PXFWFS JG UIF OFU GPSDF PO B TZTUFN JT DPOTUBOU XF DBO FBTJMZ àOE BO BOBMZUJDBM TPMVUJPO PG UIF GPSN S U = . . . $POTJEFS B TZTUFN PO XIJDI UIFSF JT B DPOTUBOU OFU GPSDF JO UIF +Y EJSFDUJPO EVSJOH UIF UJNF ΔU 'PS PSEJOBSZ TQFFET γ ≈  TP WY ≈ QY /N 6TJOH B UIF .PNFOUVN 1SJODJQMF JO JUT VQEBUF GPSN C UIF QPTJUJPO VQEBUF FRVBUJPO BOE D UIF TQFDJBM SFTVMU GPS BWFSBHF WFMPDJUZ JO UIF DBTF PG DPOTUBOU OFU GPSDF XF DBO FBTJMZ EFSJWF FRVBUJPOT GPS UIF QPTJUJPO BOE WFMPDJUZ BT B GVODUJPO PG UJNF 'PS DPOWFOJFODF XFMM TUBSU NFBTVSJOH UIF UJNF U GSPN  BOE XSJUF U UP SFQSFTFOU ΔU QYG = QYJ + 'OFU Y U 'OFU Y WYG = WYJ + U N   'OFU Y WYJ + U + WYJ N WBWH Y =   'OFU Y WBWH Y = WYJ + U  N




$IBQUFS  5IF .PNFOUVN 1SJODJQMF YG = YJ + WBWH Y U  'OFU Y  YG = YJ + WYJ U + U  N 5P FNQIBTJ[F UIBU XF OPX IBWF BO BOBMZUJDBM TPMVUJPO XF XJMM XSJUF FRVBUJPOT UIBU FYQSFTT QPTJUJPO BOE WFMPDJUZ BT B GVODUJPO PG UJNF HJWFO UIBU UIF OFU GPSDF JT DPOTUBOU "/"-:5*$"- 40-65*0/ $0/45"/5 /&5 '03$& Y U = YJ + WYJ U + WY U = WYJ +

 'OFU Y  U  N

'OFU Y U N

*G UIF OFU GPSDF IBE BDUFE JO UIF Z PS [ EJSFDUJPO UIF TPMVUJPO XPVME CF UIF TBNF FYDFQU GPS UIF TVCTDSJQUT JOEJDBUJOH EJSFDUJPO 5IF UJNF U DBO CF BSCJUSBSJMZ MBSHF BT MPOH BT UIF OFU GPSDF PO UIF TZTUFN JT DPOTUBOU EVSJOH UIF UJNF JOUFSWBM  UP U 5IF UJNF EFSJWBUJWF PG WY JT UIF Y DPNQPOFOU PG BDDFMFSBUJPO XIJDI JT DPOTUBOU TJODF 'OFU Y JT DPOTUBOU EWY 'OFU Y = = BY EU N

&9".1-&

'JHVSF  " GBO DBSU PO B USBDL "VUIPS QIPUP

4PMVUJPO x (m) 0.2

$POTUBOU /FU 'PSDF PO B 'BO $BSU "O FBTZ XBZ UP BQQMZ B OFBSMZ DPOTUBOU GPSDF JT UP NPVOU B CBUUFSZQPXFSFE GBO PO B DBSU 'JHVSF   *G UIF GBO EJSFDUT BJS CBDLXBSET UIF JOUFSBDUJPO XJUI UIF BJS QVTIFT UIF DBSU GPSXBSE XJUI B OFBSMZ DPOTUBOU GPSDF "JSCPBUT VTFE JO UIF WFSZ TIBMMPX 'MPSJEB &WFSHMBEFT BSF CVJMU JO B TJNJMBS XBZ XJUI MBSHF GBOT PO UPQ PG UIF CPBUT QSPQFMMJOH UIFN UISPVHI UIF TXBNQ 4VQQPTF UIBU ZPV IBWF B GBO DBSU XIPTF NBTT JT  LH :PV UVSO PO UIF GBO UIFO HJWF UIF DBSU B TIPWF TP UIBU XIFO JU JT BU MPDBUJPO    N JU JT USBWFMJOH XJUI WFMPDJUZ    NT "U B UJNF  TFDPOET MBUFS ZPV PCTFSWF UIBU UIF DBSU JT BU MPDBUJPO    N 8IBU XBT UIF GPSDF FYFSUFE CZ UIF BJS PO UIF GBO DBSU

4JODF CPUI OFU GPSDF BOE NPUJPO BSF QBSBMMFM UP UIF USBDL XF POMZ OFFE UP DPOTJEFS UIF Y DPNQPOFOUT PG QPTJUJPO WFMPDJUZ BOE GPSDF 6TJOH UIF FRVBUJPOT EFSJWFE BCPWF GPS UIF NPUJPO PG B TZTUFN UIBU FYQFSJFODFT B DPOTUBOU OFU GPSDF YG = YJ + WYJ U +

0.1

 'OFU Y  U  N

 N =  N +  NT  T + 0.25

0.5

0.75 Time (s)

'JHVSF  " QMPU PG Y WT U GPS UIF GBO DBSU /PUF UIBU WY U JT UIF EFSJWBUJWF PG UIJT GVODUJPO 8IFSF UIF TMPQF JT [FSP WY JT [FSP UIJT PDDVST BU UIF UJNF XIFO WY DIBOHFT TJHO GSPN QPTJUJWF UP OFHBUJWF

'OFU Y  T    LH

'OFU Y = − / 8F TFF UIBU UIF OFU GPSDF XBT JO UIF −Y EJSFDUJPO PQQPTJUF UP UIF JOJUJBM EJSFDUJPO PG NPUJPO PG UIF DBSU 8F DPODMVEF UIBU UIF DBSU JOJUJBMMZ NPWFE JO UIF +Y EJSFDUJPO TMPXJOH EPXO DPNJOH UP B TUPQ UIFO NPWJOH BU JODSFBTJOH TQFFE JO UIF −Y EJSFDUJPO " HSBQI PG Y WT U GPS UIF DBSU JT TIPXO JO 'JHVSF  8F VTFE B DPNQVUFS UP DBMDVMBUF BMM UIF Y U WBMVFT OFFEFE GPS UIJT HSBQI


 "OBMZUJDBM 1SFEJDUJPO $POTUBOU /FU 'PSDF &9".1-&



,JDLJOH B #BMM PO UIF .PPO " TPDDFS CBMM XIPTF NBTT JT  LH JT JOJUJBMMZ BU SFTU PO UIF TVSGBDF PG UIF .PPO BU MPDBUJPO    N 0O UIF TVSGBDF PG UIF .PPO UIF DPOTUBOU H ≈  /LH "O BTUSPOBVU √ LJDLT UIF CBMM HJWJOH JU BO JOJUJBM WFMPDJUZ PG    NT UIJT TQFFE PG  +  =  NT DBO CF DPNQBSFE UP UIF UPQ TQFFE PG BCPVU  NT BDIJFWFE CZ QSPGFTTJPOBM TPDDFS QMBZFST 8IBU BSF UIF DPPSEJOBUFT PG UIF IJHIFTU QPTJUJPO UIF CBMM SFBDIFT )PX MPOH JT UIF CBMM JO UIF BJS CFGPSF IJUUJOH UIF HSPVOE )PX GBS EPFT UIF CBMM USBWFM CFGPSF IJUUJOH UIF HSPVOE

4PMVUJPO

*U JT OPU JNNFEJBUFMZ PCWJPVT IPX UP à OE UIF NBYJNVN IFJHIU PG UIF CBMM CFDBVTF UIJT JT OPU B RVBOUJUZ UIBU BQQFBST JO UIF FRVBUJPOT XF EFSJWFE BCPWF *G XF UIJOL BCPVU UIF CBMMT NPUJPO XF OPUF UIBU BU UIF UPQ PG UIF CBMMT USBKFDUPSZ WZ JT JOTUBOUBOFPVTMZ [FSP 5IJT JT B VTFGVM PCTFSWBUJPO CFDBVTF XF DBO TPMWF GPS UIF UJNF BU XIJDI UIJT PDDVST BOE UIFO XF DBO DBMDVMBUF UIF Y BOE Z DPPSEJOBUFT PG UIF CBMM BU UIBU UJNF #FDBVTF UIF .PPO JT BJSMFTT UIFSF JT USVMZ OP BJS SFTJTUBODF BOE UIF OFU GPSDF PO UIF CBMM JT UIF EPXOXBSE HSBWJUBUJPOBM GPSDF  −NH  XIJDI EPFT OPU BGGFDU UIF Y DPNQPOFOU PG UIF CBMMT NPNFOUVN "HBJO TFUUJOH UIF JOJUJBM UJNF U =  BOE TVCTUJUVUJOH U GPS ΔU XF HFU −NH

U N  =  NT + − /LH U U =  T

WZG = WZJ +

 −NH  U  N ZG =  +  NT  T +  − /LH  T  ZG =  N YG = YJ + WYJ U ZG = ZJ + WZJ U +

YG =  +  NT  T

YG =  N 5IF DPPSEJOBUFT PG UIF IJHIFTU QPTJUJPO PG UIF CBMM BSF    N BTTVNJOH XF DIPPTF PVS DPPSEJOBUF BYFT TP UIBU UIF NPUJPO JT JO UIF [ =  QMBOF /PUF UIBU UIF NBYJNVN IFJHIU PG  N JT BCPVU UIF IFJHIU PG B TUPSZ CVJMEJOH /FYU XF à OE UIF UJNF PG áJHIU BOE UIF EJTUBODF UIF CBMM HPFT 0OF XBZ UP àOE UIF áJHIU UJNF JT UP OPUF UIBU PO MFWFM HSPVOE ZG =  XIFO UIF CBMM IJUT  −NH  U  N  =  +  NT U +  − /LH U U =  T

ZG = ZJ + WZJ U +

YG = YJ + WYJ U YG =  +  NT  T

YG =  N 0O &BSUI XIFSF H =  /LH FWFO JG UIFSF XFSF OP BUNPTQIFSF UIJT TPDDFS CBMM XPVME HP POMZ BCPVU  N MFTT UIBO IBMG UIF MFOHUI PG B TPDDFS Ã FME 5IF iIBOH UJNFu PG  T JT WFSZ MPOH PO &BSUI JO UIF BCTFODF PG BJS SFTJTUBODF

JU XPVME CF POMZ  T




$IBQUFS  5IF .PNFOUVN 1SJODJQMF vx

5IF BQQSPBDI XF UPPL BCPWF JT FRVJWBMFOU UP B NPSF GPSNBM DBMDVMVTCBTFE BQQSPBDI 8F DPVME EJGGFSFOUJBUF UIF FRVBUJPO HJWJOH Z BT B GVODUJPO PG U BOE àOE JUT NBYJNVN CZ TFUUJOH EZEU =  &JUIFS BQQSPBDI XJMM ZJFME B DPSSFDU BOTXFS

vx (does not change)

vix

Δt ti

tf

t

x

(SBQIT DBO CF B DPNQBDU XBZ PG SFQSFTFOUJOH QBSUJDVMBS BTQFDUT PG NPUJPO *U JT EJGàDVMU UP HSBQI BMM DPNQPOFOUT PG B TZTUFNT QPTJUJPO WFMPDJUZ PS NPNFOUVN BU PODF TP UZQJDBMMZ XF NBLF POF HSBQI GPS FBDI DPNQPOFOU 'PS FYBNQMF 'JHVSFT  BOE  TIPX HSBQIT PG QPTJUJPO BOE WFMPDJUZ DPNQPOFOUT WT UJNF GPS UIF TPDDFS CBMM JO UIF QSFDFEJOH FYBNQMF

xf

xi

x (increases at constant rate)

ti

tf

t

'JHVSF  .PUJPO HSBQIT GPS UIF LJDLFE CBMM 5PQ WY WT U CPUUPN Y WT U WY EPFT OPU DIBOHF CFDBVTF UIF OFU GPSDF BDUFE POMZ JO UIF Z EJSFDUJPO vy Rising

viy

vy

(SBQIT PG .PUJPO

26&45*0/

4JODF UIF Y DPNQPOFOU PG UIF OFU GPSDF PO UIF CBMM JT [FSP WY EPFTOU DIBOHF BOE UIF HSBQI PG WY WT U JT TJNQMZ B IPSJ[POUBM MJOF XJUI B TMPQF PG [FSP 5IF TMPQF PG WY WT U JT UIF Y DPNQPOFOU PG UIF BDDFMFSBUJPO BY  EWY /EU *O UIJT DBTF BY = 

26&45*0/

Falling

*O 'JHVSF  XIZ JT UIF HSBQI PG WY WT U B IPSJ[POUBM

MJOF

)PX JT UIF HSBQI PG WY WT UJNF SFMBUFE UP UIF HSBQI PG

Y WT UJNF de

cr ea

sin

g

ti

tf

t

5IF HSBQI PG Y WT UJNF JT B TUSBJHIU MJOF SJTJOH CFDBVTF UIF CBMM JT NPWJOH JO UIF +Y EJSFDUJPO 5IF TMPQF PG UIF Y WT U HSBQI JT FRVBM UP WY TJODF WY JT UIF SBUF PG DIBOHF PG Y WY = EY/EU  4JNJMBSMZ UIF TMPQF PG UIF Z WT U HSBQI BU BOZ UJNF JT FRVBM UP WZ BU UIBU UJNF 26&45*0/

vfy

8IBU JT UIF TJHOJàDBODF PG UIF QPJOU BU XIJDI WZ =  JO

'JHVSF 

y Falling

yf

Rising

yi ti

tf

t

8IFO WZ JT FRVBM UP [FSP UIF TMPQF PG Z WT U JT NPNFOUBSJMZ [FSP BOE Z JT B NBYJNVN "U UIJT JOTUBOU UIF CBMM JT OFJUIFS SJTJOH OPS GBMMJOH #FGPSF UIBU QPJOU UIF TMPQF JT QPTJUJWF DPSSFTQPOEJOH UP WZ >  BOE BGUFS UIBU QPJOU UIF TMPQF JT OFHBUJWF DPSSFTQPOEJOH UP WZ <  *O 'JHVSF  UIF HSBQI PG WZ IBT B DPOTUBOU OFHBUJWF TMPQF UIF UPQ HSBQI CFDBVTF UIF Z DPNQPOFOU PG UIF GPSDF IBT UIF DPOTUBOU WBMVF −NH XIJDI NBLFT UIF Z DPNQPOFOU PG NPNFOUVN EFDSFBTF BU B DPOTUBOU SBUF *O UIF OFYU TFDUJPO XF XJMM TFF UIBU UIF FRVBUJPO GPS Z JT B RVBESBUJD GVODUJPO JO UIF UJNF XIJDI JT DPOTJTUFOU XJUI UIF QBSBCPMJD GPSN PG UIF HSBQI PG Z WT UJNF U

This is y vs. time t. It is not the path.

'JHVSF  .PUJPO HSBQIT GPS UIF LJDLFE CBMM 5PQ WZ WT U CPUUPN Z WT U y

yi

4JODF BU TQFFET UIBU BSF MPX DPNQBSFE UP UIF TQFFE PG MJHIU Q ≈ NW QZ ≈ NWZ  " HSBQI PG QZ WT UJNF XPVME MPPL WFSZ TJNJMBS UP UIF HSBQI PG WZ WT UJNF CVU UIF VOJUT PG UIF Z BYJT XPVME CF EJGGFSFOU 5IF BDUVBM QBUI PG UIF CBMM UIF HSBQI PG Z WT Y 'JHVSF  JT BMTP BO JOWFSUFE QBSBCPMB 4JODF Y JODSFBTFT MJOFBSMZ XJUI U XIFUIFS XF QMPU Z WT U PS Z WT Y XFMM TFF B TJNJMBS DVSWF 5IF VOJUT PG UIF IPSJ[POUBM BYJT BSF EJGGFSFOU PG DPVSTF NFUFST JOTUFBE PG TFDPOET 

Actual path: y vs. x 〈xf , yf , 0〉

yf

〈xi, yi, 0〉 xi

xf

)PX XPVME B QMPU PG QZ WT UJNF DPNQBSF UP UIF QMPU PG WZ WT UJNF 'JHVSF 

26&45*0/

x

'JHVSF  5IF BDUVBM USBKFDUPSZ PG UIF UISPXO CBMM XJUI OFHMJHJCMF BJS SFTJTUBODF Z WT Y 

$IFDLQPJOU  " CBMM JT LJDLFE PO &BSUI GSPN B MPDBUJPO   − N PO UIF HSPVOE XJUI JOJUJBM WFMPDJUZ −  − NT /FHMFDUJOH BJS SFTJTUBODF B 8IBU JT UIF WFMPDJUZ PG UIF CBMM  T BGUFS CFJOH LJDLFE C 8IBU JT UIF MPDBUJPO PG UIF CBMM  T BGUFS CFJOH LJDLFE D 8IBU JT UIF NBYJNVN IFJHIU SFBDIFE CZ UIF CBMM


 *UFSBUJWF 1SFEJDUJPO 7BSZJOH /FU 'PSDF



1SPKFDUJPOT PG B i8PSME -JOFu

y

y vs. t y vs. x world line

x x vs. t

t

'JHVSF  JMMVTUSBUFT BO JOUFSFTUJOH XBZ UP UIJOL PG UIF HSBQIT PG Z WT Y Z WT U BOE Y WT U 5IF SFE DVSWF JO 'JHVSF  JT B HSBQIJDBM EFTDSJQUJPO PG UIF NPUJPO JO Y Z BOE U PG B CBMM UIBU NPWFT JO UIF YZ QMBOF *NBHJOF TIJOJOH B MJHIU PO UIF SFE DVSWF UP QSPKFDU B TIBEPX POUP UIF YZ QMBOF PS UIF ZU QMBOF PS UIF YU QMBOF 5IFTF iQSPKFDUJPOTu BSF TIPXO JO DZBO JO UIF à HVSF $PNQBSF UIFTF QSPKFDUJPOT XJUI UIF JOEJWJEVBM HSBQIT TIPXO JO 'JHVSFT   BOE  5IF SFE DVSWF B GVODUJPO PG Y Z BOE U JT DBMMFE B iXPSME MJOFu 8PSME MJOFT QMBZ BO JNQPSUBOU SPMF JO UFYUCPPLT PO TQFDJBM SFMBUJWJUZ *O SFBMJUZ B XPSME MJOF TIPVME CF B GVODUJPO PG Y Z [ BOE U CVU JUT WFSZ EJGàDVMU FWFO XJUI B DPNQVUFS UP ESBX B GPVSEJNFOTJPOBM DVSWF

 *5&3"5*7& 13&%*$5*0/ 7"3:*/( /&5 '03$&

'JHVSF  QSPKFDUJPOT

" iXPSME MJOFu BOE JUT

" GBNJMJBS FYBNQMF PG B GPSDF UIBU JT OPU DPOTUBOU JT UIF GPSDF FYFSUFE CZ B TQSJOH UIBU JT TUSFUDIFE PS DPNQSFTTFE 5IF NBHOJUVEF PG UIJT GPSDF EFQFOET PO UIF BNPVOU PG TUSFUDI PS DPNQSFTTJPO PG UIF TQSJOH 5IF GPSDF JT EJSFDUFE BMPOH UIF MJOF PG UIF TQSJOH

5IF 4QSJOH 'PSDF L0

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0.1

0.2 Time (s)

0.3

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0.2

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0.5 Time (s)

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5IF GPSDFT PO UIF UXP PCKFDUT BSF WFSZ EJGGFSFOU 5IF GBO DBSU JT TVCKFDU POMZ UP B DPOTUBOU GPSDF XIJMF UIF OFU GPSDF PO UIF CMPDL JT UIF TVN PG UXP GPSDFT B DPOTUBOU HSBWJUBUJPOBM GPSDF BOE B TQSJOH GPSDF UIBU WBSJFT XJUI UIF TUSFUDI PG UIF TQSJOH 5IF JOTUSVDUJPOT UIBU TQFDJGZ IPX UP DBMDVMBUF UIF TQSJOH GPSDF NJSSPS XIBU ZPV XPVME EP PO QBQFS VTJOH ZPVS DBMDVMBUPS *G S JT UIF MPDBUJPO PG UIF Ã YFE FOE PG UIF TQSJOH UIFO IFSF JT B USBOTMBUJPO PG UIF BMHFCSBJD FYQSFTTJPOT UP 71ZUIPO FYQSFTTJPOT  =S â&#x2C6;&#x2019;S -

-  CMPDLQPT  TQSJOHQPT

Ë&#x2020; = -/|  -|  -

-IBU  -NBH -

 â&#x2C6;&#x2019; - T = |-|

T  NBH -  -

Ë&#x2020;  = â&#x2C6;&#x2019;LT · T · '

'@TQSJOH  LT T -IBU

26&45*0/

 DBMDVMBUFE 8IFSF JT |-|

71ZUIPO QSPWJEFT B GVODUJPO GPS DBMDVMBUJOH UIF NBHOJUVEF PG B WFDUPS *O UIF  8F DPVME DPEF BCPWF UIF NBH GVODUJPO JT VTFE UP HFU UIF NBHOJUVEF PG - IBWF DBMDVMBUFE JU TFQBSBUFMZ JOTUFBE HJWJOH JU B OBNF TVDI BT -NBH BT JT EPOF CFMPX -NBH  TRSU -Y  -Z  -[ 

-IBU  --NBH $IFDLQPJOU  4PNF DPEF XPVME OFFE UP CF BEEFE JO GSPOU PG FBDI DPNQVUBUJPOBM MPPQ EJTDVTTFE BCPWF JO PSEFS UP NBLF B SVOOBCMF QSPHSBN 8IBU UIJOHT XPVME ZPV OFFE UP JOTUSVDU UIF DPNQVUFS UP EP CFGPSF CFHJOOJOH UIF MPPQ B $SFBUF PCKFDUT C %FÃ OF DPOTUBOUT D 4FU JOJUJBM QPTJUJPOT E 4FU JOJUJBM NPNFOUVN F $BMDVMBUF UIF OFU GPSDF

 %&3*7"5*0/ 41&$*"-$"4& "7&3"(& 7&-0$*5: vx vxf

vx increases at constant rate

)FSF BSF UXP QSPPGT POF HFPNFUSJD BOE POF BMHFCSBJD VTJOH DBMDVMVT GPS UIF GPMMPXJOH TQFDJBMDBTF SFTVMU DPODFSOJOH BWFSBHF WFMPDJUZ WJY + WGY

JG WY DIBOHFT BU B DPOTUBOU SBUF  5IF SFTVMUT BSF TJNJMBS GPS WBWH Z BOE WBWH [ 

WBWH Y = vxi Î&#x201D;t Î&#x201D;t Î&#x201D;t Î&#x201D;t Î&#x201D;t Î&#x201D;t Î&#x201D;t Î&#x201D;t Î&#x201D;t Î&#x201D;t Î&#x201D;t ti tf t 'JHVSF  (SBQI PG WY WT U DPOTUBOU GPSDF EJWJEFE JOUP OBSSPX WFSUJDBM TMJDFT FBDI PG IFJHIU WY BOE XJEUI Î&#x201D;U

(FPNFUSJD 1SPPG *G 'OFU Y JT DPOTUBOU QGY = QJY + 'OFU Y Î&#x201D;U JNQMJFT UIBU QY DIBOHFT BU B DPOTUBOU SBUF "U TQFFET TNBMM DPNQBSFE UP UIF TQFFE PG MJHIU WY â&#x2030;&#x2C6; QY /N TP B HSBQI PG WY WT UJNF JT B TUSBJHIU MJOF BT JO 'JHVSF  6TJOH UIJT HSBQI XF GPSN OBSSPX WFSUJDBM TMJDFT FBDI PG IFJHIU WY BOE OBSSPX XJEUI Î&#x201D;U 8JUIJO FBDI OBSSPX TMJDF WY DIBOHFT WFSZ MJUUMF TP UIF DIBOHF JO QPTJUJPO EVSJOH UIF CSJFG UJNF Î&#x201D;U JT BQQSPYJNBUFMZ Î&#x201D;Y = WY Î&#x201D;U 5IFSFGPSF UIF DIBOHF JO Y JT BQQSPYJNBUFMZ FRVBM UP UIF BSFB PG UIF TMJDF PG IFJHIU WY BOE XJEUI Î&#x201D;U 'JHVSF   *G XF BEE VQ UIF BSFBT PG BMM UIFTF TMJDFT XF HFU BQQSPYJNBUFMZ UIF BSFB VOEFS UIF MJOF JO 'JHVSF  BOE UIJT JT BMTP FRVBM UP UIF UPUBM EJTQMBDFNFOU Î&#x201D;Y + Î&#x201D;Y + Î&#x201D;Y + · · · = YG â&#x2C6;&#x2019; YJ  *G XF HP UP UIF MJNJU PG BO JOà OJUF OVNCFS PG TMJDFT FBDI XJUI JOàOJUFTJNBM XJEUI UIF TVN PG TMJDFT SFBMMZ JT UIF BSFB BOE UIJT




$IBQUFS  5IF .PNFOUVN 1SJODJQMF

vx (avg)

BSFB XF IBWF TIPXO UP CF FRVBM UP UIF DIBOHF JO QPTJUJPO 5IJT LJOE PG TVN PG BO JOà OJUF OVNCFS PG JOàOJUFTJNBM QJFDFT JT DBMMFE BO iJOUFHSBMu JO DBMDVMVT 5IF BSFB VOEFS UIF MJOF JT B USBQF[PJE BOE GSPN HFPNFUSZ XF LOPX UIBU UIF BSFB PG B USBQF[PJE JT UIF BWFSBHF PG UIF UXP CBTFT UJNFT UIF BMUJUVEF BSFB = Î&#x201D;t

Altitude = Î&#x201D;t

'JHVSF  0OF OBSSPX TMJDF IBT BO BSFB HJWFO BQQSPYJNBUFMZ CZ WY Î&#x201D;U 5IJT JT FRVBM UP Î&#x201D;Y UIF EJTQMBDFNFOU PG UIF PCKFDU

Top = vxi

UPQ + CPUUPN

BMUJUVEF



5VSO 'JHVSF  PO JUT TJEF BT JO 'JHVSF  BOE ZPV TFF UIBU UIF UPQ BOE CPUUPN IBWF MFOHUIT WJY BOE WGY XIJMF UIF BMUJUVEF PG UIF USBQF[PJE JT UIF UPUBM UJNF UG â&#x2C6;&#x2019; UJ  5IFSFGPSF XF IBWF UIF GPMMPXJOH SFTVMU USBQF[PJE BSFB = YG â&#x2C6;&#x2019; YJ =

WJY + WGY

UG â&#x2C6;&#x2019; UJ



%JWJEJOH CZ UG â&#x2C6;&#x2019; UJ XF IBWF UIJT YG â&#x2C6;&#x2019; YJ WJY + WGY

= UG â&#x2C6;&#x2019; U J 

Bottom = vxf

'JHVSF  5IF BSFB PG UIF XIPMF USBQF[PJE JT FRVBM UP UIF UPUBM EJTQMBDFNFOU 5IF Ã HVSF JO 'JHVSF  IBT CFFO SPUBUFE  EFHSFFT DMPDLXJTF

#Z EFÃ OJUJPO IPXFWFS UIF Y DPNQPOFOU PG BWFSBHF WFMPDJUZ JT UIF DIBOHF JO Y EJWJEFE CZ UIF UPUBM UJNF TP XF IBWF QSPWFE UIBU WBWH Y =

WJY + WGY



JG WY DIBOHFT MJOFBSMZ XJUI UJNF

5IF QSPPG EFQFOEFE DSJUJDBMMZ PO UIF TUSBJHIUMJOF iMJOFBSu DIBOHF JO WFMPDJUZ XIJDI PDDVST JG 'OFU Y JT DPOTUBOU BOE W  D  0UIFSXJTF XF XPVMEOU IBWF B USBQF[PJEBM BSFB 5IBUT XIZ UIF SFTVMU JTOU USVF JO HFOFSBM JUT BO JNQPSUBOU CVU TQFDJBM DBTF

"MHFCSBJD 1SPPG 6TJOH $BMDVMVT "O BMHFCSBJD QSPPG VTJOH DBMDVMVT DBO BMTP CF HJWFO 8F XJMM VTF UIF Y DPNQPOFOU PG UIF EFSJWBUJWF WFSTJPO PG UIF .PNFOUVN 1SJODJQMF NPSF BCPVU UIJT JO B MBUFS DIBQUFS   OFU Î&#x201D;U Î&#x201D;Q = '

JNQMJFT UIBU

Î&#x201D;Q  = 'OFU Î&#x201D;U

*O UIF MJNJU XF IBWF MJN

Î&#x201D; Uâ&#x2020;&#x2019; 

Î&#x201D;Q EQ  = = 'OFU Î&#x201D;U EU

BOE

EQY = 'OFU Y EU

*G 'OFU Y JT B DPOTUBOU UIF UJNF EFSJWBUJWF PG QY JT B DPOTUBOU TP XF IBWF QY = 'OFU Y U + QJY

TJODF QY = QJY XIFO U = 

:PV DBO DIFDL UIJT CZ UBLJOH UIF EFSJWBUJWF XJUI SFTQFDU UP UJNF U XIJDI HJWFT UIF PSJHJOBM FRVBUJPO EQY /EU = 'OFU Y  "U TQFFET TNBMM DPNQBSFE UP UIF TQFFE PG MJHIU WY â&#x2030;&#x2C6; QY /N TP XF DBO XSJUF WY =

'OFU Y U + WJY N

TJODF WY = WJY XIFO U = 


 3FMBUJWJTUJD .PUJPO



)PXFWFS UIF Y DPNQPOFOU PG WFMPDJUZ JT UIF SBUF BU XIJDI Y JT DIBOHJOH WY =

EY 'OFU Y = U + WJY EU N

/PX UIF RVFTUJPO JT DBO ZPV UIJOL PG B GVODUJPO PG Y UIBU IBT UIJT UJNF EFSJWBUJWF 4JODF UIF UJNF EFSJWBUJWF PG U JT U UIF GPMMPXJOH FRVBUJPO GPS Y IBT UIF BQQSPQSJBUF EFSJWBUJWF Y=

 'OFU Y  U + WJY U + YJ  N

TJODF Y = YJ XIFO U = 

:PV DBO DIFDL UIJT CZ UBLJOH UIF EFSJWBUJWF XJUI SFTQFDU UP U XIJDI HJWFT UIF FRVBUJPO GPS WY TJODF E  U /EU = U BOE E U /EU =  5IF BWFSBHF WFMPDJUZ UIBU XF TFFL JT UIF DIBOHF JO QPTJUJPO EJWJEFE CZ UIF UPUBM UJNF WBWH Y =

Y â&#x2C6;&#x2019; YJ   'OFU Y = U + WJY = WGY â&#x2C6;&#x2019; WJY + WJY U  N 

XIFSF XF IBWF VTFE UIF FRVBUJPO XF QSFWJPVTMZ EFSJWFE GPS UIF WFMPDJUZ WGY = WY =

'OFU Y U + WJY N

4JNQMJGZJOH UIF FYQSFTTJPO GPS WBWH Y XF IBWF UIF QSPPG WBWH Y =

WJY + WGY



JG WY DIBOHFT BU B DPOTUBOU SBUF

 3&-"5*7*45*$ .05*0/ *OFSUJBM 'SBNFT PG 3FGFSFODF /FXUPOT à STU MBX JT WBMJE POMZ JO BO iJOFSUJBM GSBNFu PG SFGFSFODF POF JO VOJGPSN NPUJPO PS BU SFTU XJUI SFTQFDU UP UIF QFSWBTJWF iDPTNJD NJDSPXBWF CBDLHSPVOEu TFF UIF PQUJPOBM EJTDVTTJPO BU UIF FOE PG $IBQUFS   4JODF UIF .PNFOUVN 1SJODJQMF JT B RVBOUJUBUJWF WFSTJPO PG /FXUPOT àSTU MBX XF FYQFDU UIF .PNFOUVN 1SJODJQMF UP CF WBMJE JO BO JOFSUJBM SFGFSFODF GSBNF CVU OPU JO B SFGFSFODF GSBNF UIBU JT OPU JO VOJGPSN NPUJPO -FUT DIFDL UIBU UIJT JT USVF *G ZPV WJFX TPNF PCKFDUT GSPN B TQBDF TIJQ UIBU JT NPWJOH VOJGPSNMZ XJUI WFMPDJUZ WT XJUI SFTQFDU UP UIF DPTNJD NJDSPXBWF CBDLHSPVOE BMM PG UIF WFMPDJUJFT PG UIPTF PCKFDUT IBWF UIF DPOTUBOU WT TVCUSBDUFE GSPN UIFN BT GBS BT ZPV BSF DPODFSOFE 'PS FYBNQMF B SPDL NPWJOH BU UIF TBNF WFMPDJUZ BT ZPVS TQBDFDSBGU XPVME IBWF WSPDL = W â&#x2C6;&#x2019;WT =  JO ZPVS SFGFSFODF GSBNF JU XPVME BQQFBS UP CF TUBUJPOBSZ BT JU DPBTUFE BMPOH CFTJEF ZPVS TQBDFDSBGU 8JUI B DPOTUBOU TQBDF TIJQ WFMPDJUZ XF IBWF Î&#x201D;WT =  BOE UIF DIBOHF PG NPNFOUVN PG UIF NPWJOH PCKFDU SFEVDFT UP UIF GPMMPXJOH GPS TQFFET TNBMM DPNQBSFE UP D  Î&#x201D;<N W â&#x2C6;&#x2019;WT > = Î&#x201D; NW â&#x2C6;&#x2019; Î&#x201D; NWT = Î&#x201D; NW TJODF Î&#x201D; NWT =  5IFSFGPSF  OFU Î&#x201D;U Î&#x201D;<N W â&#x2C6;&#x2019;WT > = Î&#x201D; NW = '




$IBQUFS  5IF .PNFOUVN 1SJODJQMF

*G UIF WFMPDJUZ WT PG UIF TQBDF TIJQ EPFTOU DIBOHF JU SFQSFTFOUT BO JOFSUJBM GSBNF PG SFGFSFODF UIF GPSN BOE WBMJEJUZ PG UIF .PNFOUVN 1SJODJQMF JT VOBGGFDUFE CZ UIF NPUJPO PG UIF TQBDF TIJQ )PXFWFS JG ZPVS TQBDF TIJQ JODSFBTFT JUT TQFFE PS DIBOHFT EJSFDUJPO Î&#x201D;WT = /  BO PCKFDUT NPUJPO SFMBUJWF UP ZPV DIBOHFT XJUIPVU BOZ GPSDF BDUJOH PO JU *O UIBU DBTF UIF .PNFOUVN 1SJODJQMF JT OPU WBMJE GPS UIF PCKFDU CFDBVTF ZPV BSF OPU JO BO JOFSUJBM GSBNF

'BTU.PWJOH 0CKFDUT "QQMZJOH UIF .PNFOUVN 1SJODJQMF UP BO PCKFDU NPWJOH BU B TQFFE OFBS UIF TQFFE PG MJHIU JT B CJU NPSF DPNQMFY BMHFCSBJDBMMZ CVU DPODFQUVBMMZ FYBDUMZ UIF TBNF BT XIBU XF IBWF EPOF QSFWJPVTMZ 5IF EFSJWBUJPO PG UIF FRVBUJPO GPS TQFFE JO UFSNT PG NPNFOUVN GPS B GBTUNPWJOH QBSUJDMF NBZ CF GPVOE JO $IBQUFS  &9".1-&

'BTU 1SPUPO % $POTUBOU /FU 'PSDF 3FMBUJWJTUJD " QSPUPO JO B QBSUJDMF BDDFMFSBUPS JT NPWJOH XJUI WFMPDJUZ D   TP UIF TQFFE JT  Ã&#x2014;  Ã&#x2014;  NT =  Ã&#x2014;  NT " DPOTUBOU FMFDUSJD  OFU =  Ã&#x2014; â&#x2C6;&#x2019;   / GPSDF JT BQQMJFE UP UIF QSPUPO UP TQFFE JU VQ ' 8IBU JT UIF QSPUPOT TQFFE BT B GSBDUJPO PG UIF TQFFE PG MJHIU BGUFS  OBOPTFDPOET  OT =  Ã&#x2014; â&#x2C6;&#x2019; T

4PMVUJPO

4ZTUFN 5IF QSPUPO 4VSSPVOEJOHT &MFDUSJD DIBSHFT JO UIF BDDFMFSBUPS 'SFFCPEZ EJBHSBN 'JHVSF  .PNFOUVN 1SJODJQMF

Proton 'JHVSF  5IF GSFFCPEZ EJBHSBN GPS UIF QSPUPO

 OFU Î&#x201D;U QG = QJ + ' QGY   = γJ NWJY   + 'OFU Y   Î&#x201D;U 'JOE UIF Y DPNQPOFOU PG UIF JOJUJBM NPNFOUVN γ = 





D â&#x2C6;&#x2019; D

 = 

QJY = γJ NWJY =  ·  Ã&#x2014; â&#x2C6;&#x2019; LH  ·  Ã&#x2014;  NT

=  Ã&#x2014; â&#x2C6;&#x2019; LH · NT 'JOE UIF Y DPNQPOFOU PG UIF JNQVMTF 'OFU Y Î&#x201D;U =  Ã&#x2014; â&#x2C6;&#x2019; / ·  Ã&#x2014; â&#x2C6;&#x2019; T =  Ã&#x2014; â&#x2C6;&#x2019; / · T "QQMZ UIF .PNFOUVN 1SJODJQMF QGY =  Ã&#x2014; â&#x2C6;&#x2019; LH · NT +  Ã&#x2014; â&#x2C6;&#x2019; / · T

=  Ã&#x2014; â&#x2C6;&#x2019; LH · NT


 .FBTVSFNFOUT BOE 6OJUT



'JOE UIF àOBM TQFFE QGY WGY =  ND Q  D GY + ND QGY  × − LH · NT =  OP VOJUT

= ND  × − LH  ×  NT

WGY  =  =√ D  +  WGY = D "MUIPVHI UIF NBHOJUVEF PG UIF NPNFOUVN JODSFBTFE GSPN  × − LH · NT UP  × − LH · NT UIF TQFFE EJEOU JODSFBTF WFSZ NVDI CFDBVTF UIF QSPUPOT JOJUJBM TQFFE D XBT BMSFBEZ DMPTF UP UIF DPTNJD TQFFE MJNJU D #FDBVTF UIF TQFFE IBSEMZ DIBOHFE UIF EJTUBODF UIF QSPUPO NPWFE EVSJOH UIF  OT XBT BQQSPYJNBUFMZ FRVBM UP  ×  ×  NT  × − T =  N

 .&"463&.&/54 "/% 6/*54 6TJOH UIF .PNFOUVN 1SJODJQMF SFRVJSFT B DPOTJTUFOU XBZ UP NFBTVSF MFOHUI UJNF NBTT BOE GPSDF BOE B DPOTJTUFOU TFU PG VOJUT 8F TUBUF UIF EFàOJUJPOT PG UIF TUBOEBSE 4ZTUÍNF *OUFSOBUJPOBMF 4* VOJUT BOE XF CSJFáZ EJTDVTT TPNF TVCUMF JTTVFT VOEFSMZJOH UIJT DIPJDF PG VOJUT

6OJUT .FUFST 4FDPOET ,JMPHSBNT $PVMPNCT BOE /FXUPOT 0SJHJOBMMZ UIF NFUFS XBT EFàOFE BT UIF EJTUBODF CFUXFFO UXP TDSBUDIFT PO B QMBUJOVN CBS JO B WBVMU JO 1BSJT BOE B TFDPOE XBT  UI PG B iNFBO TPMBS EBZu /PX IPXFWFS UIF TFDPOE JT EFàOFE JO UFSNT PG UIF GSFRVFODZ PG MJHIU FNJUUFE CZ B DFTJVN BUPN BOE UIF NFUFS JT EFàOFE BT UIF EJTUBODF MJHIU USBWFMT JO   UI PG B TFDPOE PS BCPVU ×− T  OT  5IF TQFFE PG MJHIU JT EFàOFE UP CF FYBDUMZ    NT WFSZ DMPTF UP  ×  NT  "T B SFTVMU PG UIFTF NPEFSO SFEFàOJUJPOT JU JT SFBMMZ TQFFE PG MJHIU BOE UJNF UIBU BSF UIF JOUFSOBUJPOBMMZ BHSFFEVQPO CBTJD VOJUT OPU MFOHUI BOE UJNF #Z JOUFSOBUJPOBM BHSFFNFOU POF LJMPHSBN JT UIF NBTT PG B QMBUJOVN CMPDL LFQU JO UIBU TBNF WBVMU JO 1BSJT "T B QSBDUJDBM NBUUFS PUIFS NBTTFT BSF DPNQBSFE UP UIJT TUBOEBSE LJMPHSBN CZ VTJOH B CBMBODFCFBN PS TQSJOH XFJHIJOH TDBMF NPSF BCPVU UIJT JO B NPNFOU  5IF OFXUPO UIF VOJU PG GPSDF JT EFàOFE BT UIBU GPSDF XIJDI BDUJOH GPS  T JNQBSUT UP  LH B WFMPDJUZ DIBOHF PG  NT 8F DPVME NBLF B TDBMF GPS GPSDF CZ DBMJCSBUJOH UIF BNPVOU PG TUSFUDI PG B TQSJOH JO UFSNT PG OFXUPOT 5IF DPVMPNC UIF 4* VOJU PG FMFDUSJD DIBSHF JT EFàOFE JO UFSNT PG FMFDUSJD DVSSFOUT 5IF DIBSHF PG B QSPUPO JT  × − $

4PNF 4VCUMF *TTVFT 8IBU XF IBWF KVTU TBJE BCPVU 4* VOJUT JT TVGàDJFOU GPS QSBDUJDBM QVSQPTFT UP QSFEJDU UIF NPUJPO PG PCKFDUT CVU IFSF BSF TPNF RVFTUJPOT UIBU NJHIU CPUIFS ZPV *T JU MFHJUJNBUF UP NFBTVSF UIF NBTT UIBU BQQFBST JO UIF .PNFOUVN 1SJODJQMF CZ TFFJOH IPX UIBU NBTT JT BGGFDUFE CZ HSBWJUZ PO B CBMBODFCFBN TDBMF *T JU MFHJUJNBUF UP VTF UIF .PNFOUVN 1SJODJQMF UP EFàOF UIF VOJUT PG




$IBQUFS  5IF .PNFOUVN 1SJODJQMF

GPSDF XIFO UIF DPODFQU PG GPSDF JT JUTFMG BTTPDJBUFE XJUI UIF TBNF MBX *T UIJT BMM DJSDVMBS SFBTPOJOH BOE UIF .PNFOUVN 1SJODJQMF NFSFMZ B EFàOJUJPO XJUI MJUUMF DPOUFOU )FSF JT B DIBJO PG SFBTPOJOH UIBU BEESFTTFT UIFTF JTTVFT

.FBTVSJOH *OFSUJBM .BTT 8IFO XF VTF CBMBODFCFBN PS TQSJOH XFJHIJOH TDBMFT UP NFBTVSF NBTT XIBU XFSF SFBMMZ NFBTVSJOH JT UIF iHSBWJUBUJPOBM NBTTu‡UIBU JT UIF NBTT UIBU BQQFBST JO UIF MBX PG HSBWJUBUJPO BOE JT B NFBTVSF PG IPX NVDI UIJT PCKFDU JT BGGFDUFE CZ UIF HSBWJUZ PG UIF &BSUI *O QSJODJQMF UIJT iHSBWJUBUJPOBM NBTTu DPVME CF EJGGFSFOU GSPN UIF TPDBMMFE iJOFSUJBM NBTTu‡UIF NBTT UIBU BQQFBST JO UIF EFàOJUJPO PG NPNFOUVN *U JT QPTTJCMF UP DPNQBSF UIF JOFSUJBM NBTTFT PG UXP PCKFDUT BOE XF àOE FYQFSJNFOUBMMZ UIBU JOFSUJBM BOE HSBWJUBUJPOBM NBTT TFFN UP CF FOUJSFMZ FRVJWBMFOU )FSF JT B XBZ UP DPNQBSF UXP JOFSUJBM NBTTFT EJSFDUMZ XJUIPVU JOWPMWJOH HSBWJUZ 4UBSUJOH GSPN SFTU QVMM PO UIF àSTU PCKFDU XJUI B TQSJOH TUSFUDIFE CZ TPNF BNPVOU T GPS BO BNPVOU PG UJNF ΔU BOE NFBTVSF UIF JODSFBTF PG TQFFE ΔW  5IFO TUBSUJOH GSPN SFTU QVMM PO UIF TFDPOE PCKFDU XJUI UIF TBNF TQSJOH TUSFUDIFE CZ UIF TBNF BNPVOU T GPS UIF TBNF BNPVOU PG UJNF ΔU BOE NFBTVSF UIF JODSFBTF PG TQFFE ΔW  8F EFàOF UIF SBUJP PG UIF JOFSUJBM NBTTFT BT N /N = ΔW /ΔW  4JODF POF PG UIFTF NBTTFT DPVME CF UIF TUBOEBSE LJMPHSBN LFQU JO 1BSJT XF OPX IBWF B XBZ PG NFBTVSJOH JOFSUJBM NBTT JO LJMPHSBNT )BWJOH EFàOFE JOFSUJBM NBTT UIJT XBZ XF àOE FYQFSJNFOUBMMZ UIBU UIF .PNFOUVN 1SJODJQMF JT PCFZFE CZ CPUI PG UIFTF PCKFDUT JO BMM TJUVBUJPOT OPU KVTU JO UIF POF TQFDJBM FYQFSJNFOU XF VTFE UP DPNQBSF UIF UXP NBTTFT .PSFPWFS XF àOE UP FYUSFNFMZ IJHI QSFDJTJPO UIBU UIF JOFSUJBM NBTT JO LJMPHSBNT NFBTVSFE CZ UIJT DPNQBSJTPO FYQFSJNFOU JT FYBDUMZ UIF TBNF BT UIF HSBWJUBUJPOBM NBTT JO LJMPHSBNT PCUBJOFE CZ DPNQBSJOH XJUI B TUBOEBSE LJMPHSBN PO B CBMBODFCFBN TDBMF PS VTJOH B DBMJCSBUFE TQSJOH TDBMF BOE UIBU JU EPFTOU NBUUFS XIBU UIF PCKFDUT BSF NBEF PG XPPE DPQQFS HMBTT FUD  5IJT KVTUJàFT UIF DPOWFOJFOU VTF PG PSEJOBSZ XFJHIJOH TDBMFT UP EFUFSNJOF JOFSUJBM NBTT

*T 5IJT $JSDVMBS 3FBTPOJOH 5IF EFàOJUJPOT PG GPSDF BOE NBTT NBZ TPVOE MJLF DJSDVMBS SFBTPOJOH BOE UIF .PNFOUVN 1SJODJQMF NBZ TPVOE MJLF KVTU B LJOE PG EFàOJUJPO XJUI OP SFBM DPOUFOU CVU UIFSF JT SFBM QPXFS JO UIF .PNFOUVN 1SJODJQMF 'PSHFU GPS B NPNFOU UIF EFàOJUJPO PG GPSDF JO OFXUPOT BOE NBTT JO LJMPHSBNT 5IF FYQFSJNFOUBM GBDU SFNBJOT UIBU BOZ PCKFDU JG TVCKFDUFE UP B TJOHMF GPSDF CZ B TQSJOH XJUI DPOTUBOU TUSFUDI FYQFSJFODFT B DIBOHF PG NPNFOUVN BOE WFMPDJUZ QSPQPSUJPOBM UP UIF EVSBUJPO PG UIF JOUFSBDUJPO /PUF UIBU JU JT OPU B DIBOHF PG QPTJUJPO QSPQPSUJPOBM UP UIF UJNF UIBU XPVME CF B DPOTUBOU TQFFE CVU B DIBOHF PG WFMPDJUZ 5IBUT SFBM DPOUFOU .PSFPWFS XF àOE UIBU UIF DIBOHF PG WFMPDJUZ JT QSPQPSUJPOBM UP UIF BNPVOU PG TUSFUDI PG UIF TQSJOH 5IBU UPP JT SFBM DPOUFOU 5IFO XF àOE UIBU B EJGGFSFOU PCKFDU VOEFSHPFT B EJGGFSFOU SBUF PG DIBOHF PG WFMPDJUZ XJUI UIF TBNF TQSJOH TUSFUDI CVU BGUFS XFWF NBEF POF TJOHMF DPNQBSJTPO FYQFSJNFOU UP EFUFSNJOF UIF NBTT SFMBUJWF UP UIF TUBOEBSE LJMPHSBN UIF .PNFOUVN 1SJODJQMF XPSLT JO BMM TJUVBUJPOT 5IBUT SFBM DPOUFOU 'JOBMMZ DPNF UIF EFUBJMT PG TFUUJOH TUBOEBSET GPS NFBTVSJOH GPSDF JO OFXUPOT BOE NBTT JO LJMPHSBNT BOE XF VTF UIF .PNFOUVN 1SJODJQMF JO IFMQJOH TFU UIFTF TUBOEBSET -PHJDBMMZ IPXFWFS UIJT DPNFT BGUFS IBWJOH FTUBCMJTIFE UIF MBX JUTFMG


2VFTUJPOT



46.."3: 4ZTUFN JT B QPSUJPO PG UIF 6OJWFSTF BDUFE PO CZ UIF TVSSPVOEJOHT 'PSDF JT B RVBOUJUBUJWF NFBTVSF PG JOUFSBDUJPOT VOJUT BSF OFXUPOT /   OFU JT UIF WFDUPS TVN PG BMM UIF GPSDFT BDUJOH PO /FU GPSDF ' B TZTUFN

4QFDJBMDBTF SFTVMU GPS BWFSBHF WFMPDJUZ JG OFU GPSDF JT DPOTUBOU WBWH Y =

WJY + WGY



3FTVMUT BSF TJNJMBS GPS WBWH Z BOE WBWH [  "OBMZUJDBM TPMVUJPO GPS NPUJPO XJUI DPOTUBOU OFU GPSDF

 *NQVMTF JT UIF QSPEVDU PG GPSDF UJNFT UJNF 'Î&#x201D;U NPNFOUVN DIBOHF FRVBMT OFU JNQVMTF UIF JNQVMTF EVF UP UIF OFU GPSDF  1IZTJDBM NPEFMT BSF USBDUBCMF BQQSPYJNBUJPOTJEFBMJ[BUJPOT PG UIF SFBM XPSME

WY U = WYJ +

 'OFU Y  U  N

'OFU Y U N

4QFDJÃ D GPSDFT

Ë&#x2020;  = â&#x2C6;&#x2019;LT T' VQEBUF GPSN

PS DIPPTJOH iOPXu BT UIF JOJUJBM UJNF BOE iB TIPSU UJNF JO UIF GVUVSFu BT UIF Ã OBM UJNF XF DBO XSJUF  OFU OPX Î&#x201D;U Q GVUVSF = Q OPX + ' 5IF UJNF JOUFSWBM Î&#x201D;U NVTU CF TNBMM FOPVHI UIBU UIF OFU GPSDF JT OFBSMZ DPOTUBOU EVSJOH UIJT UJNF JOUFSWBM *5&3"5*7& 13&%*$5*0/ 0' .05*0/  OFU $BMDVMBUF UIF OFU WFDUPS GPSDF ' BDUJOH PO UIF TZTUFN

3FQFBU

Y U = YJ + WYJ U +

'PSDF CZ B TQSJOH

5IF .PNFOUVN 1SJODJQMF  OFU Î&#x201D;U Î&#x201D;Q = '  OFU Î&#x201D;U QG = QJ + '

JG WY DIBOHFT MJOFBSMZ XJUI UJNF

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XIFSF H = + // LH

GPS BO PCKFDU PG NBTT N OFBS UIF TVSGBDF PG UIF &BSUI

6QEBUF UIF NPNFOUVN PG UIF TZTUFN  OFU Î&#x201D;U QG = QJ + ' 6QEBUF UIF QPTJUJPOSG =SJ +WBWH Î&#x201D;U

6TF QG /N UP BQQSPYJNBUF UIF WBWH JO FBDI TUFQ

26&45*0/4 2 #FDBVTF UIF DIBOHF PG UIF NPNFOUVN JT FRVBM UP UIF OFU JNQVMTF UIF SFMBUJPOTIJQ PG NPNFOUVN JUTFMG UP UIF OFU GPSDF JT TPNFXIBU JOEJSFDU BT DBO CF TFFO JO UIJT RVFTUJPO "O PCKFDU JT JOJUJBMMZ NPWJOH JO UIF +Y EJSFDUJPO XJUI NBHOJUVEF PG NPNFOUVN Q XJUI B OFU GPSDF PG NBHOJUVEF ' BDUJOH PO UIF PCKFDU JO FJUIFS UIF +Y PS â&#x2C6;&#x2019;Y EJSFDUJPO "GUFS B WFSZ TIPSU UJNF TBZ XIFUIFS UIF NBHOJUVEF PG UIF NPNFOUVN JODSFBTFT EFDSFBTFT PS TUBZT UIF TBNF JO FBDI PG UIF GPMMPXJOH TJUVBUJPOT B 5IF OFU GPSDF BDUT JO UIF +Y EJSFDUJPO BOE ' JT DPOTUBOU C 5IF OFU GPSDF BDUT JO UIF +Y EJSFDUJPO BOE ' JT JODSFBTJOH D 5IF OFU GPSDF BDUT JO UIF +Y EJSFDUJPO BOE ' JT EFDSFBTJOH E 5IF OFU GPSDF BDUT JO UIF â&#x2C6;&#x2019;Y EJSFDUJPO BOE ' JT DPOTUBOU F 5IF OFU GPSDF BDUT JO UIF â&#x2C6;&#x2019;Y EJSFDUJPO BOE ' JT JODSFBTJOH G 5IF OFU GPSDF BDUT JO UIF â&#x2C6;&#x2019;Y EJSFDUJPO BOE ' JT EFDSFBTJOH

2 "O PCKFDU JT NPWJOH JO UIF +Z EJSFDUJPO 8IJDI JG BOZ PG UIF GPMMPXJOH TUBUFNFOUT NJHIU CF USVF $IFDL BMM UIBU BQQMZ B 5IF OFU GPSDF PO UIF PCKFDU JT [FSP C 5IF OFU GPSDF PO UIF PCKFDU JT JO UIF â&#x2C6;&#x2019;Z EJSFDUJPO D 5IF OFU GPSDF PO UIF PCKFDU JT JO UIF +Z EJSFDUJPO 2 :PV PCTFSWF UISFF DBSUT NPWJOH UP UIF MFGU $BSU " NPWFT UP UIF MFGU BU OFBSMZ DPOTUBOU TQFFE $BSU # NPWFT UP UIF MFGU HSBEVBMMZ TQFFEJOH VQ $BSU $ NPWFT UP UIF MFGU HSBEVBMMZ TMPXJOH EPXO 8IJDI DBSU PS DBSUT JG BOZ FYQFSJFODF B OFU GPSDF UP UIF MFGU 2 *O PSEFS UP QVMM B TMFE BDSPTT B MFWFM à FME BU DPOTUBOU WFMPDJUZ ZPV IBWF UP FYFSU B DPOTUBOU GPSDF %PFTOU UIJT WJPMBUF /FXUPOT àSTU BOE TFDPOE MBXT PG NPUJPO XIJDI JNQMZ UIBU OP GPSDF JT




$IBQUFS  5IF .PNFOUVN 1SJODJQMF

SFRVJSFE UP NBJOUBJO B DPOTUBOU WFMPDJUZ &YQMBJO UIJT TFFNJOH DPOUSBEJDUJPO 2 *O B MBC FYQFSJNFOU ZPV PCTFSWF UIBU B QFOEVMVN TXJOHT XJUI B iQFSJPEu UJNF GPS POF SPVOE USJQ PG  T *O BO JUFSBUJWF DBMDVMBUJPO PG UIF NPUJPO XIJDI PG UIF GPMMPXJOH XPVME /05 CF B SFBTPOBCMF DIPJDF GPS Î&#x201D;U GPS FJUIFS IBOE PS DPNQVUFS JUFSBUJWF DBMDVMBUJPOT B  T C  T D  T E  T 2 " DPNFU QBTTFT OFBS UIF 4VO 8IFO UIF DPNFU JT DMPTFTU UP UIF 4VO JU JT  Ã&#x2014;  N GSPN UIF 4VO :PV OFFE UP DIPPTF B UJNF TUFQ UP VTF JO QSFEJDUJOH UIF DPNFUT NPUJPO 8IJDI PG UIF GPMMPXJOH XPVME CF B SFBTPOBCMF EJTUBODF GPS UIF DPNFU UP NPWF JO POF UJNF TUFQ EPJOH BO JUFSBUJWF DBMDVMBUJPO CZ IBOE B  Ã&#x2014;  N C  Ã&#x2014;  N D  Ã&#x2014;  N E  Ã&#x2014;  N

2 " CBMM NPWFT JO UIF EJSFDUJPO PG UIF BSSPX MBCFMFE D JO 'JHVSF  5IF CBMM JT TUSVDL CZ B TUJDL UIBU CSJFáZ FYFSUT B GPSDF PO UIF CBMM JO UIF EJSFDUJPO PG UIF BSSPX MBCFMFE F 8IJDI BSSPX CFTU EFTDSJCFT UIF EJSFDUJPO PG Î&#x201D;Q UIF DIBOHF JO UIF CBMMT NPNFOUVN

h

a

b

g

c f

d e j zero magnitude 'JHVSF 

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rr1 " 1JOH1POH CBMM JT BDUFE VQPO CZ UIF &BSUI BJS SFTJTUBODF BOE B TUSPOH XJOE )FSF BSF UIF QPTJUJPOT PG UIF CBMM BU TFWFSBM UJNFT &BSMZ UJNF JOUFSWBM "U U =  T UIF QPTJUJPO XBT   â&#x2C6;&#x2019; N "U U =  T UIF QPTJUJPO XBT   â&#x2C6;&#x2019; N

UIF TUBSU PG UIF FBSMZ UJNF JOUFSWBM UP U =  T UIF TUBSU PG UIF MBUF UJNF JOUFSWBM XIBU XBT UIF BWFSBHF OFU GPSDF BDUJOH PO UIF CBMM &YQSFTT ZPVS SFTVMU BT B WFDUPS rr1 "  LH CMPDL PG JDF JT TMJEJOH CZ ZPV PO B WFSZ TMJQQFSZ áPPS BU  NT "T JU HPFT CZ ZPV HJWF JU B LJDL QFSQFOEJDVMBS UP JUT QBUI :PVS GPPU JT JO DPOUBDU XJUI UIF JDF CMPDL GPS  T 5IF CMPDL FWFOUVBMMZ TMJEFT BU BO BOHMF PG  EFHSFFT GSPN JUT PSJHJOBM EJSFDUJPO 5IF PWFSIFBE WJFX TIPXO JO 'JHVSF  JT BQQSPYJNBUFMZ UP TDBMF 5IF BSSPX SFQSFTFOUT UIF BWFSBHF GPSDF ZPVS UPF BQQMJFT CSJFáZ UP UIF CMPDL PG JDF B 8IJDI PG UIF QPTTJCMF QBUIT TIPXO JO UIF EJBHSBN DPSSFTQPOET UP UIF DPSSFDU PWFSIFBE WJFX PG UIF CMPDLT QBUI C 8IJDI DPNQPOFOUT PG UIF CMPDLT NPNFOUVN BSF DIBOHFE CZ UIF JNQVMTF BQQMJFE CZ ZPVS GPPU $IFDL BMM UIBU BQQMZ 5IF EJBHSBN TIPXT B UPQ WJFX MPPLJOH EPXO PO UIF Y[ QMBOF

Ë&#x2020; JO UIF EJSFDUJPO PG UIF CMPDLT D 8IBU JT UIF VOJU WFDUPS Q NPNFOUVN BGUFS UIF LJDL E 8IBU JT UIF Y DPNQPOFOU PG UIF CMPDLT NPNFOUVN BGUFS UIF LJDL F 3FNFNCFS UIBU Q = |Q|Ë&#x2020; Q 8IBU JT UIF NBHOJUVEF PG UIF CMPDLT NPNFOUVN BGUFS UIF LJDL G 6TF ZPVS BOTXFST UP UIF QSFDFEJOH RVFTUJPOT UP Ã OE UIF [ DPNQPOFOU PG UIF CMPDLT NPNFOUVN BGUFS UIF LJDL ESBXJOH B EJBHSBN JT IFMQGVM  H 8IBU XBT UIF NBHOJUVEF PG UIF BWFSBHF GPSDF ZPV BQQMJFE UP UIF CMPDL

22º A

22º

B

-BUF UJNF JOUFSWBM "U U =  T UIF QPTJUJPO XBT  â&#x2C6;&#x2019; â&#x2C6;&#x2019; N "U U =  T UIF QPTJUJPO XBT  â&#x2C6;&#x2019; â&#x2C6;&#x2019; N B *O UIF FBSMZ UJNF JOUFSWBM GSPN U =  T UP U =  T XIBU XBT UIF BWFSBHF NPNFOUVN PG UIF CBMM 5IF NBTT PG UIF 1JOH1POH CBMM JT  HSBNT  Ã&#x2014; â&#x2C6;&#x2019; LH  &YQSFTT ZPVS SFTVMU BT B WFDUPS C *O UIF MBUF UJNF JOUFSWBM GSPN U =  T UP U =  T XIBU XBT UIF BWFSBHF NPNFOUVN PG UIF CBMM &YQSFTT ZPVS SFTVMU BT B WFDUPS D *O UIF UJNF JOUFSWBM GSPN U =  T

C 22º 'JHVSF  rr1 *O PVUFS TQBDF B SPDL PG NBTT  LH JT BDUFE PO CZ B DPOTUBOU OFU GPSDF  â&#x2C6;&#x2019;  / EVSJOH B  T UJNF JOUFSWBM "U UIF FOE PG UIJT UJNF JOUFSWBM UIF SPDL IBT B WFMPDJUZ PG


1SPCMFNT    NT 8IBU XBT UIF SPDLT WFMPDJUZ BU UIF CFHJOOJOH PG UIF UJNF JOUFSWBM 4FDUJPO  rr1 " TUFFM TBGF XJUI NBTT  LH GBMMT POUP DPODSFUF +VTU CFGPSF IJUUJOH UIF DPODSFUF JUT TQFFE JT  NT BOE JU TNBTIFT XJUIPVU SFCPVOEJOH BOE FOET VQ CFJOH  N TIPSUFS UIBO CFGPSF 8IBU JT UIF BQQSPYJNBUF NBHOJUVEF PG UIF GPSDF FYFSUFE PO UIF TBGF CZ UIF DPODSFUF )PX EPFT UIJT DPNQBSF XJUI UIF HSBWJUBUJPOBM GPSDF PG UIF &BSUI PO UIF TBGF &YQMBJO ZPVS BOBMZTJT DBSFGVMMZ BOE KVTUJGZ ZPVS FTUJNBUFT PO QIZTJDBM HSPVOET rr1 *O B DSBTI UFTU B USVDL XJUI NBTT  LH USBWFMJOH BU  NT TNBTIFT IFBEPO JOUP B DPODSFUF XBMM XJUIPVU SFCPVOEJOH 5IF GSPOU FOE DSVNQMFT TP NVDI UIBU UIF USVDL JT  N TIPSUFS UIBO CFGPSF B 8IBU JT UIF BWFSBHF TQFFE PG UIF USVDL EVSJOH UIF DPMMJTJPO UIBU JT EVSJOH UIF JOUFSWBM CFUXFFO à STU DPOUBDU XJUI UIF XBMM BOE DPNJOH UP B TUPQ C "CPVU IPX MPOH EPFT UIF DPMMJTJPO MBTU 5IBU JT IPX MPOH JT UIF JOUFSWBM CFUXFFO àSTU DPOUBDU XJUI UIF XBMM BOE DPNJOH UP B TUPQ D 8IBU JT UIF NBHOJUVEF PG UIF BWFSBHF GPSDF FYFSUFE CZ UIF XBMM PO UIF USVDL EVSJOH UIF DPMMJTJPO E *U JT JOUFSFTUJOH UP DPNQBSF UIJT GPSDF UP UIF XFJHIU PG UIF USVDL $BMDVMBUF UIF SBUJP PG UIF GPSDF PG UIF XBMM UP UIF HSBWJUBUJPOBM GPSDF NH PO UIF USVDL 5IJT MBSHF SBUJP TIPXT XIZ B DPMMJTJPO JT TP EBNBHJOH F 8IBU BQQSPYJNBUJPOT EJE ZPV NBLF JO ZPVS BOBMZTJT rr1 " UFOOJT CBMM IBT B NBTT PG  LH " QSPGFTTJPOBM UFOOJT QMBZFS IJUT UIF CBMM IBSE FOPVHI UP HJWF JU B TQFFE PG  NT BCPVU  NJI  5IF CBMM IJUT B XBMM BOE CPVODFT CBDL XJUI BMNPTU UIF TBNF TQFFE  NT  "T JOEJDBUFE JO 'JHVSF  IJHITQFFE QIPUPHSBQIZ TIPXT UIBU UIF CBMM JT DSVTIFE  DN  N BU UIF JOTUBOU XIFO JUT TQFFE JT NPNFOUBSJMZ [FSP CFGPSF SFCPVOEJOH .BLJOH UIF WFSZ SPVHI BQQSPYJNBUJPO UIBU UIF MBSHF GPSDF UIBU UIF XBMM FYFSUT PO UIF CBMM JT BQQSPYJNBUFMZ DPOTUBOU EVSJOH DPOUBDU EFUFSNJOF UIF BQQSPYJNBUF NBHOJUVEF PG UIJT GPSDF )JOU 5IJOL BCPVU UIF BQQSPYJNBUF BNPVOU PG UJNF JU UBLFT GPS UIF CBMM UP DPNF NPNFOUBSJMZ UP SFTU 'PS DPNQBSJTPO OPUF UIBU UIF HSBWJUBUJPOBM GPSDF PO UIF CBMM JT RVJUF TNBMM POMZ BCPVU  LH  // LH â&#x2030;&#x2C6;  / " GPSDF PG  / JT BQQSPYJNBUFMZ UIF TBNF BT B GPSDF PG POF QPVOE

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Matter & Interactions, chapter 2  

Matter & Interactions, Fourth Edition by Chabay • Sherwood

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