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CHAPTER 3 Kinematics in Two Dimensions; Vectors I: actor 2 dest with mation along astialgh ie, We mow conser te FOURE 3-1. Cs 2c Fate ee seesy FIGURE 5-8. perm vals 0 Oba ca Str crt Tae ew glares reed Byoe By sal ‘Desig «dap of a particular plgsea! tuations alga Leip i poyses snd ies rapes yrs when dang wth vecons ina dagrem, ch felon reprised hy an eto The oto lve dre sta pin the diveton of se Yeesr quate i apresete The leigh w the $1106 is inn prapetianal othe magni of The secs qa Le Semple, Fy totyatccn aos have been daw toprcaitiag che velo oa at ‘rious paves otra a ce Terie of te seecty at ese Point Eun he fou pif Fp. 3-1 by msesuring ne engi af the soresparding ae sad usingie seve shows Lem Dk ‘Wien we unite thesis farce aye ne ace pe ih sting are ores cle spubol Tous for veoste we wu ADRS ass conse fly whe ragie nt Selo, We Wa We ply aos a8 we Tor other smb EEE) Addition of vectors—Graphical Methods Liecuuse veslom ae quanbilics Uh have sinesvon as nell 8 maamiuase tex faust be acded in epeval way. Ti tis Chaptes, as ail deal saaly 9th Uixplcsmest wel rsh ae nn use mb Ha lea ect, But hs ests will op or othe: ects Ne easountes tk ‘o tee ame artamae for aig sels Some gr hmoie ca lo he used for acing years Hf bey ure inthe sare deat, Fer eam i petsen cles 4k eis: abe das ard Gkin eat the next dag. te pevsen fe how 1k = Bik est ofthe fal tpn wy athe ut ‘esi daplcemeat i kart the east (Fe, 3-24) hon the oe hed, the femim welds sem cat na Hs de ane cy West (Mm the Fens ditzon Unde second da, ct ste person sil ene up 24m for: the erin 3-20). $2 the tesutm dsplacemenc i 2k to te east Im sas case, the Fesltait ‘apbecnent ie ublainal by sureties, Am kar = 2 he tsp ithe cast bed if the eo wetor are ao lng he same tin. For stl apse # poor sks tear sess then wa ke SI ont. Tiese Usplaements crt be repreetie! ota gh in whic ote postive 1's pnts nth snd peste st pele cor, Tg. 3-3 On ns grace se anton bal Do aonsent Ihe dp ese eet he IN Gaplcemest to ti ots Then we de” 9 second roa Dy, to repsesent 40 Si agplissment the orn ot wets afew to laa Fg. 8 Aller kine ths wale petson is ro WM exe, and Sk woth the pony of origin "Ine sesame daplarement i teprernied hy the a5 labeled Oy in Fg 2-4. Unga als ad a pofeter, yu cae iespure Cit rf the pers 1 km con ite igh akan anale #92" rth si Th outer Words (4 vssulant dip eeemewt 2c" has a agrituse of Tiion aud ais cn-aiale ¥ 27° wih tbe postive sauie The walode Heng) a0 con san he abaned sing t36theenene ot thagare nis esses Dy Dey and Dy fons sot simile win Dy ax the byponce Ts By = VBP OE suo + Soba = 9 2Ehn? = Ab ‘You cau ave the Prukagowsen hecrean, of eve, aly wan the eters ate 13. Kinematis in Two Dimensions; Veetare ‘Te cult dsslecsment veto, By, is hesum wf the eaten Dy snd B +p. Ths ior ane saan, A pura esas aking He san Uh a rt clang te sare line i that se weg f the "estan vetoes mot eq fete stim of the magnadcs sf the to spars vostore ur eee Le Than emit cso a alg ne am it] Tear sxanple (Re. 3-81 Oh thm, waereas | Dy oquels Sem. Nowe ho has we eat et Dy eql wo H2km, bese we bw yet eyo nd 12k eons pat of se rant wet is rngite We coud ete se Unig Hee vn thous By = By) Bem 12h Nu Fy EXERGIBE A Uses ntti cant mgt oe Fire 34 Ulises the pmcral len on raphy aig be wes tone together, no matte: We: engles they wee to ge ei sup, THs cles le a6 1. On eins, dea one of he Wsoters elt, —to eae 2 Newt dry the eecans seston, 9 see. pling He i 2 the Hp of 36 ine vost and bung a ieditstom ool '8 Tre aro dren tro the tal ofthe ft west to dhe op othe secon ses repeats then rsa, of he On Yar The length ofthe sasulaas Cctor wepsesnss ts naga, Note that weous {am br tramsbied orale to thenacteer (ring, The rms length nat fagle) wo atconpliy tess maiptlatiors The lea of the subarea be Frese With ster ct eommptred te eal Angi che sae Bith protrvtor This mothad is kc the eltp method uf a¥ding weston Tis ot imeostant im whieh older the vests ate adces. For expe, & Aspovernent a lt sent, bach fete» dpe a Tul ker ea, Selb esuteat of ILZhn cad aie 8 — 27" Ge Fip. 3-4), the seme as {Bren hey were ade eae oncer (Fig 13. Pnat Wee vey, wer 1] ‘Tae wi Mop wed ef adding vectors eat be exe 0 tees ot ne \wetnee Tae eau teas tom Ie fl tre ts ecto hep of Te Tas. one added. An cxampe is show fs Fi. 5, he ave yet eal: pre so destasoreats (noche, sth, mest) or periape eves forees Chek for ourwet sat yo ae ie aoe selon ie mle mwah onder si Eee vezers FIGURE 3-5 TMaersutant of see seats By =F 4 T+, i SECTION 2 Adiisen of Yoetare Graphics) Methods 47 eto A secon way ald wud the walldograne meth. [fly ise emt to the i fp meta is ethos the fas sector ane daa Shreng fom gormmion nig, and parllsgam i som-trcted wing Pe two vets os adjacnt aides sin ia Pi 3b. The valiant the ial vw enn she seammem engi 9 Tig, Yn he ep roto shen, adi heart th netbeds yilthe FIGURE 3-6 ves caution 1h is 4 entwra ctor tn ara the aim weston #8 the eve rita twee the te of het vest, a in Fh 306 Phe dco SF pease the ny of the two veers (Ca fast represente ther der ee anaes wl sce inthe nent Section) [CONCEPTUAL EXAMPLEE-T] Range of vattor tong! estors ec tae el Soy Wo the cnge ot pole ‘ectr fepresenig the Sm ofthe wa? goal mening Supp 1% as fo REGPONGE The sun sa tice on ay vel from 6 {-30 + 20) whste the estore por in a seme snetion, ta H(=-32) 40) whem 7 Seas ane ania, Given a Yoo Vows define He main of Is secon (W) ta hea weer oi of f£ Ihe sree magni o8¥ but opp in tection, 1p. 3S homer that fi secure wner negate in oy sense ols man tue the magma of cy tector postive Rater, wins sign cll us aout ie dicetion nay — Windies os —- That the diecamse bere tas esters fs equal te the sum af the fist plus the neg of the second Tver ces for dion a Wertracan he ape sow in Fig. 2-8 using te a-tip mete, Jf -.ef + a: 48 CHAPTERS Kiremtieg in Two Dimensions: Verte IQURE 2-8 Sliasing ta _ A soci ca be mused by asia We define hei radu tht 8 has the same dreston os Band ane magne ch That is malipiaiion s ‘eet hem poses soo ar «changes the megninds 2 the Wael Py a tacor © That cows ater she disston Me fos nog scala he maaicade of Ce provins eV i alc feithun, she mine fan) Se ae cnecten precy Spots total WS Fi 5 YL. few Bese eli Ei] Adding vectors by Components Adding wsstrsgrephicely using a ler and preter afen na salient cetuile amc fot inal for vector hee dinero We discs ga & tore pomerial mw prover melhed tor adcing sector Ii de net ong Haprisa thes —they ate alts wet fr siuaing. fo cece our rath ah for geting the cares renal CConsase it a yocsr W dh. es fou particular plan We canbe expressed the sua of two oer veto called the compouents oft origual seer the tampons anetavallycheyen’ Ui fe tlong Ime perpemticalar rep {ors The proces of oding the sompoens i Knot os resolv the reetor fewer intoitsrompanents, Aa examples sayin Vig 3 Tle the veetorV cl he fsleplaomeat veetor that pein at ap unals = 31° yon. of est, where tee nve chose Se postive a cas co be wo We east aed che ans tovih This vevlor V iy reseed inlets antl 9 compares hg deg, Ged Last ovr fron she tip (A) of she vector liges AB and AC) makeing thom pespendiewar tthe 1 ane y-aen Tae the bee UH aed epee Sen. dhe © andy componemns of Y, respectively as shown in Fi 310b Those vedlor components ate pritien V, ard ¥,. Sve generally show ¥ector Seopomenic av ty, be nocnty baaishs The sera samme, Fe find Vere nusrbecs, with uns, that cre iver a pasiive or nsgate sgh fepening wn whetaer rey pein alone 1 postive ot negate tae yA ‘Avan be soon in Pe 3-10, ¥, + ¥, = V by dhe cualelogram metho! of aldig vectors, FIGURE $-10 writes soe Fiat Space is mede up of the distensions cand somstines it i necessary to reinive a wetor iis eeenponents lang three ral perpeadic cisstnne. Ie resangalar evrdinaies the somporents are ¥,V sud Vo, Reson of vector in thre dinsesiome i mecly a extnsion of the cbove wehtioae, We till airy he coaccrcd wilh into bn Which the Seclorcare va pre ne to component are cll tha: ave neces. “To tld vcstore wing th method. nf @umponenis, we wees so URE EE Usigomoneti Tune on sine, cosine aa hast ich we sh even SECTION 3 4 Adcing vectors by Gompenenss 48 GURE 9-11 Sinting sith ancnste ae Eanes sontcr ig taaghs ‘bie te sh act de “aed FROURE 2-12. Find e Given any ane an Fig 3. (Na sigh wine a bo eomtewesad by ening ine peapendicnar other of ie isee ein Ing 3011. Te engest SMe of aht Ul, syste ser gh sl eall th hypotenins wa te liber I tte opposite the aagle 0 isabel, and the sae ace s Iaboied 0, We les anand repro the hag of the widen respectively Wer aim dation the Mince flgranratie hinctows sie. toting ad Target Fasbreiad got ania Wrst the igh ame te ols Aids oppseie PRE ycoterase coup —Sealieens on or i sony SE 0pE=Ie 8 Side adjacent "0 of one side to the alles, or uf ave ide tthe Mjputcne, ceraing the seme Tha os in Tig 3 Te we howe: ng ais ack ood ane This the sth of site, cine ana argent Sn not cope or how bik the angles Thes depen! vals om the sae of the angle, The values of fine swine, ome ange! for deren! angle cam be fous eng 9 wen faleaton ve fon the Table i Appenie ‘vel tegenomietsc en Fe venta ea ‘whist follows lor he Pthgctean therm (oa? = ia Ra 3 3D, Mas wre reate EM [Boe aso Appar & or oer dates om wganamein fretions an ets) “The ise ot tgomarei neti for Hig the rape ox vet itwsrsted in ig, 3-evwhere sector and te conspensets are thowaht 38 tanking sah wimg We then sel Use dca pe a fivon the Figure, fe mle he seinen of in by/7 by ¥en bos fos 92 2h ¥, = Vain. rary inlay foo the define ot cor, we aban Wy = Foo an Note hut 4 is chossn (by convertion) te be dhe angle thatthe woctor anes (wor the pele a eine Bos 3-3, eeu valle Yd ¥ for uy vest such as that ls eatadin Fg 3 Dora 3-12 Supoots V cepresercs 2 dsplacement of 0 13. Kinematis in Two Dimensions; Veetare in advection sin Fig 2°12.Then = St, Fa Fenelon or Tables, sn — HS) and em a — O85, Le VY, = W cae = cbs} = 499m 4ere, ¥, = sine ~ [S02 m}(0500) ~ 250m (aor. “Thorsate co weps 0 scecfys weerr ina given eootdictesystons 1. Ween aise to eumponents Ve and Y ioe nit 2 Wesnn give is mrgmitice Von the sngle it makes ih the posse sass 2 ‘We canst irom ome eessipsin w the wher using Eup 3-2, md fr Ge ravers, by aking the thorein of Pyéagors! andthe definzon of targest vovn_¥, ( z cmyovene and sae ow po ‘We sar now discuss how to add veetnra using gompoaenta, Te fies ten is VF Ve. isnplics teat won nal Hy Vie My BA Riera Thal ihe sum of the + enmopnenic eqs the component ofthe elim, nd sets fry, Thet tis us con be verified by ecanealcxnination of Fig 1 ur nora shat we eet a ner enponsnts coger Tt te tonporent of the resubanc aid we a2 athe Poorsponuat ogsthe ges Ce P campos of the restne Wed vt 48d conypanent > comisonent. he vas diction ofthe realand wlan ane exes Uy ext be abtined wing Ege 3-4 SECTION 3 4 Adcing Vecters by Compenenss. 51 URE 2-19. Exanpe 5-2 {9 Ths wo uiglaere es Badd. 0D. srsovstino ‘ead graph lly kas sete of ing the stor ede tn Exile “The company of a wives vton wl Be fern fir differen csc of recuse the work invohied in -udsing xcators by a good etoies af exee—far inp elaning one of theses be he ine diet a5 one of Ue tector: thea that eter wil hyve only one nonzero empenent [SCL an covers aeptacement. 0 rural avi corer waves Tho pet oles ene drvse 22 0km na tribes scan, She Ca dvs © incon (OOF south of easfor Dk (Fg # se), Whether dclececent Irons the pt oles APPROAGH We resale e3c1 voter nine sel» compenents, We st the {eomponentstogthor ane then th ¥ smporentsingsiner ovine He the fy components Une resallan. Be chow the pve 4 ais ho be el fms the pete the marth snc tse ae the compe sien wes SOLUTION fesse cash dipiacement vector inn is conponent. a howe in Fa 3-15b, Since B, fas esiivde 250 na pits sth, has val 2 © component By, ~220km 1 has both ard y somone Dy. SFO km yose tO) — HCA Okw(DSOD — +225K p, (AFoka (0366) 407k (a7. sin" Nocee th Ba, is mustive besase this vestor corporat puis along dhe Degaivey tare renlla vecte has crmpoens D,=Dy+ y= uhm + ZSkm ~ —25.54n0 Dy ~ D+ Oy, ~ 0k. + (-a0.7Ra) ~ —187 km, “Tre epoiis the rosin weer comm eit D,-235ka, DB, —187 im, ‘We can al aieity Che sesultant ester by giving its simutade ad angle toe Las 4 DVB ~ VG mFS CIF — kw ton — Br = TESTA yg, D> 25k A exten wh am ING TAK, an ARE THN oF A TANT! te lan "385. Tae nels mens # = 285° beow the use Fa 3 vhs rosutast doplacenent fe 30k dicted at 385° ba e NOTE Alwuye be eutenche ubout the yucdrumt in which Uh resulan: seer Tash electron caleulatr dows at falls ge ths agora, bus © eee he sem af tngnametne ftom dara on wht “ca” th ane fala: or exact the tage: estxe he fst ara tid ouacrans (ns {sp rend IN tS), Wr azgeo the san Ht Koh queda AAppenuie A-7.,The bet way seep ties uf wales ud to che an veer isles to daw a est diagram. A vector digiar gies fou sates tn sl wher snalyn pdt nd proven tere the rx “The following Prblcm Soleing Bor shou Got be sonsidered« presi Ho. Herne ea summery of Tags a 0 et fou ig es nee the protien a 13. Kinematis in Two Dimensions: Veetare EEN Adding vectors ler 5 ie! camry of how bs a S00 oF mee ‘ectrs thing components 1 Dm a diagram. aig "te vacons orp By lute te parallogzum or a-to tp melbos. Caos: ay anes Chose then ia ea. psi et i mals Str worl eer, [or expt ean ony avs usr tte dsaion of spe of he Wet 32 at seta wl aeons one compenent) Hesalve each vertae inte is a 9 eam rents showing vacn ccmponsn. clone. is apprusrale ery} ats go. {dashed} arom, Caenate acts component (Sen net isen) using Sines und ooines If 5, the angle that weir 9) ees wi theron ren ¥i,— Heine Cy = Fens, he cattle sige ay compra tat he negatve aor ¥ exis gets a san Snore dunn me emigonens ofthe restitan. Disfr Vo Vig + = any others Wen Nt We any ters This isthe answers he eompuneas ofthe realm ‘clo Check Sans to se i hey Ec the quedrant Stes in you dara (psi | she). 6 yo sant o koow the maguitade aud deufon of Ihe salam yoko Lage Ss ¥ vevi Ihe vecor sing sat seas dre eps atthe sonnet pron {quail} Wes Tinos short mips. Aa alan: tip inves thee es the wend sun (4°) fo 40 and a id ot 3 val srs fo 350m acho. Wha ihe st or deme APPROACH Wo li theses se Ptr Sal SOLUTION 1 Deaw adage sich ss Fg. 3-16 here, reife revisor errant Che tu: Ava can Fe 2 Rouivegeapmorse in pcaie ldap fawe The sparsia ae coe in Tig. 3 Io Inca ding a ie eters cary tam End D, eoresent te tsybeement sth tra splocement has regis 969 a and pois SU Ges axe (Soul of ssy.a8 as show in our orgie sketch Fi © imme nights we did m Mg S10, fee we dae hem als ip Sie wha seca all si me tate neace soe 44 Culedte he conponts Dy +Dane ln By: = #Docost8* = + ka) = 4312 em By = “Dang? = ~ 040900907] = ~31. km Buda —DysusS8" —SHAMNNOD) —22.km y= danse = ‘soikmeney) =e 6 Werave wen mins so each none iv 3-16 pois ho pQURES-18 ae 3-8 Tie rary decom ihe component a sha nthe ree ner {Ath compenratr We ade componente epee and wee he} tipanaes jel lou te aed composts ts vase, Yar Components Batt Doe thn —31ltm— a50ken = 350k. Bh 8 Ura anyexmpert re si em and asim pon oapsety a BE am tae wt an ut Ts ron nay wo eve De uno th sa 6. Maunitudy and dirvetiva: We eur el By cy 750 SECTION 3 4 Adcing Yecters by Compenenss 59 EB) Projectile Motion 1 Chapter 2, we sulied de wot of bfects i ane deusion bi tne of Felise bovies sedergcingacesLtaton dae to guests Now We een Ue ve [peer icon nf anjerta ring ‘arcugh Ihe sin tan mensions a Ms Ean surfeg, such esa sib urs sf Cate sascha, ick foal, sd speeding Suet These areal xarpls at projet moti sce Fi. 3-17), trick wc cam serie akg pla om dren Ah iseen ingevtant, i way eases select can be fared, ana: wll zane it hich the bet iowa or preted, Ne ons vnlys-mosn eer has been ploeeced, ans tere Hands of is ceught Ut iy we lize our 1 gesty clove Thea the aeclcttion ef the chee: shat due to gre wish URE 3-17 Toa sie si ™ * ‘Gallee wes the fr to destsibe projectile tien gecanately He shows Hsqunytend that iccoule e undstsitod ny analginge Rorzosl and oe eal somponets siswinitom wr vie maton sepataicl Far seers, se snore sal the not, Des BL soa onan fgge r= fb atthe ign ota iy coondinace ase! 129 3 FIGURE 8-18 Pj nnia taal all iva doi nthe banrontl {a} dicen, ewe sheysiy asap. porns Uke dicen ofthe alls oti at UL ist a ys ‘ungenct the ath ollomiog Galles Mach we wea the or aul aad vert renponena ole elas ad ep, separatls. are ecm ay the kine ‘uitions Ey 21a teush 2-10) tothe x aud yconcoxcats a te tution ee ese 2) the bal eases the bles wp {2 = 7, hs ens an eomaponent of vel ‘Once the all leaves the tei (40), experiance @ very dost sctelertai gle neaccradi hs rama Tso sally 20 =O frat nerenes conzisaly in the dowaward ivssien (ustil the ball fits the brome, Tes take y to repre upon Caen ay ga eo Eg.2-1ig we sin wrle = nah sine we sl by 0, The veces snsise rca geen Oy i See Fis 318 where nobel 3. Airemsties in Two Dimensions: Vetore