Motion in One Dimension 2

Transcription

1 The curren bsolue lnd speed record holder is he Briish designed ThrusSSC, win urbofn-powered cr which chieved 763 miles per hour (1,8 km/h) for he mile (1.6 km), breking he sound brrier. The cr ws driven by Andy Green (UK) on 10/15/1997 in he Blck Rock Deser in Gerlch, Nevd. AP Phoo/Ben Mrgo Moion in One Dimension Life is moion. Our muscles coordine moion microscopiclly o enble us o wlk nd jog. Our hers pump irelessly for decdes, moving blood hrough our bodies. Cell wll mechnisms move selec oms nd molecules in nd ou of cells. From he prehisoric chse of nelopes cross he svnn o he pursui of sellies in spce, msery of moion hs been criicl o our survivl nd success s species. The sudy of moion nd of physicl conceps such s force nd mss is clled dynmics. The pr of dynmics h describes moion wihou regrd o is cuses is clled kinemics. In his chper he focus is on kinemics in one dimension: moion long srigh line. This kind of moion nd, indeed, ny moion involves he conceps of displcemen, velociy, nd ccelerion. Here, we use hese conceps o sudy he moion of objecs undergoing consn ccelerion. In Chper 3 we will repe his discussion for objecs moving in wo dimensions. The firs recorded evidence of he sudy of mechnics cn be rced o he people of ncien Sumeri nd Egyp, who were ineresed primrily in undersnding he moions of hevenly bodies. The mos sysemic nd deiled erly sudies of he hevens were conduced by he Greeks from bou 300 B.C. o A.D Ancien scieniss nd lypeople regrded he Erh s he cener of he Universe. This geocenric model ws cceped by such nobles s Arisole (384 3 B.C.) nd Cludius Polemy (bou A.D. 140). Lrgely becuse of he uhoriy of Arisole, he geocenric model becme he cceped heory of he Universe unil he seveneenh cenury. Abou 50 B.C., he Greek philosopher Arisrchus worked ou he deils of model of he solr sysem bsed on sphericl Erh h roed on is xis nd revolved round he Sun. He proposed h he sky ppered o urn weswrd becuse he Erh ws urning eswrd. This model wsn given much considerion becuse i ws believed h urning Erh would genere powerful winds s i moved hrough he ir. We now know h he Erh crries he ir nd everyhing else wih i s i roes. The Polish sronomer Nicolus Copernicus ( ) is credied wih iniiing he revoluion h finlly replced he geocenric model. In his sysem, clled he heliocenric model, Erh nd he oher plnes revolve in circulr orbis round he Sun..1 Displcemen. Velociy.3 Accelerion.4 Moion Digrms.5 One-Dimensionl Moion wih Consn Accelerion.6 Freely Flling Objecs 5

2 6 CHAPTER Moion in One Dimension This erly knowledge formed he foundion for he work of Glileo Glilei ( ), who snds ou s he dominn fcilior of he enrnce of physics ino he modern er. In 1609 he becme one of he firs o mke sronomicl observions wih elescope. He observed mounins on he Moon, he lrger sellies of Jupier, spos on he Sun, nd he phses of Venus. Glileo s observions convinced him of he correcness of he Copernicn heory. His quniive sudy of moion formed he foundion of Newon s revoluionry work in he nex cenury..1 Displcemen Moion involves he displcemen of n objec from one plce in spce nd ime o noher. Describing moion requires some convenien coordine sysem nd specified origin. A frme of reference is choice of coordine xes h defines he sring poin for mesuring ny quniy, n essenil firs sep in solving virully ny problem in mechnics (Fig..1). In Acive Figure., for exmple, cr moves long he x-xis. The coordines of he cr ny ime describe is posiion in spce nd, more impornly, is displcemen some given ime of ineres. Definiion of displcemen c Tip.1 A Displcemen Isn Disnce! The displcemen of n objec is no he sme s he disnce i rvels. Toss ennis bll up nd cch i. The bll rvels disnce equl o wice he mximum heigh reched, bu is displcemen is zero. The displcemen Dx of n objec is defined s is chnge in posiion nd is given by Dx ; x f x i [.1] where x i is he coordine of he iniil posiion of he cr nd x f is he coordine of he cr s finl posiion. (The indices i nd f snd for iniil nd finl, respecively.) SI uni: meer (m) We will use he Greek leer del, D, o denoe chnge in ny physicl quniy. From he definiion of displcemen, we see h Dx (red del ex ) is posiive if x f is greer hn x i nd negive if x f is less hn x i. For exmple, if he cr moves from poin o poin so h he iniil posiion is x i 5 30 m nd he finl posiion is x f 5 5 m, he displcemen is Dx 5 x f x i 5 5 m 30 m 5 1 m. However, if he cr moves from poin o poin, hen he iniil posiion is x i 5 38 m nd he finl posiion is x f 5 53 m, nd he displcemen is Dx 5 x f x i 5 53 m 38 m 5 91 m. A posiive nswer indices displcemen in NASA/USGS Figure.1 () How lrge is he cnyon? Wihou frme of reference, i s hrd o ell. (b) The cnyon is Vlles Mrineris on Mrs, nd wih frme of reference provided by superposed ouline of he Unied Ses, is size is esily grsped. NASA/USGS b

3 . Velociy 7 The cr moves o he righ beween posiions nd. x (m) x (m) The cr moves o he lef beween posiions nd. x (m) b x (s) Acive Figure. () A cr moves bck nd forh long srigh line ken o be he x-xis. Becuse we re ineresed only in he cr s rnslionl moion, we cn model i s pricle. (b) Grph of posiion vs. ime for he moion of he pricle. he posiive x-direcion, wheres negive nswer indices displcemen in he negive x-direcion. Acive Figure.b displys he grph of he cr s posiion s funcion of ime. Becuse displcemen hs boh mgniude (size) nd direcion, i s vecor quniy, s re velociy nd ccelerion. In generl, vecor quniy is chrcerized by hving boh mgniude nd direcion. By conrs, sclr quniy hs mgniude, bu no direcion. Sclr quniies such s mss nd emperure re compleely specified by numeric vlue wih pproprie unis; no direcion is involved. Vecor quniies will be usully denoed in boldfce ype wih n rrow over he op of he leer. For exmple, v S represens velociy nd S denoes n ccelerion, boh vecor quniies. In his chper, however, i won be necessry o use h noion becuse in one-dimensionl moion n objec cn only move in one of wo direcions, nd hese direcions re esily specified by plus nd minus signs. Tip. Vecors Hve Boh Mgniude nd Direcion Sclrs hve size. Vecors, oo, hve size, bu hey lso indice direcion.. Velociy In everydy usge he erms speed nd velociy re inerchngeble. In physics, however, here s cler disincion beween hem: Speed is sclr quniy, hving only mgniude, wheres velociy is vecor, hving boh mgniude nd direcion. Why mus velociy be vecor? If you wn o ge o own 70 km wy in n hour s ime, i s no enough o drive speed of 70 km/h; you mus rvel in he correc direcion s well. Th s obvious, bu i shows h velociy gives considerbly more informion hn speed, s will be mde more precise in he forml definiions. The verge speed of n objec over given ime inervl is he lengh of he ph i rvels divided by he ol elpsed ime: ph lengh Averge speed ; elpsed ime SI uni: meer per second (m/s) b Definiion of verge speed

4 8 CHAPTER Moion in One Dimension In symbols his equion migh be wrien v 5 d/, where v represens he verge speed (no verge velociy), d represens he ph lengh, nd represens he elpsed ime during he moion. The ph lengh is ofen clled he ol disnce, bu h cn be misleding, becuse disnce hs differen, precise mhemicl mening bsed on differences in he coordines beween he iniil nd finl poins. Disnce (neglecing ny curvure of he surfce) is given by he Pyhgoren heorem, Ds 5!1x f x i 1 1y f y i, which depends only on he endpoins, (x i, y i ) nd (x f, y f ), nd no on wh hppens in beween. The sme equion gives he mgniude of displcemen. The srigh-line disnce from Aln, Georgi, o S. Peersburg, Florid, for exmple, is bou 500 miles. If someone drives cr h disnce in 10 h, he cr s verge speed is 500 mi/10 h 5 50 mi/h, even if he cr s speed vries grely during he rip. If he driver kes scenic deours off he direc roue long he wy, however, or doubles bck for while, he ph lengh increses while he disnce beween he wo ciies remins he sme. A side rip o Jcksonville, Florid, for exmple, migh dd 100 miles o he ph lengh, so he cr s verge speed would hen be 600 mi/10 h 5 60 mi/h. The mgniude of he verge velociy, however, would remin 50 mi/h. EXAMPLE.1 The Toroise nd he Hre GOAL Apply he concep of verge speed. PROBLEM A urle nd rbbi engge in foorce over disnce of 4.00 km. The rbbi runs km nd hen sops for 90.0-min np. Upon wkening, he remembers he rce nd runs wice s fs. Finishing he course in ol ime of 1.75 h, he rbbi wins he rce. () Clcule he verge speed of he rbbi. (b) Wh ws his verge speed before he sopped for np? Assume no deours or doubling bck. STRATEGY Finding he overll verge speed in pr () is jus mer of dividing he ph lengh by he elpsed ime. Pr (b) requires wo equions nd wo unknowns, he ler urning ou o be he wo differen verge speeds: v 1 before he np nd v fer he np. One equion is given in he semen of he problem (v 5 v 1 ), wheres he oher comes from he fc he rbbi rn for only 15 minues becuse he npped for 90 minues. SOLUTION () Find he rbbi s overll verge speed. Apply he equion for verge speed: Averge speed ; ph lengh elpsed ime 5.9 km/h km 1.75 h (b) Find he rbbi s verge speed before his np. Sum he running imes, nd se he sum equl o 0.5 h: h Subsiue 1 5 d 1 /v 1 nd 5 d /v : (1) d 1 v 1 1 d v h Subsiue v 5 v 1 nd he vlues of d 1 nd d ino Equion (1): () km v km v h Solve Equion () for v 1 : v km/h REMARKS As seen in his exmple, verge speed cn be clculed regrdless of ny vriion in speed over he given ime inervl. QUESTION.1 Does doubling of n objec s verge speed lwys double he mgniude of is displcemen in given moun of ime? Explin. EXERCISE.1 Esime he verge speed of he Apollo spcecrf in meers per second, given h he crf ook five dys o rech he Moon from Erh. (The Moon is m from Erh.) ANSWER, 900 m/s

5 . Velociy 9 Unlike verge speed, verge velociy is vecor quniy, hving boh mgniude nd direcion. Consider gin he cr of Figure., moving long he rod (he x-xis). Le he cr s posiion be x i some ime i nd x f ler ime f. In he ime inervl D 5 f i, he displcemen of he cr is Dx 5 x f x i. The verge velociy v during ime inervl D is he displcemen Dx divided by D: SI uni: meer per second (m/s) v ; Dx D 5 x f x i f i [.] Unlike he verge speed, which is lwys posiive, he verge velociy of n objec in one dimension cn be eiher posiive or negive, depending on he sign of he displcemen. (The ime inervl D is lwys posiive.) In Figure., for exmple, he verge velociy of he cr is posiive in he upper illusrion, posiive sign indicing moion o he righ long he x-xis. Similrly, negive verge velociy for he cr in he lower illusrion of he figure indices h i moves o he lef long he x-xis. As n exmple, we cn use he d in Tble.1 o find he verge velociy in he ime inervl from poin o poin (ssume wo digis re significn): v 5 Dx 5 m 30 m 5 5. m/s D 10 s 0 s Aside from meers per second, oher common unis for verge velociy re fee per second (f/s) in he U.S. cusomry sysem nd cenimeers per second (cm/s) in he cgs sysem. To furher illusre he disincion beween speed nd velociy, suppose we re wching drg rce from sionry blimp. In one run we see cr follow he srigh-line ph from o shown in Figure.3 during he ime inervl D, nd in second run cr follows he curved ph during he sme inervl. From he definiion in Equion., he wo crs hd he sme verge velociy becuse hey hd he sme displcemen Dx 5 x f x i during he sme ime inervl D. The cr king he curved roue, however, rveled greer ph lengh nd hd he higher verge speed. b Definiion of verge velociy Tble.1 Posiion of he Cr Vrious Times Posiion (s) x (m) x i Figure.3 A drg rce viewed from sionry blimp. One cr follows he rus-colored srigh-line ph from o, nd second cr follows he blue curved ph. x f x Quick Quiz.1 Figure.4 shows he unusul ph of confused foobll plyer. Afer receiving kickoff his own gol, he runs downfield o wihin inches of ouchdown, hen reverses direcion nd rces bck unil he s ckled he exc locion where he firs cugh he bll. During his run, which ook 5 s, wh is () he ph lengh he rvels, (b) his displcemen, nd (c) his verge velociy in he x-direcion? (d) Wh is his verge speed? FOOTBALL FOOTBALL yd 50 yd 100 yd Figure.4 (Quick Quiz.1) The ph followed by confused foobll plyer. Grphicl Inerpreion of Velociy If cr moves long he x-xis from o o, nd so forh, we cn plo he posiions of hese poins s funcion of he ime elpsed since he sr of he moion. The resul is posiion vs. ime grph like hose of Figure.5 (pge 30). In Figure.5, he grph is srigh line becuse he cr is moving consn

6 30 CHAPTER Moion in One Dimension Acive Figure.5 () Posiion vs. ime grph for he moion of cr moving long he x-xis consn velociy. (b) Posiion vs. ime grph for he moion of cr wih chnging velociy, using he d in Tble.1. x (m) (s) 50 x (m) The verge velociy beween ny wo poins equls he slope of he blue line connecing he poins (s) 50 b Tip.3 Slopes of Grphs The word slope is ofen used in reference o he grphs of physicl d. Regrdless of he ype of d, he slope is given by chnge in vericl xis Slope 5 chnge in horizonl xis Slope crries unis. Tip.4 Averge Velociy vs. Averge Speed Averge velociy is no he sme s verge speed. If you run from x 5 0 m o x 5 5 m nd bck o your sring poin in ime inervl of 5 s, he verge velociy is zero, wheres he verge speed is 10 m/s. Definiion of insnneous c velociy velociy. The sme displcemen Dx occurs in ech ime inervl D. In his cse, he verge velociy is lwys he sme nd is equl o Dx/D. Figure.5b is grph of he d in Tble.1. Here, he posiion vs. ime grph is no srigh line becuse he velociy of he cr is chnging. Beween ny wo poins, however, we cn drw srigh line jus s in Figure.5, nd he slope of h line is he verge velociy Dx/D in h ime inervl. In generl, he verge velociy of n objec during he ime inervl D is equl o he slope of he srigh line joining he iniil nd finl poins on grph of he objec s posiion versus ime. From he d in Tble.1 nd he grph in Figure.5b, we see h he cr firs moves in he posiive x-direcion s i rvels from o, reches posiion of 5 m ime 5 10 s, hen reverses direcion nd heds bckwrds. In he firs 10 s of is moion, s he cr rvels from o, is verge velociy is. m/s, s previously clculed. In he firs 40 seconds, s he cr goes from o, is displcemen is Dx 5 37 m (30 m) 5 67 m. So he verge velociy in his inervl, which equls he slope of he blue line in Figure.5b from o, is v 5 Dx/D 5(67 m)/(40 s) m/s. In generl, here will be differen verge velociy beween ny disinc pir of poins. Insnneous Velociy Averge velociy doesn ke ino ccoun he deils of wh hppens during n inervl of ime. On cr rip, for exmple, you my speed up or slow down number of imes in response o he rffic nd he condiion of he rod, nd on rre occsions even pull over o ch wih police officer bou your speed. Wh is mos imporn o he police (nd o your own sfey) is he speed of your cr nd he direcion i ws going priculr insn in ime, which ogeher deermine he cr s insnneous velociy. So in driving cr beween wo poins, he verge velociy mus be compued over n inervl of ime, bu he mgniude of insnneous velociy cn be red on he cr s speedomeer. The insnneous velociy v is he limi of he verge velociy s he ime inervl D becomes infiniesimlly smll: SI uni: meer per second (m/s) Dx v ; lim D S0 D [.3]

7 . Velociy 31 Tble. Posiions of Cr Specific Insns of Time (s) x (m) Tble.3 Clculed Vlues of he Time Inervls, Displcemens, nd Averge Velociies of he Cr of Tble. Time Inervl (s) D (s) Dx (m) v 1m/s 1.00 o o o o o o The noion lim D S0 mens h he rio Dx/D is repeedly evlued for smller nd smller ime inervls D. As D ges exremely close o zero, he rio Dx/D ges closer nd closer o fixed number, which is defined s he insnneous velociy. To beer undersnd he forml definiion, consider d obined on our vehicle vi rdr (Tble.). A s, he cr is x m, nd s, i s x m. The verge velociy compued for his inervl Dx/D 5 (5.5 m 5.00 m)/(3.00 s 1.00 s) m/s. This resul could be used s n esime for he velociy s, bu i wouldn be very ccure becuse he speed chnges considerbly in he -second ime inervl. Using he res of he d, we cn consruc Tble.3. As he ime inervl ges smller, he verge velociy more closely pproches he insnneous velociy. Using he finl inervl of only s, we find h he verge velociy is v 5Dx/D m/ s m/s. Becuse s is very shor ime inervl, he cul insnneous velociy is probbly very close o his ler verge velociy, given he limis on he cr s biliy o ccelere. Finlly using he conversion fcor on he fron endshees of he book, we see h his is 105 mi/h, likely violion of he speed limi. As cn be seen in Figure.6, he chords formed by he blue lines grdully pproch ngen line s he ime inervl becomes smller. The slope of he line ngen o he posiion vs. ime curve given ime is defined o be he insnneous velociy h ime. The insnneous speed of n objec, which is sclr quniy, is defined s he mgniude of he insnneous velociy. Like verge speed, insnneous speed (which we will usully cll, simply, speed ) hs no direcion ssocied wih i nd hence crries no lgebric sign. For exmple, if one objec hs n b Definiion of insnneous speed x (m) The slopes of he blue lines re verge velociies which pproch he slope of he green ngen line, n insnneous velociy. Figure.6 Grph represening he moion of he cr from he d in Tble (s)

8 3 CHAPTER Moion in One Dimension insnneous velociy of 115 m/s long given line nd noher objec hs n insnneous velociy of 15 m/s long he sme line, boh hve n insnneous speed of 15 m/s. EXAMPLE. Slowly Moving Trin GOAL Obin verge nd insnneous velociies from grph. PROBLEM A rin moves slowly long srigh porion of rck ccording o he grph of posiion versus ime in Figure.7. Find () he verge velociy for he ol rip, (b) he verge velociy during he firs 4.00 s of moion, (c) he verge velociy during he nex 4.00 s of moion, (d) he insnneous velociy 5.00 s, nd (e) he insnneous velociy s. x (m) STRATEGY The verge velociies cn be obined by subsiuing he d ino he definiion. The insnneous velociy Figure.7 () (Exmple.) (b) (Exercise.) 5.00 s is he sme s he verge velociy h poin becuse he posiion vs. ime grph is srigh line, indicing consn velociy. Finding he insnneous velociy when s requires skeching line ngen o he curve h poin nd finding is slope x (m) (s) 1 4 b 6 8 (s) 10 1 SOLUTION () Find he verge velociy from o. Clcule he slope of he dshed blue line: (b) Find he verge velociy during he firs 4 seconds of he rin s moion. Agin, find he slope: (c) Find he verge velociy during he nex 4 seconds. Here, here is no chnge in posiion s he rin moves from o, so he displcemen Dx is zero: (d) Find he insnneous velociy 5.00 s. This is he sme s he verge velociy found in (b), becuse he grph is srigh line: (e) Find he insnneous velociy s. The ngen line ppers o inercep he x-xis (3.0 s, 0 m) nd grze he curve (9.0 s, 4.5 m). The insnneous velociy s equls he slope of he ngen line hrough hese poins: v 5 Dx D m m/s 1.0 s v 5 Dx D m m/s 4.00 s v 5 Dx D 5 0 m 4.00 s 5 0 m/s v m/s v 5 Dx D m 0 m 9.0 s 3.0 s m/s REMARKS From he origin o, he rin moves consn speed in he posiive x-direcion for he firs 4.00 s, becuse he posiion vs. ime curve is rising sedily owrd posiive vlues. From o, he rin sops x m for 4.00 s. From o, he rin rvels incresing speed in he posiive x-direcion. QUESTION. Would vericl line in grph of posiion versus ime mke sense? Explin. EXERCISE. Figure.7b grphs noher run of he rin. Find () he verge velociy from o ; (b) he verge nd insnneous velociies from o ; (c) he pproxime insnneous velociy s; nd (d) he verge nd insnneous velociy s. ANSWERS () 0 m/s (b) boh re 10.5 m/s (c) m/s (d) boh re.5 m/s

9 .3 Accelerion 33.3 Accelerion i v i f v f Going from plce o plce in your cr, you rrely rvel long disnces consn velociy. The velociy of he cr increses when you sep hrder on he gs pedl nd decreses when you pply he brkes. The velociy lso chnges when you round curve, lering your direcion of moion. The chnging of n objec s velociy wih ime is clled ccelerion. Figure.8 A cr moving o he righ cceleres from velociy of v i o velociy of v f in he ime inervl D 5 f i. Averge Accelerion A cr moves long srigh highwy s in Figure.8. A ime i i hs velociy of v i, nd ime f is velociy is v f, wih Dv 5 v f v i nd D 5 f i. The verge ccelerion during he ime inervl D is he chnge in velociy Dv divided by D: b Definiion of verge ccelerion SI uni: meer per second per second (m/s ) ; Dv D 5 v f v i f i [.4] For exmple, suppose he cr shown in Figure.8 cceleres from n iniil velociy of v i m/s o finl velociy of v f 5 10 m/s in ime inervl of s. (Boh velociies re owrd he righ, seleced s he posiive direcion.) These vlues cn be insered ino Equion.4 o find he verge ccelerion: 5 Dv D 5 0 m/s 10 m/s s 515 m/s Accelerion is vecor quniy hving dimensions of lengh divided by he ime squred. Common unis of ccelerion re meers per second per second ((m/s)/s, which is usully wrien m/s ) nd fee per second per second (f/s ). An verge ccelerion of 15 m/s mens h, on verge, he cr increses is velociy by 5 m/s every second in he posiive x-direcion. For he cse of moion in srigh line, he direcion of he velociy of n objec nd he direcion of is ccelerion re reled s follows: When he objec s velociy nd ccelerion re in he sme direcion, he speed of he objec increses wih ime. When he objec s velociy nd ccelerion re in opposie direcions, he speed of he objec decreses wih ime. To clrify his poin, suppose he velociy of cr chnges from 10 m/s o 0 m/s in ime inervl of s. The minus signs indice h he velociies of he cr re in he negive x-direcion; hey do no men h he cr is slowing down! The verge ccelerion of he cr in his ime inervl is 5 Dv D 5 0 m/s 110 m/s s 55 m/s The minus sign indices h he ccelerion vecor is lso in he negive x-direcion. Becuse he velociy nd ccelerion vecors re in he sme direcion, he speed of he cr mus increse s he cr moves o he lef. Posiive nd negive ccelerions specify direcions relive o chosen xes, no speeding up or slowing down. The erms speeding up or slowing down refer o n increse nd decrese in speed, respecively. Tip.5 Negive Accelerion Negive ccelerion doesn necessrily men n objec is slowing down. If he ccelerion is negive nd he velociy is lso negive, he objec is speeding up! Tip.6 Decelerion The word decelerion mens reducion in speed, slowing down. Some confuse i wih negive ccelerion, which cn speed somehing up. (See Tip.5.) Quick Quiz. True or Flse? () A cr mus lwys hve n ccelerion in he sme direcion s is velociy. (b) I s possible for slowing cr o hve posiive ccelerion. (c) An objec wih consn nonzero ccelerion cn never sop nd remin res.

10 34 CHAPTER Moion in One Dimension An objec wih nonzero ccelerion cn hve velociy of zero, bu only insnneously. When bll is ossed srigh up, is velociy is zero when i reches is mximum heigh. Grviy sill cceleres he bll h poin, however; oherwise, i wouldn fll down. Insnneous Accelerion The vlue of he verge ccelerion ofen differs in differen ime inervls, so i s useful o define he insnneous ccelerion, which is nlogous o he insnneous velociy discussed in Secion.. Definiion of insnneous c ccelerion The insnneous ccelerion is he limi of he verge ccelerion s he ime inervl D goes o zero: Dv ; lim D S0 D [.5] SI uni: meer per second per second (m/s ) x i f v v i v v f v The cr moves wih differen velociies poins nd. The slope of he green line is he insnneous ccelerion of he cr poin (Eq..5). Here gin, he noion lim D S0 mens h he rio Dv/D is evlued for smller nd smller vlues of D. The closer D ges o zero, he closer he rio ges o fixed number, which is he insnneous ccelerion. Figure.9, velociy vs. ime grph, plos he velociy of n objec gins ime. The grph could represen, for exmple, he moion of cr long busy sree. The verge ccelerion of he cr beween imes i nd f cn be found by deermining he slope of he line joining poins nd. If we imgine h poin is brough closer nd closer o poin, he line comes closer nd closer o becoming ngen. The insnneous ccelerion of n objec given ime equls he slope of he ngen o he velociy vs. ime grph h ime. From now on, we will use he erm ccelerion o men insnneous ccelerion. In he specil cse where he velociy vs. ime grph of n objec s moion is srigh line, he insnneous ccelerion of he objec ny poin is equl o is verge ccelerion. Th lso mens h he ngen line o he grph overlps he grph iself. In h cse, he objec s ccelerion is sid o be uniform, which mens h i hs consn vlue. Consn ccelerion problems re imporn in kinemics nd will be sudied exensively in his nd he nex chper. v f v i i v f Quick Quiz.3 Prs (), (b), nd (c) of Figure.10 represen hree grphs of he velociies of differen objecs moving in srigh-line phs s funcions of ime. The possible ccelerions of ech objec s funcions of ime re shown in prs (d), (e), nd (f). Mch ech velociy vs. ime grph wih he ccelerion vs. ime grph h bes describes he moion. b The slope of he blue line connecing nd is he verge ccelerion of he cr during he ime inervl f i (Eq..4). Acive Figure.10 (Quick Quiz.3) Mch ech velociy vs. ime grph o is corresponding ccelerion vs. ime grph. v v v b c Figure.9 () A cr, modeled s pricle, moving long he x-xis from o, hs velociy v xi 5 i nd velociy v xf 5 f. (b) Velociy vs. ime grph for n objec moving in srigh line. d e f

11 .4 Moion Digrms 35 EXAMPLE.3 Cching Fly Bll GOAL Apply he definiion of insnneous ccelerion. PROBLEM A bsebll plyer moves in srigh-line ph in order o cch fly bll hi o he oufield. His velociy s funcion of ime is shown in Figure.11. Find his insnneous ccelerion poins,, nd. STRATEGY A ech poin, he velociy vs. ime grph is srigh line segmen, so he insnneous ccelerion will be he slope of h segmen. Selec wo poins on ech segmen nd use hem o clcule he slope. SOLUTION Accelerion. The ccelerion equls he slope of he line connecing he poins (0 s, 0 m/s) nd (.0 s, 4.0 m/s): Accelerion. Dv 5 0, becuse he segmen is horizonl: Accelerion. The ccelerion equls he slope of he line connecing he poins (3.0 s, 4.0 m/s) nd (4.0 s,.0 m/s): v (m/s) O (s) v (m/s) O Figure.11 () (Exmple.3) (b) (Exercise.3) 5 Dv 4.0 m/s m/s D.0 s 0 5 Dv 4.0 m/s 4.0 m/s m/s D 3.0 s.0 s 5 Dv.0 m/s 4.0 m/s m/s D 4.0 s 3.0 s b (s) REMARKS Assume he plyer is iniilly moving in he posiive x-direcion. For he firs.0 s, he bllplyer moves in he posiive x-direcion (he velociy is posiive) nd sedily cceleres (he curve is sedily rising) o mximum speed of 4.0 m/s. He moves for 1.0 s sedy speed of 4.0 m/s nd hen slows down in he ls second (he v vs. curve is flling), sill moving in he posiive x-direcion (v is lwys posiive). QUESTION.3 Cn he ngen line o velociy vs. ime grph ever be vericl? Explin. EXERCISE.3 Repe he problem, using Figure.11b. ANSWER The ccelerions,, nd re 3.0 m/s, 1.0 m/s, nd 0 m/s, respecively..4 Moion Digrms Velociy nd ccelerion re someimes confused wih ech oher, bu hey re very differen conceps, s cn be illusred wih he help of moion digrms. A moion digrm is represenion of moving objec successive ime inervls, wih velociy nd ccelerion vecors skeched ech posiion, red for velociy vecors nd viole for ccelerion vecors, s in Acive Figure.1 (pge 36). The ime inervls beween djcen posiions in he moion digrm re ssumed equl. A moion digrm is nlogous o imges resuling from sroboscopic phoogrph of moving objec. Ech imge is mde s he srobe ligh flshes. Acive Figure.1 represens hree ses of srobe phoogrphs of crs moving long srigh rodwy from lef o righ. The ime inervls beween flshes of he sroboscope re equl in ech digrm. In Acive Figure.1, he imges of he cr re eqully spced: The cr moves he sme disnce in ech ime inervl. This mens h he cr moves wih consn posiive velociy nd hs zero ccelerion. The red rrows re ll he sme lengh (consn velociy) nd here re no viole rrows (zero ccelerion). In Acive Figure.1b, he imges of he cr become frher pr s ime progresses nd he velociy vecor increses wih ime, becuse he cr s displcemen

12 36 CHAPTER Moion in One Dimension Acive Figure.1 Moion digrms of cr moving long srigh rodwy in single direcion. The velociy ech insn is indiced by red rrow, nd he consn ccelerion is indiced by purple rrow. This cr moves consn velociy (zero ccelerion). This cr hs consn ccelerion in he direcion of is velociy. b v v This cr hs consn ccelerion in he direcion opposie is velociy. c v beween djcen posiions increses s ime progresses. The cr is moving wih posiive velociy nd consn posiive ccelerion. The red rrows re successively longer in ech imge, nd he viole rrows poin o he righ. In Acive Figure.1c, he cr slows s i moves o he righ becuse is displcemen beween djcen posiions decreses wih ime. In his cse, he cr moves iniilly o he righ wih consn negive ccelerion. The velociy vecor decreses in ime (he red rrows ge shorer) nd evenully reches zero, s would hppen when he brkes re pplied. Noe h he ccelerion nd velociy vecors re no in he sme direcion. The cr is moving wih posiive velociy, bu wih negive ccelerion. Try consrucing your own digrms for vrious problems involving kinemics. Quick Quiz.4 The hree grphs in Figure.13 represen he posiion vs. ime for objecs moving long he x-xis. Which, if ny, of hese grphs is no physiclly possible?.5 Figure.14 is digrm of muliflsh imge of n ir puck moving o he righ on horizonl surfce. The imges skeched re sepred by equl ime inervls, nd he firs nd ls imges show he puck res. () In Figure.14b, which color grph bes shows he puck s posiion s funcion of ime? (b) In Figure.14c, which color grph bes shows he puck s velociy s funcion of ime? (c) In Figure.14d, which color grph bes shows he puck s ccelerion s funcion of ime? x x x b c d + O + x O + O v Figure.14 (Quick Quiz.5) Choose he correc grphs. Mny pplicions of mechnics involve objecs moving wih consn ccelerion. This ype of moion is imporn becuse i pplies o numerous objecs in nure, such s n objec in free fll ner Erh s surfce (ssuming ir resisnce cn be negleced). A grph of ccelerion versus ime for moion wih consn cceler b c Figure.13 (Quick Quiz.4) Which posiion vs. ime curve is impossible?.5 One-Dimensionl Moion wih Consn Accelerion

13 .5 One-Dimensionl Moion wih Consn Accelerion 37 ion is shown in Acive Figure.15. When n objec moves wih consn ccelerion, he insnneous ccelerion ny poin in ime inervl is equl o he vlue of he verge ccelerion over he enire ime inervl. Consequenly, he velociy increses or decreses he sme re hroughou he moion, nd plo of v versus gives srigh line wih eiher posiive, zero, or negive slope. Becuse he verge ccelerion equls he insnneous ccelerion when is consn, we cn elimine he br used o denoe verge vlues from our defining equion for ccelerion, wriing 5, so h Equion.4 becomes 5 v f v i f i The observer iming he moion is lwys libery o choose he iniil ime, so for convenience, le i 5 0 nd f be ny rbirry ime. Also, le v i 5 v 0 (he iniil velociy 5 0) nd v f 5 v (he velociy ny rbirry ime ). Wih his noion, we cn express he ccelerion s or 5 v v 0 v 5 v 0 1 (for consn ) [.6] Equion.6 ses h he ccelerion sedily chnges he iniil velociy v 0 by n moun. For exmple, if cr srs wih velociy of 1.0 m/s o he righ nd cceleres o he righ wih m/s, i will hve velociy of 114 m/s fer.0 s hve elpsed: v 5 v m/s 1 (6.0 m/s )(.0 s) m/s The grphicl inerpreion of v is shown in Acive Figure.15b. The velociy vries linerly wih ime ccording o Equion.6, s i should for consn ccelerion. Becuse he velociy is incresing or decresing uniformly wih ime, we cn express he verge velociy in ny ime inervl s he rihmeic verge of he iniil velociy v 0 nd he finl velociy v: v 5 v 0 1 v (for consn ) [.7] Remember h his expression is vlid only when he ccelerion is consn, in which cse he velociy increses uniformly. We cn now use his resul long wih he defining equion for verge velociy, Equion., o obin n expression for he displcemen of n objec s funcion of ime. Agin, we choose i 5 0 nd f 5, nd for convenience, we wrie Dx 5 x f x i 5 x x 0. This resuls in Dx 5 v 5 v 0 1 v b v 0 v x b x 0 Slope v 0 c Slope 0 Slope Slope v v 0 Acive Figure.15 A pricle moving long he x-xis wih consn ccelerion. () he ccelerion vs. ime grph, (b) he velociy vs. ime grph, nd (c) he posiion vs. ime grph. v Dx 5 1 1v 0 1 v (for consn ) [.8] We cn obin noher useful expression for displcemen by subsiuing he equion for v (Eq..6) ino Equion.8: Dx 5 1 1v 0 1 v 0 1 Dx 5 v (for consn ) [.9] This equion cn lso be wrien in erms of he posiion x, since Dx 5 x x 0. Acive Figure.15c shows plo of x versus for Equion.9, which is reled o he grph of velociy vs. ime: The re under he curve in Acive Figure.15b is equl o v 0 1 1, which is equl o he displcemen Dx. In fc, he re under he grph of v versus for ny objec is equl o he displcemen Dx of he objec.

14 38 CHAPTER Moion in One Dimension Tble.4 Equions for Moion in Srigh Line Under Consn Accelerion Equion v 5 v 0 1 Dx 5 v v 5 v 0 1 Dx Informion Given by Equion Velociy s funcion of ime Displcemen s funcion of ime Velociy s funcion of displcemen Noe: Moion is long he x-xis. A 5 0, he velociy of he pricle is v 0. Finlly, we cn obin n expression h doesn conin ime by solving Equion.6 for nd subsiuing ino Equion.8, resuling in Dx 5 1 1v 1 v 0 v v 0 b 5 v v 0 v 5 v 0 1 Dx (for consn ) [.10] Equions.6 nd.9 ogeher cn solve ny problem in one-dimensionl moion wih consn ccelerion, bu Equions.7,.8, nd, especilly,.10 re someimes convenien. The hree mos useful equions Equions.6,.9, nd.10 re lised in Tble.4. The bes wy o gin confidence in he use of hese equions is o work number of problems. There is usully more hn one wy o solve given problem, depending on which equions re seleced nd wh quniies re given. The difference lies minly in he lgebr. PROBLEM-SOLVING STRATEGY Tip.7 Pigs Don Fly Afer solving problem, you should hink bou your nswer nd decide wheher i seems resonble. If i isn, look for your miske! Moion in One Dimension Consn Accelerion The following procedure is recommended for solving problems involving ccelered moion. 1. Red he problem.. Drw digrm, choosing coordine sysem, lbeling iniil nd finl poins, nd indicing direcions of velociies nd ccelerions wih rrows. 3. Lbel ll quniies, circling he unknowns. Conver unis s needed. 4. Equions from Tble.4 should be seleced nex. All kinemics problems in his chper cn be solved wih he firs wo equions, nd he hird is ofen convenien. 5. Solve for he unknowns. Doing so ofen involves solving wo equions for wo unknowns. 6. Check your nswer, using common sense nd esimes. Mos of hese problems reduce o wriing he kinemic equions from Tble.4 nd hen subsiuing he correc vlues ino he consns, v 0, nd x 0 from he given informion. Doing his produces wo equions one liner nd one qudric for wo unknown quniies. EXAMPLE.4 The Dyon 500 GOAL Apply he bsic kinemic equions. v 0 = 0 v =? PROBLEM () A rce cr sring from res cceleres consn re of 5.00 m/s. Wh is he velociy of he cr fer i hs rveled f? (b) How much ime hs elpsed? (c) Clcule he verge velociy wo differen wys. STRATEGY () We ve red he problem, drwn he digrm in Figure.16, nd chosen coordine sysem (seps 1 nd ). We d like o find he velociy v fer cerin Figure.16 (Exmple.4) known displcemen Dx. The ccelerion is lso known, s is he iniil velociy v 0 (sep 3, lbeling, is complee), so he hird equion in Tble.4 looks mos useful for solving pr (). Given he velociy, x = 0 x = 30.5 m + x

15 .5 One-Dimensionl Moion wih Consn Accelerion 39 he firs equion in Tble.4 cn hen be used o find he ime in pr (b). Pr (c) requires subsiuion ino Equions. nd.7, respecively. SOLUTION () Conver unis of Dx o SI, using he informion in he inside fron cover. Wrie he kinemics equion for v (sep 4): Solve for v, king he posiive squre roo becuse he cr moves o he righ (sep 5): Subsiue v 0 5 0, m/s, nd Dx m: f f 1 m 3.8 f b m v 5 v 0 1 Dx v 5 "v 0 1 Dx v 5 "v 0 1 Dx 5 " m/s m m/s (b) How much ime hs elpsed? Apply he firs equion of Tble.4: v 5 1 v 0 Subsiue vlues nd solve for ime : 17.5 m/s 5 (5.00 m/s ) 17.5 m/s m/s s (c) Clcule he verge velociy in wo differen wys. Apply he definiion of verge velociy, Equion.: v 5 x f x i f i m 3.50 s m/s Apply he definiion of verge velociy in Equion.7: v 5 v 0 1 v m/s m/s REMARKS The nswers re esy o check. An lerne echnique is o use Dx 5 v o find nd hen use he equion v 5 v 0 1 o find v. Noice h he wo differen equions for clculing he verge velociy, due o rounding, give slighly differen nswers. QUESTION.4 Wh is he finl speed if he displcemen is incresed by fcor of 4? EXERCISE.4 Suppose he driver in his exmple now slms on he brkes, sopping he cr in 4.00 s. Find () he ccelerion, (b) he disnce he cr rvels while brking, ssuming he ccelerion is consn, nd (c) he verge velociy. ANSWERS () 4.38 m/s (b) 35.0 m (c) 8.75 m/s EXAMPLE.5 Cr Chse GOAL Solve problem involving wo objecs, one moving consn ccelerion nd he oher consn velociy. PROBLEM A cr rveling consn speed of 4.0 m/s psses rooper hidden behind billbord, s in Figure.17. One second fer he speeding cr psses he billbord, he rooper ses off in chse wih consn ccelerion of 3.00 m/s. () How long does i ke he rooper o overke he speeding cr? (b) How fs is he rooper going h ime? STRATEGY Solving his problem involves wo simulneous kinemics equions of posiion, one for he rooper nd he oher for he cr. Choose 5 0 o correspond o he ime he rooper kes 1.00 s 0? Figure.17 (Exmple.5) A speeding cr psses hidden rooper. When does he rooper cch up o he cr? up he chse, when he cr is x cr m becuse of is hed sr (4.0 m/s s). The rooper cches up wih he cr when heir posiions re he sme, which suggess seing x rooper 5 x cr nd solving for ime, which cn hen be used o find he rooper s speed in pr (b). (Coninued)

16 40 CHAPTER Moion in One Dimension SOLUTION () How long does i ke he rooper o overke he cr? Wrie he equion for he cr s displcemen: Dx cr 5 x cr x 0 5 v cr Tke x m, v m/s, nd cr 5 0. Solve for x cr : Wrie he equion for he rooper s posiion, king x 0 5 0, v 0 5 0, nd rooper m/s : Se x rooper 5 x cr, nd solve he qudric equion. (The qudric formul ppers in Appendix A, Equion A.8.) Only he posiive roo is meningful. x cr 5 x 0 1 v m 1 (4.0 m/s) x rooper 5 1 rooper m/s m/s (1.50 m/s ) m 1 (4.0 m/s) (1.50 m/s ) (4.0 m/s) 4.0 m s (b) Find he rooper s speed h ime. Subsiue he ime ino he rooper s velociy equion: v rooper 5 v 0 1 rooper (3.00 m/s )(16.9 s) m/s REMARKS The rooper, rveling bou wice s fs s he cr, mus swerve or pply his brkes srongly o void collision! This problem cn lso be solved grphiclly by ploing posiion versus ime for ech vehicle on he sme grph. The inersecion of he wo grphs corresponds o he ime nd posiion which he rooper overkes he cr. QUESTION.5 The grphicl soluion corresponds o finding he inersecion of wh wo ypes of curves in he x-plne? EXERCISE.5 A mooris wih n expired license g is rveling 10.0 m/s down sree, nd policemn on moorcycle, king noher 5.00 s o finish his donu, gives chse n ccelerion of.00 m/s. Find () he ime required o cch he cr nd (b) he disnce he rooper rvels while overking he mooris. ANSWERS () 13.7 s (b) 188 m EXAMPLE.6 Runwy Lengh GOAL Apply kinemics o horizonl moion wih wo phses. PROBLEM A ypicl jeliner lnds speed of mi/h nd deceleres he re of (10.0 mi/h)/s. If he plne rvels consn speed of mi/h for 1.00 s fer lnding before pplying he brkes, wh is he ol displcemen of he ircrf beween ouchdown on he runwy nd coming o res? STRATEGY See Figure.18. Firs, conver ll quniies o SI unis. The problem mus be solved in wo prs, or phses, corresponding o he iniil cos fer ouchdown, followed by brking. Using he kinemic equions, find he displcemen during ech pr nd dd he wo displcemens. Origin v Cosing disnce v 0 = 71.5 m/s = 0 = 1.00 s Brking disnce v v 0 = 71.5 m/s v f = 0 = 4.47 m/s Figure.18 (Exmple.6) Cosing nd brking disnces for lnding jeliner. +x SOLUTION Conver unis of speed nd ccelerion o SI: Tking 5 0, v m/s, nd s, find he displcemen while he plne is cosing: Use he ime-independen kinemic equion o find he displcemen while he plne is brking. v m/s mi/h b m/s 1.00 mi/h m/s mi/h /s b m/s 1.00 mi/h Dx cosing 5 v m/s11.00 s m v 5 v 0 1 Dx brking

17 .5 One-Dimensionl Moion wih Consn Accelerion 41 Tke m/s nd v m/s. The negive sign on mens h he plne is slowing down. Sum he wo resuls o find he ol displcemen: Dx brking 5 v v m/s m/s 5 57 m Dx cosing 1 Dx brking m 1 57 m m REMARKS To find he displcemen while brking, we could hve used he wo kinemics equions involving ime, nmely, Dx 5 v nd v 5 v 0 1, bu becuse we weren ineresed in ime, he ime-independen equion ws esier o use. QUESTION.6 How would he nswer chnge if he plne cosed for.00 s before he pilo pplied he brkes? EXERCISE.6 A je lnds 80.0 m/s, he pilo pplying he brkes.00 s fer lnding. Find he ccelerion needed o sop he je wihin m fer ouchdown. ANSWER 9.41 m/s EXAMPLE.7 The Acel: The Porsche of Americn Trins GOAL Find ccelerions nd displcemens from velociy vs. ime grph. PROBLEM The sleek high-speed elecric rin known s he Acel (pronounced hh-sell-h) is currenly in service on he Wshingon-New York-Boson run. The Acel consiss of wo power crs nd six coches nd cn crry 304 pssengers speeds up o 170 mi/h. In order o negoie curves comforbly high speeds, he rin crriges il s much s 6 from he vericl, prevening pssengers from being pushed o he side. A velociy vs. ime grph for he Acel is shown in Figure.19 (pge 4). () Describe he moion of he Acel. (b) Find he pek ccelerion of he Acel in miles per hour per second ((mi/h)/s) s he rin speeds up from 45 mi/h o 170 mi/h. (c) Find he rin s displcemen in miles beween 5 0 nd 5 00 s. (d) Find he verge ccelerion of he Acel nd is displcemen in miles in he inervl from 00 s o 300 s. (The rin hs regenerive brking, which mens h i feeds energy bck ino he uiliy lines ech ime i sops!) (e) Find he ol displcemen in he inervl from 0 o 400 s. STRATEGY For pr (), remember h he slope of he ngen line ny poin of he velociy vs. ime grph gives he ccelerion h ime. To find he pek ccelerion in pr (b), sudy he grph nd loce he poin which he slope is seepes. In prs (c) hrough (e), esiming he re under he curve gives he displcemen during given period, wih res below he ime xis, s in pr (e), subrced from he ol. The verge ccelerion in pr (d) cn be obined by subsiuing numbers ken from he grph ino he definiion of verge ccelerion, 5 Dv/D. SOLUTION () Describe he moion. From bou 50 s o 50 s, he Acel cruises consn velociy in he 1x-direcion. Then he rin cceleres in he 1x-direcion from 50 s o 00 s, reching op speed of bou 170 mi/h, whereupon i brkes o res 350 s nd reverses, sedily gining speed in he x-direcion. (b) Find he pek ccelerion. Clcule he slope of he seepes ngen line, which connecs he poins (50 s, 50 mi/h) nd (100 s, 150 mi/h) (he ligh blue line in Figure.19b): (c) Find he displcemen beween 0 s nd 00 s. Using ringles nd recngles, pproxime he re in Figure.19c (see pge 4): 5 slope 5 Dv D mi/h s 5.0 (mi/h)/s Dx 0 S 00 s 5 re 1 1 re 1 re 3 1 re 4 1 re 5 < ( mi/h)( s) 1 ( mi/h)( s) 1 ( mi/h)( s) s mi/h s mi/h mi/h (mi/h)s (Coninued)

18 4 CHAPTER Moion in One Dimension v (mi/h) (s) v (mi/h) v (s) b v (mi/h) (s) v (mi/h) (s) c d Figure.19 (Exmple.7) () Velociy vs. ime grph for he Acel. (b) The slope of he seepes ngen blue line gives he pek ccelerion, nd he slope of he green line is he verge ccelerion beween 00 s nd 300 s. (c) The re under he velociy vs. ime grph in some ime inervl gives he displcemen of he Acel in h ime inervl. (d) (Exercise.7). Conver unis o miles by convering hours o seconds: (d) Find he verge ccelerion from 00 s o 300 s, nd find he displcemen. The slope of he green line is he verge ccelerion from 00 s o 300 s (Fig..19b): The displcemen from 00 s o 300 s is equl o re 6, which is he re of ringle plus he re of very nrrow recngle beneh he ringle: Dx 0 S 00 s < mi # s h 1 h b mi s 5 slope 5 Dv D mi/h s (mi/h)/s Dx 00 S 300 s < s mi/h 1 ( mi/h)( s) (mi/h)(s) 5.5 mi (e) Find he ol displcemen from 0 s o 400 s. The ol displcemen is he sum of ll he individul displcemens. We sill need o clcule he displcemens for he ime inervls from 300 s o 350 s nd fromv 350 s o 400 s. The ler is negive becuse i s below he ime xis. REMARKS There re number of wys o find he pproxime re under grph. Choice of echnique is personl preference. QUESTION.7 According o he grph in Figure.19, wh differen imes is he ccelerion zero? EXERCISE.7 Suppose he velociy vs. ime grph of noher rin is given in Figure.19d. Find () he mximum insnneous ccelerion nd (b) he ol displcemen in he inervl from 0 s o s. ANSWERS () 1.0 (mi/h)/s (b) 4.7 mi Dx 300 S 350 s < s mi/h (mi/h)(s) Dx 350 S 400 s < s mi/h (mi/h)(s) Find he ol displcemen by summing he prs: Dx 0 S 400 s < )(mi/h)(s) mi

19 .6 Freely Flling Objecs 43.6 Freely Flling Objecs When ir resisnce is negligible, ll objecs dropped under he influence of grviy ner Erh s surfce fll owrd Erh wih he sme consn ccelerion. This ide my seem obvious ody, bu i wsn unil bou 1600 h i ws cceped. Prior o h ime, he echings of he gre philosopher Arisole (384 3 b.c.) hd held h hevier objecs fell fser hn ligher ones. According o legend, Glileo discovered he lw of flling objecs by observing h wo differen weighs dropped simulneously from he Lening Tower of Pis hi he ground pproximely he sme ime. Alhough i s unlikely h his priculr experimen ws crried ou, we know h Glileo performed mny sysemic experimens wih objecs moving on inclined plnes. In his experimens he rolled blls down sligh incline nd mesured he disnces hey covered in successive ime inervls. The purpose of he incline ws o reduce he ccelerion nd enble Glileo o mke ccure mesuremens of he inervls. (Some people refer o his experimen s diluing grviy. ) By grdully incresing he slope of he incline he ws finlly ble o drw mhemicl conclusions bou freely flling objecs, becuse flling bll is equivlen o bll going down vericl incline. Glileo s chievemens in he science of mechnics pved he wy for Newon in his developmen of he lws of moion, which we will sudy in Chper 4. Try he following experimen: Drop hmmer nd feher simulneously from he sme heigh. The hmmer his he floor firs becuse ir drg hs greer effec on he much ligher feher. On Augus, 1971, his sme experimen ws conduced on he Moon by sronu Dvid Sco, nd he hmmer nd feher fell wih excly he sme ccelerion, s expeced, hiing he lunr surfce he sme ime. In he idelized cse where ir resisnce is negligible, such moion is clled free fll. The expression freely flling objec doesn necessrily refer o n objec dropped from res. A freely flling objec is ny objec moving freely under he influence of grviy lone, regrdless of is iniil moion. Objecs hrown upwrd or downwrd nd hose relesed from res re ll considered freely flling. We denoe he mgniude of he free-fll ccelerion by he symbol g. The vlue of g decreses wih incresing liude, nd vries slighly wih liude s well. A Erh s surfce, he vlue of g is pproximely 9.80 m/s. Unless sed oherwise, we will use his vlue for g in doing clculions. For quick esimes, use g < 10 m/s. If we neglec ir resisnce nd ssume h he free-fll ccelerion doesn vry wih liude over shor vericl disnces, hen he moion of freely flling objec is he sme s moion in one dimension under consn ccelerion. This mens h he kinemics equions developed in Secion.5 cn be pplied. I s convenionl o define up s he 1 y-direcion nd o use y s he posiion vrible. In h cse he ccelerion is 5 g m/s. In Chper 7, we sudy he vriion in g wih liude. Norh Wind Archive Glileo Glilei Ilin Physicis nd Asronomer ( ) Glileo formuled he lws h govern he moion of objecs in free fll. He lso invesiged he moion of n objec on n inclined plne, esblished he concep of relive moion, invened he hermomeer, nd discovered h he moion of swinging pendulum could be used o mesure ime inervls. Afer designing nd consrucing his own elescope, he discovered four of Jupier s moons, found h our own Moon s surfce is rough, discovered sunspos nd he phses of Venus, nd showed h he Milky Wy consiss of n enormous number of srs. Glileo publicly defended Nicolus Copernicus s sserion h he Sun is he cener of he Universe (he heliocenric sysem). He published Dilogue Concerning Two New World Sysems o suppor he Copernicn model, view he Church declred o be hereicl. Afer being ken o Rome in 1633 on chrge of heresy, he ws senenced o life imprisonmen nd ler ws confined o his vill Arceri, ner Florence, where he died in 164. Quick Quiz.6 A ennis plyer on serve osses bll srigh up. While he bll is in free fll, does is ccelerion () increse, (b) decrese, (c) increse nd hen decrese, (d) decrese nd hen increse, or (e) remin consn?.7 As he ennis bll of Quick Quiz.6 rvels hrough he ir, does is speed () increse, (b) decrese, (c) decrese nd hen increse, (d) increse nd hen decrese, or (e) remin he sme?.8 A skydiver jumps ou of hovering helicoper. A few seconds ler, noher skydiver jumps ou, so hey boh fll long he sme vericl line relive o he helicoper. Boh skydivers fll wih he sme ccelerion. Does he vericl disnce beween hem () increse, (b) decrese, or (c) sy he sme? Does he difference in heir velociies (d) increse, (e) decrese, or (f) sy he sme? (Assume g is consn.)

20 44 CHAPTER Moion in One Dimension EXAMPLE.8 No Bd Throw for Rookie! GOAL Apply he kinemic equions o freely flling objec wih nonzero iniil velociy. PROBLEM A bll is hrown from he op of building wih n iniil velociy of 0.0 m/s srigh upwrd, n iniil heigh of 50.0 m bove he ground. The bll jus misses he edge of he roof on is wy down, s shown in Figure.0. Deermine () he ime needed for he bll o rech is mximum heigh, (b) he mximum heigh, (c) he ime needed for he bll o reurn o he heigh from which i ws hrown nd he velociy of he bll h insn, (d) he ime needed for he bll o rech he ground, nd (e) he velociy nd posiion of he bll s. Neglec ir drg. STRATEGY The digrm in Figure.0 esblishes coordine sysem wih y he level which he bll is relesed from he hrower s hnd, wih y posiive upwrd. Wrie he velociy nd posiion kinemic equions for he bll, nd subsiue he given informion. All he nswers come from hese wo equions by using simple lgebr or by jus subsiuing he ime. In pr (), for exmple, he bll comes o res for n insn is mximum heigh, so se v 5 0 his poin nd solve for ime. Then subsiue he ime ino he displcemen equion, obining he mximum heigh m SOLUTION () Find he ime when he bll reches is mximum heigh. 0 y 0 0 v m/s.04 s y mx 0.4 m v s y 0 v 0.0 m/s 5.00 s y.5 m v 9.0 m/s 5.83 s y 50.0 m v 37.1 m/s Figure.0 (Exmple.8) A bll is hrown upwrd wih n iniil velociy of v m/s. Posiions nd velociies re given for severl imes. Wrie he velociy nd posiion kinemic equions: v 5 1 v 0 Dy 5 y y 0 5 v Subsiue m/s, v m/s, nd y ino he preceding wo equions: Subsiue v 5 0, he velociy mximum heigh, ino Equion (1) nd solve for ime: (1) v 5 (9.80 m/s ) m/s () y 5 (0.0 m/s) (4.90 m/s ) 0 5 (9.80 m/s ) m/s 0.0 m/s m/s 5.04 s (b) Deermine he bll s mximum heigh. Subsiue he ime 5.04 s ino Equion (): y mx 5 (0.0 m/s)(.04 s) (4.90 m/s )(.04 s) m (c) Find he ime he bll kes o reurn o is iniil posiion, nd find he velociy of he bll h ime. Se y 5 0 in Equion () nd solve : 0 5 (0.0 m/s) (4.90 m/s ) 5 (0.0 m/s 4.90 m/s ) s

21 .6 Freely Flling Objecs 45 Subsiue he ime ino Equion (1) o ge he velociy: v m/s 1 (9.80 m/s )(4.08 s) m/s (d) Find he ime required for he bll o rech he ground. In Equion (), se y m: 50.0 m 5 (0.0 m/s) (4.90 m/s ) Apply he qudric formul nd ke he posiive roo: s (e) Find he velociy nd posiion of he bll s. Subsiue vlues ino Equions (1) nd (): v 5 (9.80 m/s )(5.00 s) m/s m/s y 5 (0.0 m/s)(5.00 s) (4.90 m/s )(5.00 s) 5.5 m REMARKS Noice how everyhing follows from he wo kinemic equions. Once hey re wrien down nd he consns correcly idenified s in Equions (1) nd (), he res is relively esy. If he bll were hrown downwrd, he iniil velociy would hve been negive. QUESTION.8 How would he nswer o pr (b), he mximum heigh, chnge if he person hrowing he bll jumped upwrd he insn he relesed he bll? EXERCISE.8 A projecile is lunched srigh up 60.0 m/s from heigh of 80.0 m, he edge of sheer cliff. The projecile flls, jus missing he cliff nd hiing he ground below. Find () he mximum heigh of he projecile bove he poin of firing, (b) he ime i kes o hi he ground he bse of he cliff, nd (c) is velociy impc. ANSWERS () 184 m (b) 13.5 s (c) 7.3 m/s EXAMPLE.9 Mximum Heigh Derived GOAL Find he mximum heigh of hrown projecile using symbols. PROBLEM Refer o Exmple.8. Use symbolic mnipulion o find () he ime mx i kes he bll o rech is mximum heigh nd (b) n expression for he mximum heigh h doesn depend on ime. Answers should be expressed in erms of he quniies v 0, g, nd y 0 only. STRATEGY When he bll reches is mximum heigh, is velociy is zero, so for pr () solve he kinemics velociy equion for ime nd se v 5 0. For pr (b), subsiue he expression for ime found in pr () ino he displcemen equion, solving i for he mximum heigh. SOLUTION () Find he ime i kes he bll o rech is mximum heigh. Wrie he velociy kinemics equion: v 5 1 v 0 Move v 0 o he lef side of he equion: Divide boh sides by : Turn he equion round so h is on he lef nd subsiue v 5 0, corresponding o he velociy mximum heigh: v v 0 5 v v (1) 5 v 0 Replce by mx nd subsiue 5 g : () mx 5 v 0 g (b) Find he mximum heigh. Wrie he equion for he posiion y ny ime: y 5 y 0 1 v (Coninued)

22 46 CHAPTER Moion in One Dimension Subsiue 5 v 0 /, which corresponds o he ime i kes o rech y mx, he mximum heigh: y mx 5 y 0 1 v 0 v 0 b 1 1 v 0 b 5 y 0 v 0 1 v 1 0 Combine he ls wo erms nd subsiue 5 g : (3) y mx 5 y 0 1 v 0 g REMARKS Noice h g m/s, so he second erm is posiive overll. Equions (1) (3) re much more useful hn numericl nswer becuse he effec of chnging one vlue cn be seen immediely. For exmple, doubling he iniil velociy v 0 qudruples he displcemen bove he poin of relese. Noice lso h y mx could be obined more redily from he ime-independen equion, v v 0 5 Dy. QUESTION.9 By wh fcor would he mximum displcemen bove he roofop be incresed if he building were rnspored o he Moon, where 5 1 6g? EXERCISE.9 () Using symbols, find he ime E i kes for bll o rech he ground on Erh if relesed from res heigh y 0. (b) In erms of E, how much ime M would be required if he building were on Mrs, where g? ANSWERS () E 5 Å y 0 g (b) M E EXAMPLE.10 GOAL Solve problem involving powered scen followed by free-fll moion. PROBLEM A rocke moves srigh upwrd, sring from res wih n ccelerion of 19.4 m/s. I runs ou of fuel he end of 4.00 s nd coninues o cos upwrd, reching mximum heigh before flling bck o Erh. () Find he rocke s velociy nd posiion he end of 4.00 s. (b) Find he mximum heigh he rocke reches. (c) Find he velociy he insn before he rocke crshes on he ground. STRATEGY Tke y 5 0 he lunch poin nd y posiive upwrd, s in Figure.1. The problem consiss of wo phses. In phse 1 he rocke hs ne upwrd ccelerion of 9.4 m/s, nd we cn use he kinemic equions wih consn A Rocke Goes Bllisic Rocke fuel burns ou y = 0 Phse = 9.80 m/s Lunch ccelerion o find he heigh nd velociy of he rocke he end of phse 1, when he fuel is burned up. In phse he rocke is in free fll nd hs n ccelerion of 9.80 m/s, wih iniil velociy nd posiion given by he resuls of phse 1. Apply he kinemic equions for free fll. SOLUTION () Phse 1: Find he rocke s velociy nd posiion fer 4.00 s. +y Phse 1 = 9.4 m/s Mximum heigh y mx v = 0 Rocke crshes fer flling from y mx Figure.1 (Exmple.10) Two linked phses of moion for rocke h is lunched, uses up is fuel, nd crshes. Wrie he velociy nd posiion kinemic equions: (1) v 5 v 0 1 () Dy 5 y y 0 5 v 0 1 1

23 Summry 47 Adp hese equions o phse 1, subsiuing m/s, v 0 5 0, nd y 0 5 0: Subsiue s ino Equions (3) nd (4) o find he rocke s velociy v nd posiion y he ime of burnou. These will be clled v b nd y b, respecively. (b) Phse : Find he mximum heigh he rocke ins. Adp Equions (1) nd () o phse, subsiuing m/s, v 0 5 v b m/s, nd y 0 5 y b 5 35 m: Subsiue v 5 0 (he rocke s velociy mximum heigh) in Equion (5) o ge he ime i kes he rocke o rech is mximum heigh: Subsiue s ino Equion (6) o find he rocke s mximum heigh: (3) v 5 (9.4 m/s ) (4) y m/s m/s v b m/s nd y b 5 35 m (5) v 5 (9.8 m/s ) m/s (6) y 5 35 m m/s m/s m/s m/s S m/s 9.80 m/s s y mx 5 35 m 1 (118 m/s)(1.0 s) (4.90 m/s )(1.0 s) m (c) Phse : Find he velociy of he rocke jus prior o impc. Find he ime o impc by seing y 5 0 in Equion (6) nd using he qudric formul: Subsiue his vlue of ino Equion (5): m 1 (118 m/s) (4.90 m/s) s v 5 (9.80 m/s )(5.9 s) m/s m/s REMARKS You my hink h i is more nurl o brek his problem ino hree phses, wih he second phse ending he mximum heigh nd he hird phse free fll from mximum heigh o he ground. Alhough his pproch gives he correc nswer, i s n unnecessry complicion. Two phses re sufficien, one for ech differen ccelerion. QUESTION.10 If, insed, some fuel remins, wh heigh should he engines be fired gin o brke he rocke s fll nd llow perfecly sof lnding? (Assume he sme ccelerion s during he iniil scen.) EXERCISE.10 An experimenl rocke designed o lnd uprigh flls freely from heigh of m, sring res. A heigh of 80.0 m, he rocke s engines sr nd provide consn upwrd ccelerion unil he rocke lnds. Wh ccelerion is required if he speed on ouchdown is o be zero? (Neglec ir resisnce.) ANSWER 14.7 m/s SUMMARY.1 Displcemen The displcemen of n objec moving long he x-xis is defined s he chnge in posiion of he objec, Dx ; x f x i [.1] where x i is he iniil posiion of he objec nd x f is is finl posiion. A vecor quniy is chrcerized by boh mgniude nd direcion. A sclr quniy hs mgniude only.. Velociy The verge speed of n objec is given by ph lengh Averge speed ; elpsed ime The verge velociy v during ime inervl D is he displcemen Dx divided by D. v ; Dx D 5 x f x i f i [.] The verge velociy is equl o he slope of he srigh line joining he iniil nd finl poins on grph of he posiion of he objec versus ime. The slope of he line ngen o he posiion vs. ime curve some poin is equl o he insnneous velociy h ime. The insnneous speed of n objec is defined s he mgniude of he insnneous velociy..3 Accelerion The verge ccelerion of n objec undergoing chnge in velociy Dv during ime inervl D is ; Dv D 5 v f v i f i [.4]

24 48 CHAPTER Moion in One Dimension The insnneous ccelerion of n objec cerin ime equls he slope of velociy vs. ime grph h insn..5 One-Dimensionl Moion wih Consn Accelerion The mos useful equions h describe he moion of n objec moving wih consn ccelerion long he x-xis re s follows: v 5 v 0 1 [.6] Dx 5 v [.9] v 5 v 0 1 Dx [.10] All problems cn be solved wih he firs wo equions lone, he ls being convenien when ime doesn explicily ener he problem. Afer he consns re properly idenified, mos problems reduce o one or wo equions in s mny unknowns..6 Freely Flling Objecs An objec flling in he presence of Erh s grviy exhibis free-fll ccelerion direced owrd Erh s cener. If ir fricion is negleced nd if he liude of he flling objec is smll compred wih Erh s rdius, hen we cn ssume h he free-fll ccelerion g m/s is consn over he rnge of moion. Equions.6,.9, nd.10 pply, wih 5 g. MULTIPLE-CHOICE QUESTIONS The muliple-choice quesions in his chper my be ssigned online in Enhnced WebAssign. 1. An rrow is sho srigh up in he ir n iniil speed of 15.0 m/s. Afer how much ime is he rrow heding downwrd speed of 8.00 m/s? () s (b) 1.4 s (c) 1.87 s (d).35 s (e) 3. s. One drop of oil flls srigh down ono he rod from he engine of moving cr every 5 s. Figure MCQ. shows he pern of he drops lef behind on he pvemen. Wh is he verge speed of he cr over his secion of is moion? () 0 m/s (b) 4 m/s (c) 30 m/s (d) 100 m/s (e) 10 m/s. 600 m Figure MCQ. 3. When pplying he equions of kinemics for n objec moving in one dimension, which of he following semens mus be rue? () The velociy of he objec mus remin consn. (b) The ccelerion of he objec mus remin consn. (c) The velociy of he objec mus increse wih ime. (d) The posiion of he objec mus increse wih ime. (e) The velociy of he objec mus lwys be in he sme direcion s is ccelerion. 4. A juggler hrows bowling pin srigh up in he ir. Afer he pin leves his hnd nd while i is in he ir, which semen is rue? () The velociy of he pin is lwys in he sme direcion s is ccelerion. (b) The velociy of he pin is never in he sme direcion s is ccelerion. (c) The ccelerion of he pin is zero. (d) The velociy of he pin is opposie is ccelerion on he wy up. (e) The velociy of he pin is in he sme direcion s is ccelerion on he wy up. 5. A rcing cr srs from res nd reches finl speed v in ime. If he ccelerion of he cr is consn during his ime, which of he following semens mus be rue? () The cr rvels disnce v. (b) The verge speed of he cr is v/. (c) The ccelerion of he cr is v/. (d) The velociy of he cr remins consn. (e) None of hese 6. When he pilo reverses he propeller in bo moving norh, he bo moves wih n ccelerion direced souh. Assume he ccelerion of he bo remins consn in mgniude nd direcion. Wh hppens o he bo? () I evenully sops nd remins sopped. (b) I evenully sops nd hen speeds up in he norhwrd direcion. (c) I evenully sops nd hen speeds up in he souhwrd direcion. (d) I never sops bu loses speed more nd more slowly forever. (e) I never sops bu coninues o speed up in he norhwrd direcion. 7. An objec moves long he x-xis, is posiion mesured ech insn of ime. The d re orgnized ino n ccure grph of x vs.. Which of he following quniies cnno be obined from his grph? () he velociy ny insn (b) he ccelerion ny insn (c) he displcemen during some ime inervl (d) he verge velociy during some ime inervl (e) he speed of he pricle ny insn 8. A skeborder srs from res nd moves down hill wih consn ccelerion in srigh line, rveling for 6 s. In second ril, he srs from res nd moves long he sme srigh line wih he sme ccelerion for only s. How does his displcemen from his sring poin in his second ril compre wih he firs ril? () one-hird s lrge (b) hree imes lrger (c) one-ninh s lrge (d) nine imes lrger (e) 1/!3 imes s lrge 9. Rces re imed o n ccurcy of 1/1 000 of second. Wh disnce could person in-line sking speed of 8.5 m/s rvel in h period of ime? () 85 mm (b) 85 cm (c) 8.5 m (d) 8.5 mm (e) 8.5 km

25 Concepul Quesions A suden he op of building hrows red bll upwrd wih speed v 0 nd hen hrows blue bll downwrd wih he sme iniil speed v 0. Immediely before he wo blls rech he ground, which of he following semens re rue? (Choose ll correc semens; neglec ir fricion.) () The speed of he red bll is less hn h of he blue bll. (b) The speed of he red bll is greer hn h of he blue bll. (c) Their velociies re equl. (d) The speed of ech bll is greer hn v 0. (e) The ccelerion of he blue bll is greer hn h of he red bll. 11. A bll is hrown downwrd from he op of 40.0 m ower wih n iniil speed of 1 m/s. Assuming negligible ir resisnce, wh is he speed of he bll jus before hiing he ground? () 8 m/s (b) 30 m/s (c) 56 m/s (d) 784 m/s (e) More informion is needed. 1. A bll is hrown srigh up in he ir. For which siuion re boh he insnneous velociy nd he ccelerion zero? () on he wy up (b) he op of he fligh ph (c) on he wy down (d) hlfwy up nd hlfwy down (e) none of hese CONCEPTUAL QUESTIONS The concepul quesions in his chper my be ssigned online in Enhnced WebAssign. Chrles D. Winers/Cengge Lerning b c Figure CQ.6 1. If he velociy of pricle is nonzero, cn he pricle s ccelerion be zero? Explin.. If he velociy of pricle is zero, cn he pricle s ccelerion be nonzero? Explin. 3. If cr is rveling eswrd, cn is ccelerion be weswrd? Explin. 4. () Cn he equions in Tble.4 be used in siuion where he ccelerion vries wih ime? (b) Cn hey be used when he ccelerion is zero? 5. Two crs re moving in he sme direcion in prllel lnes long highwy. A some insn, he velociy of cr A exceeds he velociy of cr B. Does h men h he ccelerion of A is greer hn h of B h insn? Explin. 6. Figure CQ.6 shows srobe phoogrphs ken of disk moving from lef o righ under differen condiions. The ime inervl beween imges is consn. Tking he direcion o he righ o be posiive, describe he moion of he disk in ech cse. For which cse is () he ccelerion posiive? (b) he ccelerion negive? (c) he velociy consn? 7. () Cn he insnneous velociy of n objec n insn of ime ever be greer in mgniude hn he verge velociy over ime inervl conining h insn? (b) Cn i ever be less? 8. A bll is hrown vericlly upwrd. () Wh re is velociy nd ccelerion when i reches is mximum liude? (b) Wh is he ccelerion of he bll jus before i his he ground? 9. Consider he following combinions of signs nd vlues for he velociy nd ccelerion of pricle wih respec o one-dimensionl x-xis: Velociy Accelerion. Posiive Posiive b. Posiive Negive c. Posiive Zero d. Negive Posiive e. Negive Negive f. Negive Zero g. Zero Posiive h. Zero Negive Describe wh he pricle is doing in ech cse nd give rel-life exmple for n uomobile on n es wes one-dimensionl xis, wih es considered he posiive direcion. 10. A bll rolls in srigh line long he horizonl direcion. Using moion digrms (or muliflsh phoogrphs), describe he velociy nd ccelerion of he bll for ech of he following siuions: () The bll moves o he righ consn speed. (b) The bll moves from righ o lef nd coninully slows down. (c) The bll moves from righ o lef nd coninully speeds up. (d) The bll moves o he righ, firs speeding up consn re nd hen slowing down consn re.

26 50 CHAPTER Moion in One Dimension PROBLEMS The problems in his chper my be ssigned online in Enhnced WebAssign. Seleced problems lso hve Wch I video soluions. 1. denoes srighforwrd problem;. denoes inermedie problem; 3. denoes chllenging problem 1. denoes full soluion vilble in Suden Soluions Mnul/ Sudy Guide 1. denoes problems mos ofen ssigned in Enhnced WebAssign denoes biomedicl problems denoes guided problems denoes Mser I uoril vilble in Enhnced WebAssign denoes sking for quniive nd concepul resoning denoes symbolic resoning problem.1 Displcemen. Velociy 1. The speed of nerve impulse in he humn body is bou 100 m/s. If you ccidenlly sub your oe in he drk, esime he ime i kes he nerve impulse o rvel o your brin.. Ligh rvels speed of bou m/s. () How mny miles does pulse of ligh rvel in ime inervl of 0.1 s, which is bou he blink of n eye? (b) Compre his disnce o he dimeer of Erh. 3. A person rvels by cr from one ciy o noher wih differen consn speeds beween pirs of ciies. She drives for 30.0 min 80.0 km/h, 1.0 min 100 km/h, nd 45.0 min 40.0 km/h nd spends 15.0 min eing lunch nd buying gs. () Deermine he verge speed for he rip. (b) Deermine he disnce beween he iniil nd finl ciies long he roue. 4. The curren indoor world record ime in he 00-m rce is 19.9 s, held by Frnk Fredericks of Nmibi (1996), while he indoor record ime in he one-mile rce is 8.5 s, held by Hichm El Guerrouj of Morroco (1997). Find he men speed in meers per second corresponding o hese record imes for () he 00-m even nd (b) he one-mile even. 5. Two bos sr ogeher nd rce cross 60-km-wide lke nd bck. Bo A goes cross 60 km/h nd reurns 60 km/h. Bo B goes cross 30 km/h, nd is crew, relizing how fr behind i is geing, reurns 90 km/h. Turnround imes re negligible, nd he bo h complees he round rip firs wins. () Which bo wins nd by how much? (Or is i ie?) (b) Wh is he verge velociy of he winning bo? 6. A grph of posiion versus ime for cerin pricle moving long he x-xis is shown in Figure P.6. Find he verge velociy in he ime inervls from () 0 o.00 s, (b) 0 o 4.00 s, (c).00 s o 4.00 s, (d) 4.00 s o 7.00 s, nd (e) 0 o 8.00 s. x (m) (s) Figure P.6 Problems 6 nd A mooris drives norh for 35.0 minues 85.0 km/h nd hen sops for 15.0 minues. He hen coninues norh, rveling 130 km in.00 h. () Wh is his ol displcemen? (b) Wh is his verge velociy? 8. A ennis plyer moves in x (m) srigh-line ph s shown 4 in Figure P.8. Find her verge velociy in he (s) ime inervls from () 0 o s, (b) 0 o 4.0 s, (c) 1.0 s o 5.0 s, nd (d) 0 o 5.0 s. Figure P.8 9. A je plne hs keoff speed of v o 5 75 m/s nd cn move long he runwy n verge ccelerion of 1.3 m/s. If he lengh of he runwy is.5 km, will he plne be ble o use his runwy sfely? Defend your nswer. 10. Two crs rvel in he sme direcion long srigh highwy, one consn speed of 55 mi/h nd he oher 70 mi/h. () Assuming hey sr he sme poin, how much sooner does he fser cr rrive desinion 10 mi wy? (b) How fr mus he fser cr rvel before i hs 15-min led on he slower cr? 11. The cheeh cn rech op speed of 114 km/h (71 mi/h). While chsing is prey in shor sprin, cheeh srs from res nd runs 45 m in srigh line, reching finl speed of 7 km/h. () Deermine he cheeh s verge ccelerion during he shor sprin, nd (b) find is displcemen s. 1. An hlee swims he lengh L of pool in ime 1 nd mkes he reurn rip o he sring posiion in ime. If she is swimming iniilly in he posiive x-direcion, deermine her verge velociies symboliclly in () he firs hlf of he swim, (b) he second hlf of he swim, nd (c) he round rip. (d) Wh is her verge speed for he round rip? 13. A person kes rip, driving wih consn speed of 89.5 km/h, excep for.0-min res sop. If he person s verge speed is 77.8 km/h, () how much ime is spen on he rip nd (b) how fr does he person rvel? 14. A oroise cn run wih speed of 0.10 m/s, nd hre cn run 0 imes s fs. In rce, hey boh sr he sme ime, bu he hre sops o res for.0 minues. The oroise wins by shell (0 cm). () How long does he rce ke? (b) Wh is he lengh of he rce?

27 Problems To qulify for he finls in rcing even, rce cr mus chieve n verge speed of 50 km/h on rck wih ol lengh of m. If priculr cr covers he firs hlf of he rck n verge speed of 30 km/h, wh minimum verge speed mus i hve in he second hlf of he even in order o qulify? 16. One hlee in rce running on long, srigh rck wih consn speed v 1 is disnce d behind second hlee running wih consn speed v. () Under wh circumsnces is he firs hlee ble o overke he second hlee? (b) Find he ime i kes he firs hlee o overke he second hlee, in erms of d, v 1, nd v. (c) A wh minimum disnce d from he leding hlee mus he finish line be loced so h he riling hlee cn les ie for firs plce? Express d in erms of d, v 1, nd v by using he resul of pr (b). 17. A grph of posiion versus ime for cerin pricle moving long he x-xis is shown in Figure P.6. Find he insnneous velociy he insns () s, (b) s, (c) s, nd (d) s. 18. A rce cr moves such h is posiion fis he relionship x 5 (5.0 m/s) 1 (0.75 m/s 3 ) 3 where x is mesured in meers nd in seconds. () Plo grph of he cr s posiion versus ime. (b) Deermine he insnneous velociy of he cr s, using ime inervls of 0.40 s, 0.0 s, nd 0.10 s. (c) Compre he verge velociy during he firs 4.0 s wih he resuls of pr (b). 19. Runner A is iniilly 4.0 mi wes of flgpole nd is running wih consn velociy of 6.0 mi/h due es. Runner B is iniilly 3.0 mi es of he flgpole nd is running wih consn velociy of 5.0 mi/h due wes. How fr re he runners from he flgpole when hey mee? (m/s.3 Accelerion ) 0. A pricle srs from res nd cceleres s shown in Figure P.0. Deermine () he pricle s speed s nd s, nd (b) he disnce rveled in he firs 0.0 s. 1. A 50.0-g Super Bll rveling 5.0 m/s bounces off brick wll nd rebounds.0 m/s. A high-speed cmer records his even. If he bll is in conc wih he wll for 3.50 ms, wh is he mgniude of he verge ccelerion of he bll during his ime inervl?. The verge person psses ou n ccelerion of 7g (h is, seven imes he grviionl ccelerion on Erh). Suppose cr is designed o ccelere his re. How much ime would be required for he (s) Figure P.0 cr o ccelere from res o 60.0 miles per hour? (The cr would need rocke boosers!) 3. A cerin cr is cpble of ccelering re of 0.60 m/s. How long does i ke for his cr o go from speed of 55 mi/h o speed of 60 mi/h? 4. The velociy vs. ime grph for n objec moving long srigh ph is shown in Figure P.4. (i) Find he verge ccelerion of he objec during he ime inervls () 0 o 5.0 s, (b) 5.0 s o 15 s, nd (c) 0 o 0 s. (ii) Find he insnneous ccelerion ().0 s, (b) 10 s, nd (c) 18 s. v (m/s) Figure P (s) 5. A sem cpul lunches je ircrf from he ircrf crrier John C. Sennis, giving i speed of 175 mi/h in.50 s. () Find he verge ccelerion of he plne. (b) Assuming he ccelerion is consn, find he disnce he plne moves..5 One-Dimensionl Moion wih Consn Accelerion 6. Solve Exmple.5, Cr Chse by grphicl mehod. On he sme grph, plo posiion versus ime for he cr nd he rooper. From he inersecion of he wo curves, red he ime which he rooper overkes he cr. 7. An objec moving wih uniform ccelerion hs velociy of 1.0 cm/s in he posiive x-direcion when is x-coordine is 3.00 cm. If is x coordine.00 s ler is 5.00 cm, wh is is ccelerion? 8. In 1865 Jules Verne proposed sending men o he Moon by firing spce cpsule from 0-m-long cnnon wih finl speed of km/s. Wh would hve been he unrelisiclly lrge ccelerion experienced by he spce rvelers during heir lunch? (A humn cn snd n ccelerion of 15g for shor ime.) Compre your nswer wih he free-fll ccelerion, 9.80 m/s. 9. A ruck covers 40.0 m in 8.50 s while uniformly slowing down o finl velociy of.80 m/s. () Find he ruck s originl speed. (b) Find is ccelerion. 30. A speedbo increses is speed uniformly from v i m/s o v f m/s in disnce of m. () Drw coordine sysem for his siuion

28 5 CHAPTER Moion in One Dimension nd lbel he relevn quniies, including vecors. (b) For he given informion, wh single equion is mos pproprie for finding he ccelerion? (c) Solve he equion seleced in pr (b) symboliclly for he bo s ccelerion in erms of v f, v i, nd Dx. (d) Subsiue given vlues, obining h ccelerion. (e) Find he ime i kes he bo o rvel he given disnce. 31. A Cessn ircrf hs lifoff speed of 10 km/h. () Wh minimum consn ccelerion does he ircrf require if i is o be irborne fer keoff run of 40 m? (b) How long does i ke he ircrf o become irborne? 3. An objec moves wih consn ccelerion 4.00 m/s nd over ime inervl reches finl velociy of 1.0 m/s. () If is originl velociy is 6.00 m/s, wh is is displcemen during he ime inervl? (b) Wh is he disnce i rvels during his inervl? (c) If is originl velociy is 6.00 m/s, wh is is displcemen during his inervl? (d) Wh is he ol disnce i rvels during he inervl in pr (c)? 33. In es run, cerin cr cceleres uniformly from zero o 4.0 m/s in.95 s. () Wh is he mgniude of he cr s ccelerion? (b) How long does i ke he cr o chnge is speed from 10.0 m/s o 0.0 m/s? (c) Will doubling he ime lwys double he chnge in speed? Why? 34. A je plne lnds wih speed of 100 m/s nd cn ccelere mximum re of 5.00 m/s s i comes o res. () From he insn he plne ouches he runwy, wh is he minimum ime needed before i cn come o res? (b) Cn his plne lnd on smll ropicl islnd irpor where he runwy is km long? 35. Speedy Sue, driving 30.0 m/s, eners one-lne unnel. She hen observes slow-moving vn 155 m hed rveling 5.00 m/s. Sue pplies her brkes bu cn ccelere only.00 m/s becuse he rod is we. Will here be collision? Se how you decide. If yes, deermine how fr ino he unnel nd wh ime he collision occurs. If no, deermine he disnce of closes pproch beween Sue s cr nd he vn. 36. A record of rvel long srigh ph is s follows: 1. Sr from res wih consn ccelerion of.77 m/s for 15.0 s.. Minin consn velociy for he nex.05 min. 3. Apply consn negive ccelerion of 9.47 m/s for 4.39 s. () Wh ws he ol displcemen for he rip? (b) Wh were he verge speeds for legs 1,, nd 3 of he rip, s well s for he complee rip? 37. A rin is rveling down srigh rck 0 m/s when he engineer pplies he brkes, resuling in n ccelerion of 1.0 m/s s long s he rin is in moion. How fr does he rin move during 40-s ime inervl sring he insn he brkes re pplied? 38. A cr cceleres uniformly from res o speed of 40.0 mi/h in 1.0 s. Find () he disnce he cr rvels during his ime nd (b) he consn ccelerion of he cr. 39. A cr srs from res nd rvels for 5.0 s wih uniform ccelerion of 11.5 m/s. The driver hen pplies he brkes, cusing uniform ccelerion of.0 m/s. If he brkes re pplied for 3.0 s, () how fs is he cr going he end of he brking period, nd (b) how fr hs he cr gone? 40. A cr srs from res nd rvels for 1 seconds wih uniform ccelerion 1. The driver hen pplies he brkes, cusing uniform ccelerion. If he brkes re pplied for seconds, () how fs is he cr going jus before he beginning of he brking period? (b) How fr does he cr go before he driver begins o brke? (c) Using he nswers o prs () nd (b) s he iniil velociy nd posiion for he moion of he cr during brking, wh ol disnce does he cr rvel? Answers re in erms of he vribles 1,, 1, nd. 41. In he Dyon 500 uo rce, Ford Thunderbird nd Mercedes Benz re moving side by side down srighwy 71.5 m/s. The driver of he Thunderbird relizes h she mus mke pi sop, nd she smoohly slows o sop over disnce of 50 m. She spends 5.00 s in he pi nd hen cceleres ou, reching her previous speed of 71.5 m/s fer disnce of 350 m. A his poin, how fr hs he Thunderbird fllen behind he Mercedes Benz, which hs coninued consn speed? 4. A cerin cble cr in Sn Frncisco cn sop in 10 s when rveling mximum speed. On one occsion, he driver sees dog disnce d in fron of he cr nd slms on he brkes insnly. The cr reches he dog 8.0 s ler, nd he dog jumps off he rck jus in ime. If he cr rvels 4.0 m beyond he posiion of he dog before coming o sop, how fr ws he cr from he dog? (Hin: You will need hree equions.) 43. A hockey plyer is snding on his skes on frozen pond when n opposing plyer, moving wih uniform speed of 1 m/s, skes by wih he puck. Afer 3.0 s, he firs plyer mkes up his mind o chse his opponen. If he cceleres uniformly 4.0 m/s, () how long does i ke him o cch his opponen, nd (b) how fr hs he rveled in h ime? (Assume he plyer wih he puck remins in moion consn speed.) 44. A rin 400 m long is moving on srigh rck wih speed of 8.4 km/h. The engineer pplies he brkes crossing, nd ler he ls cr psses he crossing wih speed of 16.4 km/h. Assuming consn ccelerion, deermine how long he rin blocked he crossing. Disregrd he widh of he crossing..6 Freely Flling Objecs 45. A bll is hrown vericlly upwrd wih speed of 5.0 m/s. () How high does i rise? (b) How long does

29 Problems 53 i ke o rech is highes poin? (c) How long does he bll ke o hi he ground fer i reches is highes poin? (d) Wh is is velociy when i reurns o he level from which i sred? 46. A bll is hrown direcly downwrd wih n iniil speed of 8.00 m/s, from heigh of 30.0 m. Afer wh ime inervl does i srike he ground? 47. A cerin freely flling objec, relesed from res, requires 1.50 s o rvel he ls 30.0 m before i his he ground. () Find he velociy of he objec when i is 30.0 m bove he ground. (b) Find he ol disnce he objec rvels during he fll. Couresy U.S. Air Force coper in erms of g nd? (c) Wh re he nswers o prs () nd (b) if he helicoper is rising sedily he sme speed? 53. A model rocke is lunched srigh upwrd wih n iniil speed of 50.0 m/s. I cceleres wih consn upwrd ccelerion of.00 m/s unil is engines sop n liude of 150 m. () Wh cn you sy bou he moion of he rocke fer is engines sop? (b) Wh is he mximum heigh reched by he rocke? (c) How long fer lifoff does he rocke rech is mximum heigh? (d) How long is he rocke in he ir? 54. A bsebll is hi so h i rvels srigh upwrd fer being sruck by he b. A fn observes h i kes 3.00 s for he bll o rech is mximum heigh. Find () he bll s iniil velociy nd (b) he heigh i reches. Addiionl Problems 55. A ruck rcor pulls wo rilers, one behind he oher, consn speed of 100 km/h. I kes s for he big rig o compleely pss ono bridge 400 m long. For wh durion of ime is ll or pr of he ruck riler combinion on he bridge? 56. Colonel John P. Spp, USAF, priciped in sudying wheher je pilo could survive emergency ejecion. On Mrch 19, 1954, he rode rockepropelled sled h moved down rck speed of 63 mi/h (see Fig. P.56). He nd he sled were sfely brough o res in 1.40 s. Deermine in SI unis () he negive ccelerion he experienced nd (b) he disnce he rveled during his negive ccelerion. 48. An cker he bse of csle wll 3.65 m high hrows rock srigh up wih speed 7.40 m/s heigh of 1.55 m bove he ground. () Will he rock rech he op of he wll? (b) If so, wh is he rock s speed he op? If no, wh iniil speed mus he rock hve o rech he op? (c) Find he chnge in he speed of rock hrown srigh down from he op of he wll n iniil speed of 7.40 m/s nd moving beween he sme wo poins. (d) Does he chnge in speed of he downwrd-moving rock gree wih he mgniude of he speed chnge of he rock moving upwrd beween he sme elevions? Explin physiclly why or why no. 49. Trumic brin injury such s concussion resuls when he hed undergoes very lrge ccelerion. Generlly, n ccelerion less hn 800 m/s lsing for ny lengh of ime will no cuse injury, wheres n ccelerion greer hn m/s lsing for les 1 ms will cuse injury. Suppose smll child rolls off bed h is 0.40 m bove he floor. If he floor is hrdwood, he child s hed is brough o res in pproximely.0 mm. If he floor is crpeed, his sopping disnce is incresed o bou 1.0 cm. Clcule he mgniude nd durion of he decelerion in boh cses, o deermine he risk of injury. Assume he child remins horizonl during he fll o he floor. Noe h more compliced fll could resul in hed velociy greer or less hn he speed you clcule. 50. A smll milbg is relesed from helicoper h is descending sedily 1.50 m/s. Afer.00 s, () wh is he speed of he milbg, nd (b) how fr is i below he helicoper? (c) Wh re your nswers o prs () nd (b) if he helicoper is rising sedily 1.50 m/s? 51. A ennis plyer osses ennis bll srigh up nd hen cches i fer.00 s he sme heigh s he poin of relese. () Wh is he ccelerion of he bll while i is in fligh? (b) Wh is he velociy of he bll when i reches is mximum heigh? Find (c) he iniil velociy of he bll nd (d) he mximum heigh i reches. 5. A pckge is dropped from helicoper h is descending sedily speed v 0. Afer seconds hve elpsed, () wh is he speed of he pckge in erms of v 0, g, nd? (b) Wh disnce d is i from he heli Figure P A bulle is fired hrough bord 10.0 cm hick in such wy h he bulle s line of moion is perpendiculr o he fce of he bord. If he iniil speed of he bulle is 400 m/s nd i emerges from he oher side of he bord wih speed of 300 m/s, find () he ccelerion of he bulle s i psses hrough he bord nd (b) he ol ime he bulle is in conc wih he bord. 58. A speedbo moving 30.0 m/s pproches no-wke buoy mrker 100 m hed. The pilo slows he bo wih consn ccelerion of 3.50 m/s by reducing he hrole. () How long does i ke he bo o rech Phori, Inc. b

30 54 CHAPTER Moion in One Dimension he buoy? (b) Wh is he velociy of he bo when i reches he buoy? 59. A suden hrows se of keys vericlly upwrd o his frerniy broher, who is in window 4.00 m bove. The broher s ousreched hnd cches he keys 1.50 s ler. () Wih wh iniil velociy were he keys hrown? (b)? Wh ws he velociy of he keys jus before hey were cugh? 60. A suden hrows se of keys vericlly upwrd o his frerniy broher, who is in window disnce h bove. The broher s ousreched hnd cches he keys on heir wy up ime ler. () Wih wh iniil velociy were he keys hrown? (b) Wh ws he velociy of he keys jus before hey were cugh? (Answers should be in erms of h, g, nd.) 61. I hs been climed h n insec clled he froghopper (Philenus spumrius) is he bes jumper in he niml kingdom. This insec cn ccelere m/s over disnce of.0 mm s i srighens is specilly designed jumping legs. () Assuming uniform ccelerion, wh is he velociy of he insec fer i hs ccelered hrough his shor disnce, nd (b) how long did i ke o rech h velociy? (c) How high would he insec jump if ir resisnce could be ignored? Noe h he cul heigh obined is bou 0.7 m, so ir resisnce is imporn here. 6. Drw moion digrms (see Secion.5) for () n objec moving o he righ consn speed, (b) n objec moving o he righ nd speeding up consn re, (c) n objec moving o he righ nd slowing down consn re, (d) n objec moving o he lef nd speeding up consn re, nd (e) n objec moving o he lef nd slowing down consn re. (f) How would your drwings chnge if he chnges in speed were no uniform; h is, if he speed were no chnging consn re? 63. A bll is hrown upwrd from he ground wih n iniil speed of 5 m/s; he sme insn, noher bll is dropped from building 15 m high. Afer how long will he blls be he sme heigh? 64. To pss physicl educion clss universiy, suden mus run 1.0 mi in 1 min. Afer running for 10 min, she sill hs 500 yd o go. If her mximum ccelerion is 0.15 m/s, cn she mke i? If he nswer is no, deermine wh ccelerion she would need o be successful. 65. In Chper 5 we will define he cener of mss of n objec. The cener of mss moves wih consn ccelerion when consn forces c on he objec. A gymns jumps srigh up, wih her cener of mss moving.80 m/s s she leves he ground. How high bove his poin is her cener of mss () s, (b) 0.00 s, (c) s, nd (d) s herefer? 66. Two sudens re on blcony disnce h bove he sree. One suden hrows bll vericlly down- wrd speed v 0 ; he sme ime, he oher suden hrows bll vericlly upwrd he sme speed. Answer he following symboliclly in erms of v 0, g, h, nd. () Wrie he kinemic equion for he y- coordine of ech bll. (b) Se he equions found in pr () equl o heigh 0 nd solve ech for symboliclly using he qudric formul. Wh is he difference in he wo blls ime in he ir? (c) Use he ime-independen kinemics equion o find he velociy of ech bll s i srikes he ground. (d) How fr pr re he blls ime fer hey re relesed nd before hey srike he ground? 67. You drop bll from window on n upper floor of building nd i is cugh by friend on he ground when he bll is moving wih speed v f. You now repe he drop, bu you hve friend on he sree below hrow noher bll upwrd speed v f excly he sme ime h you drop your bll from he window. The wo blls re iniilly sepred by 8.7 m. () A wh ime do hey pss ech oher? (b) A wh locion do hey pss ech oher relive he window? 68. The driver of ruck slms on he brkes when he sees ree blocking he rod. The ruck slows down uniformly wih n ccelerion of 5.60 m/s for 4.0 s, mking skid mrks 6.4 m long h end he ree. Wih wh speed does he ruck hen srike he ree? 69. Emily chllenges her husbnd, Dvid, o cch $1 bill s follows. She holds he bill vericlly s in Figure P.69, wih he cener of he bill beween Dvid s index finger nd humb. Dvid mus Figure P.69 cch he bill fer Emily releses i wihou moving his hnd downwrd. If his recion ime is 0. s, will he succeed? Explin your resoning. (This chllenge is good rick you migh wn o ry wih your friends.) 70. A mounin climber snds he op of 50.0-m cliff h overhngs clm pool of wer. She hrows wo sones vericlly downwrd 1.00 s pr nd observes h hey cuse single splsh. The firs sone hd n iniil velociy of.00 m/s. () How long fer relese of he firs sone did he wo sones hi he wer? (b) Wh iniil velociy mus he second sone hve hd, given h hey hi he wer simulneously? (c) Wh ws he velociy of ech sone he insn i hi he wer? 71. An ice sled powered by rocke engine srs from res on lrge frozen lke nd cceleres 140 f/s. Afer some ime 1, he rocke engine is shu down nd he sled moves wih consn velociy v for ime. If Cengge Lerning/George Semple