CHAPTER 2: Describing Motion: Kinematics in One Dimension

Transcription

1 CHAPTER : Describing Moion: Kinemics in One Dimension Answers o Quesions A cr speeomeer mesures only spee I oes no gie ny informion bou he irecion, n so oes no mesure elociy By efiniion, if n objec hs consn elociy, hen boh he objec's spee n is irecion of moion re consn Therefore he objec CANNOT he rying spee if is elociy is consn 3 When n objec moes wih consn elociy, is erge elociy oer ny ime inerl is excly equl o is insnneous elociy ll imes 4 For boh crs, he ime elpse is he isnce rele iie by he erge elociy Since boh crs rel he sme isnce, he cr wih he lrger erge elociy will he he smller elpse ime Consier his scenrio Assume h one cr hs consn ccelerion own he rck Then grph of is spee ersus ime woul look like line "A" on he firs grph The she re of he grph represens he isnce rele, n he grph is ploe o such ime h he she re represens he lengh of he rck The ime for his cr o finish he rce is lbele " " ime " A " Now le he secon cr he much smller ccelerion iniilly, bu wih n incresing ccelerion A grph of is elociy, superimpose on he boe grph n lbele "B", migh look like he secon igrm " A " ime " B " I is seen h he ime when he firs cr finishe he rce, he secon cr is going fser hn he firs cr, becuse he hey line is higher on he grph hn he line represening he firs cr Howeer, he re uner he "B" line (he isnce h he secon cr hs rele) is smller hn he she re, n so is less hn he full rck lengh For he re uner he "B" line o be he sme s he re uner he "A" line, he grph woul nee o look like he hir igrm, inicing longer ime for he secon cr o finish he rce " A " ime " B " 5 There is no generl relionship beween he mgniue of spee n he mgniue of ccelerion For exmple, one objec my he lrge bu consn spee The ccelerion of h objec is hen Anoher objec my he smll spee bu be gining spee, n herefore he posiie ccelerion So in his cse he objec wih he greer spee hs he lesser ccelerion Or consier wo objecs h re roppe from res ifferen imes If we ignore ir resisnce, hen he objec roppe firs will lwys he greer spee hn he objec roppe secon, bu boh will he he sme ccelerion of 98 m/s 6 The ccelerion of boh he moorcycle n he bicycle re he sme, since he sme chnge in elociy occurre uring he sme ime inerl If you o furher clculion, you will fin h he isnce rele by he moorcycle uring he ccelerion is 7 imes he isnce rele by he bicycle 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he

2 Gincoli Physics: Principles wih Applicions, 6 h Eiion 7 If n objec is reling o he norh bu slowing own, i hs norhwr elociy n souhwr ccelerion 8 The elociy of n objec cn be negie when is ccelerion is posiie If we efine he posiie irecion o be o he righ, hen n objec reling o he lef h is hing reucion in spee will he negie elociy wih posiie ccelerion If gin we efine he posiie irecion o be o he righ, hen n objec reling o he righ h is hing reucion in spee will he posiie elociy n negie ccelerion 9 If norh is efine s he posiie irecion, hen n objec reling o he souh n incresing in spee hs boh negie elociy n negie ccelerion Or, if up is efine s he posiie irecion, hen n objec flling ue o griy hs boh negie elociy n negie ccelerion If he wo crs emerge sie by sie, hen he one moing fser is pssing he oher one Thus cr A is pssing cr B Wih he ccelerion gien for he problem, he ensuing moion woul be h cr A woul pull wy from cr B for ime, bu eenully cr B woul cch up o n pss cr A Assume h norh is he posiie irecion If cr is moing souh n gining spee n incresing re, hen he ccelerion will be geing lrger in mgniue Howeer, since he ccelerion is irece souhwrs, he ccelerion is negie, n is geing more negie Th is ecresing ccelerion s he spee increses Anoher exmple woul be n objec flling WITH ir resisnce As he objec flls, i gins spee, he ir resisnce increses As he ir resisnce increses, he ccelerion of he flling objec ecreses, n i gins spee less quickly he longer i flls Assuming h he ccher cches he bll he sme heigh which i lef he b, hen he bll will be reling wih spee of km/h when cugh This is proen in problem 4 3 As freely flling objec spees up, is ccelerion ue o griy sys he sme If ir resisnce is consiere, hen he ccelerion of he objec is ue o boh griy n ir resisnce The ol ccelerion ges smller s he objec spees up, unil he objec reches erminl elociy, which ime is ol ccelerion is zero Therefer is spee remins consn 4 To esime he heigh, hrow he bll upwr n ime he fligh from hrowing o cching Then, ignoring ir resisnce, he ime of rising woul be hlf of he ime of fligh Wih h "hlf" ime, ssuming h he origin is he op of he ph n h ownwr is posiie, knowing h he bll sre from he op of he ph wih spee of, use he equion y g wih h ime n he ccelerion ue o griy o fin he isnce h he bll fell Wih he sme "hlf" ime, we know h he op of he ph, he spee is Tking he upwr irecion s posiie, use he equion g g o fin he hrowing spee 5 The erge spee is NOT 8 km/h Since he wo isnces rele were he sme, he imes of rel were unequl The ime o rel from A o B 7 km/h is longer hn he ime o rel from B o C 9 km/h Thus we cnno simply erge he spee numbers To fin he erge spee, we nee o clcule (ol isnce) / (ol ime) We ssume he isnce from A o B n he isnce from B o C re boh km The ime o rel isnce wih spee is = / 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 3

3 Chper Describing Moion: Kinemics in One Dimension AB AB BC BC km km km km 7 km h 9 km h 7875 km h The erge spee is 7875 km/h 6 The souns will no occur equl ime inerls becuse he longer ny priculr nu flls, he fser i will be going Wih equl isnces beween nus, ech successie nu, hing fllen longer ime when is preecessor reches he ple, will he higher erge elociy n hus rel he iner-nu isnce in shorer perios of ime Thus he souns will occur wih smller n smller inerls beween souns To her he souns equl inerls, he nus woul he o be ie isnces corresponing o equl ime inerls Since for ech nu he isnce of fll n ime of fll re rele by g, ssume h g If we wn, 3 3, 4 4,, hen 3 g 3 9, 4 g 4 6, ec g 4, 7 The eleor moing from he secon floor o he fifh floor is NOT n exmple of consn ccelerion The eleor cceleres upwr ech ime i srs o moe, n i cceleres ownwr ech ime i sops Ignoring ir resisnce, rock flling from cliff woul he consn ccelerion (If ir resisnce is inclue, hen he ccelerion will be ecresing s he rock flls) A ish resing on ble hs n ccelerion of, so he ccelerion is consn 8 As n objec rises WITH ir resisnce, he ccelerion is lrger in mgniue hn g, becuse boh griy n ir resisnce will be cusing ownwr ccelerion As he objec FALLS wih ir resisnce, he ccelerion will be smller in mgniue hn g, becuse griy n resisnce will be opposing ech oher Becuse of he smller ccelerion being pplie oer he sme isnce, he reurn spee will be slower hn he lunch spee 9 If n objec is he insn of reersing irecion (like n objec hrown upwr, he op of is ph), i insnneously hs zero elociy n non-zero ccelerion he sme ime A person he exc boom of bungee cor plunge lso hs n insnneous elociy of zero bu nonzero (upwr) ccelerion he sme ime An objec moing wih consn elociy hs non-zero elociy n zero ccelerion he sme ime So cr riing consn spee on srigh, leel rowy woul mee his coniion The objec srs wih consn elociy in he posiie irecion A bou = 7 s, when he objec is he 5 meer posiion, i begins o gin spee i hs posiie ccelerion A bou = 7 s, when he objec is bou he m posiion, i begins o slow own i hs negie ccelerion The objec insnneously sops bou = 37 s, reching is mximum isnce from he origin of m The objec hen reerses irecion, gining spee while moing bckwrs A bou = 47 s, when he objec is gin bou he m posiion, he objec srs o slow own, n ppers o sop = 5 s, m from he sring poin Iniilly, he objec moes in he posiie irecion wih consn ccelerion, unil bou = 45 s, when i hs elociy of bou 37 m/s in he posiie irecion The ccelerion hen ecreses, reching n insnneous ccelerion of bou = 5 s, when he objec hs is mximum spee of bou 38 m/s The objec hen begins o slow own, bu coninues o moe in he posiie 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 4 i i

4 Gincoli Physics: Principles wih Applicions, 6 h Eiion irecion The objec sops moing = 9 s n sys res unil bou = 8 s Then he objec begins o moe o he righ gin, firs wih lrge ccelerion, n hen lesser ccelerion A he en of he recore moion, he objec is sill moing o he righ n gining spee Soluions o Problems The erge spee is gien by: 35 km 35 h 73 km h The ime of rel cn be foun by rerrnging he erge spee equion 5 km 5 km h 6 h 36 min 3 The isnce of rel (isplcemen) cn be foun by rerrnging he erge spee equion Also noe h he unis of he elociy n he ime re no he sme, so he spee unis will be conere h km h s 6 km 6 m 36 s 4 () 35mi h 35mi h 6km mi 56 km h (b) 35 mi h 35 mi h 6 m mi h 36 s 6 m s (c) 35mi h 35mi h 58f mi h 36 s 5f s 5 The erge elociy is gien by x 4 cm 34 cm 76 cm 6s 3 s 3s 5cm s 6 The erge elociy is gien by x 85 cm 34 cm 5cm 78 cm s 45 s s 65s The erge spee cnno be clcule To clcule he erge spee, we woul nee o know he cul isnce rele, n i is no gien 7 The ime for he firs pr of he rip is clcule from he iniil spee n he firs isnce 3 km e spee 95 km h 37 h 8 min The ime for he secon pr of he rip is herefore o 333 h 37 h 96 h 8 min The isnce for he secon pr of he rip is clcule from he erge spee for h pr of he rip n he ime for h pr of he rip e spee 65 km h 96 h 75 km 3 km () The ol isnce is hen ol 3 km 75 km 575 km 6 km (b) The erge spee is NOT he erge of he wo spees Use he efiniion of erge spee 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 5

5 Chper Describing Moion: Kinemics in One Dimension e spee ol ol 575 km 333 h 77 km h 8 The spee of soun is inime in he problem s mile per 5 secons The spee is clcule by: isnce mi 6 m spee 3 m s ime 5s mi Noe h only significn figure is gien, (5 sec), n so only significn figure is jusifie in he resul 9 The isnce rele is miles 8 lps 5 mi lp The isplcemen is becuse he ening poin is he sme s he sring poin () Aerge spee = (b) Aerge elociy = mi mi 6 m min 5 min 5 min mi 6 s x m s 43 m s The isnce rele is 6 km 6 km 74 km, n he isplcemen is 6 km 6 km 58 km The ol ime is 4 s + 48 s = 88 s 74 m () Aerge spee = 96 m s 88 s x 58 m (b) Aerge elociy = 3m s 88 s Since he locomoies he he sme spee, hey ech rel hlf he isnce, 45 km Fin he ime of rel from he erge spee 45 km 6 min e spee 447 h 68 min 7 min 95 km h h Boh objecs will he he sme ime of rel If he ruck rels isnce, hen he isnce ruck he cr rels will be m Using he equion for erge spee, cr ruck, sole for ime, n eque he wo imes m ruck cr ruck ruck 75 km h 88 km h ruck ruck cr Soling for gies ruck ruck m 75 km h 88 km h 75 km h 6346 m The ime of rel is 6346 m 6 min ruck 577 min 346 s 3 s 75 m h h Also noe h cr cr 6346 m m 6 min 88 m h h 577 min 346 s 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 6

6 Gincoli Physics: Principles wih Applicions, 6 h Eiion ALTERNATE SOLUTION: The spee of he cr relie o he ruck is 88 km h 75 km h 3 km h In he reference frme of he ruck, he cr mus rel m o cch i km 36 s 3km h h 346 s 3 The erge spee for ech segmen of he rip is gien by For he firs segmen, For he secon segmen, 3 km 394 h 79 km h 8 km 88 h 99 km h, so Thus he ol ime is 394 h 88 h 675 h 68 h o The erge spee of he plne for he enire rip is 3 km 8 km o km h 675 h o for ech segmen 4 The isnce rele is 5 km (5 km ougoing, 5 km reurn, keep significn figures) The isplcemen x is becuse he ening poin is he sme s he sring poin () To fin he erge spee, we nee he isnce rele (5 km) n he ol ime elpse 5 km During he ougoing porion, n so 63 h During he reurn 95 km h porion,, n so 5 km 4545 h 55 km h 877 h Aerge spee ol lunch (b) Aerge elociy = x ol Thus he ol ime, incluing lunch, is ol 5 km 6km h 877 h 5 The erge spee of soun is gien by, n so he ime for he soun o rel from he 65 m en of he lne bck o he bowler is 485 s Thus he ime for he soun 34 m s soun bll o rel from he bowler o he en of he lne is gien by 5 s 485 s 455 s An so he spee of he bll is: bll ol soun 65 m 673m s bll 455 s bll 6 The erge ccelerion is gien by m s 95 km h 95 km h km h 36 km h 6 s 6 s 43m s 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 7

7 Chper Describing Moion: Kinemics in One Dimension 7 () The erge ccelerion of he spriner is (b) km 36 s 74m s 96 km h m h 4 m s m s 35 s 74m s 8 The ime cn be foun from he erge ccelerion, m s 3 km h km h 8 km h 36 km h 6 m s 6 m s 58s 5 s 9 The iniil elociy of he cr is he erge spee of he cr before i cceleres m m s 5 s The finl spee is, n he ime o sop is 4 s Use Eq - o fin he ccelerion m s g 55 m s 55 m s 56 g 's 4 s 98 m s To esime he elociy, fin he erge elociy oer ech ime inerl, n ssume h he cr h h elociy he mipoin of he ime inerl To esime he ccelerion, fin he erge ccelerion oer ech ime inerl, n ssume h he cr h h ccelerion he mipoin of he ime inerl A smple of ech clculion is shown From s o 5 s, for erge elociy: 5 s s 5 s mi x 379 m 855 m 54 m 48 m s 5 s s 5 s From 5 s o 75 s, for erge ccelerion: 5 s 75 s 5 s mi 34 m s 48 m s 66 m s 53 m s 75 s 5 s 5 s Tble of Clculions (s) x (m) (s) (m/s) (s) (m/s ) Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 8

8 Gincoli Physics: Principles wih Applicions, 6 h Eiion (m/s) Grph of he elociy (s) (m/s ) Grph of he ccelerion (s) 5 m s 3 m s By efiniion, he ccelerion is 6 s The isnce of rel cn be foun from Eq -b x x m s 3m s 6 s m s 6 s 4 m The ccelerion cn be foun from Eq (-c) 3 m s 3m s x x x x 85 m 3 Assume h he plne srs from res The runwy isnce is foun by soling Eq -c for x x 33 m s x x x x 8 m 3 m s 4 The spriner srs from res The erge ccelerion is foun from Eq -c 5 m s 448 m s 44m s x x x x 5 m The elpse ime is foun by soling Eq - for ime 5 m s 6 s 448 m s 5 The wors slowing own uniformly implies h he cr hs consn ccelerion The isnce of rel is foun form combining Eqs -7 n -8 m s m s x x 6 sec 63 m 6 The finl elociy of he cr is zero The iniil elociy is foun from Eq -c wih n soling for x x x x 7 m s 9 m 36 m s 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 9

9 Chper Describing Moion: Kinemics in One Dimension 7 The finl elociy of he rier is zero The ccelerion is foun from Eq -c wih n soling for 85 km h x x 8 m Conering o "g's": 3484 m s 98 m s m s 36 km h g 36 g's 3484 m s 35 m s 8 The origin is he locion of he cr he beginning of he recion ime The iniil spee of he cr is m s 95 km h 639 m s The locion where he brkes re pplie is foun from 36 km h he equion for moion consn elociy: x 639 m s s 639 m This is R now he sring locion for he pplicion of he brkes In ech cse, he finl spee is () Sole Eq -c for he finl locion 639 m s x x x x 639 m 3 m 4 m s (b) Sole Eq -c for he finl locion wih he secon ccelerion x x 639 m s 639 m 7 m 8 m s 9 The origin is he locion of he cr he beginning of he recion ime The locion where he brkes re pplie is foun from he equion for moion consn elociy: x R This is he sring locion for he pplicion of he brkes Sole Eq -c for he finl locion of he cr, wih x x R 3 The criicl coniion is h he ol isnce coere by he pssing cr n he pproching cr mus be less hn 4 m so h hey o no collie The pssing cr hs ol isplcemen compose of seerl iniiul prs These re: i) he m of cler room he rer of he ruck, ii) he m lengh of he ruck, iii) he m of cler room he fron of he ruck, n i) he isnce he ruck rels Since he ruck rels spee of 5 m s, he ruck will he isplcemen of x 5 m s ruck x 4 m 5 m s pssing cr Thus he ol isplcemen of he cr uring pssing is To express he moion of he cr, we choose he origin o be he locion of he pssing cr when he ecision o pss is me For he pssing cr, we he n iniil elociy of 5 m s n n ccelerion of m s Fin x from Eq -b pssing x x x 5 m s m s pssing c cr cr 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he

10 Gincoli Physics: Principles wih Applicions, 6 h Eiion Se he wo expressions for x equl o ech oher in orer o fin he ime require o pss pssing cr 4 m 5 m s 5 m s m s 4 m m s pss pss pss pss pss 8s 894 s Clcule he isplcemens of he wo crs uring his ime x 4 m 5 m s 894 s 64 m pssing cr x pproching cr pproching cr 5 m s 894 s 4 m Thus he wo crs ogeher he coere ol isnce of 488 m, which is more hn llowe The cr shoul no pss 3 During he finl pr of he rce, he runner mus he isplcemen of m in ime of 8 s (3 min) Assume h he sring spee for he finl pr is he sme s he erge spee hus fr 89 m Aerge spee = 5494 m s 7 6 s The runner will ccomplish his by ccelering from spee o spee for secons, coering isnce, n hen running consn spee of for 8 secons, coering isnce We he hese relionships: 8 8 o o m 8 m 8 8 o m 8 s 5494 m s 8 s m s m s s, 3 s Since we mus he 8 s, he soluion is 3 s m s 3 The cr's iniil spee is 45 km h 5 m s o 36 km h Cse I: rying o sop The consrin is, wih he brking ecelerion of he cr 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 58 m s, cn he cr sop in 8 m isplcemen? The secons hs no relion o his pr of he problem Using equion (-c), he isnce rele uring brking is x x 5 m s 58 m s 35 m She cn sop he cr in ime Cse II: crossing he inersecion The consrin is, wih he ccelerion of he cr 65 km h 45 km h m s 6 s 36 km h 959 m s, cn she ge hrough he inersecion (rel 43 meers) in he secons before he ligh urns re? Using equion (b), he isnce rele uring he sec is x x 5 m s s 97 m s s 69 m She shoul sop

11 Chper Describing Moion: Kinemics in One Dimension 33 Choose ownwr o be he posiie irecion, n ke y he op of he cliff The iniil elociy is (-b), wih x replce by y, n he ccelerion is 98 m s The isplcemen is foun from equion y y y 98 m s 35 s y 58 m 34 Choose ownwr o be he posiie irecion The iniil elociy is m s 85 km h 36m s, n he ccelerion is 36 km h foun by soling Eq - for he ime 36m s 4 s 98 m s 98 m s, he finl elociy is The ime cn be 35 Choose ownwr o be he posiie irecion, n ke y o be he op of he Empire Se Builing The iniil elociy is, n he ccelerion is 98 m s () The elpse ime cn be foun from Eq -b, wih x replce by y y y y 38 m 98 m s (b) The finl elociy cn be foun from equion (-) 98 m s 886 s 86 m s 886s 88 s 36 Choose upwr o be he posiie irecion, n ke y o be he heigh where he bll ws hi For he upwr ph, m s, he op of he ph, n 98 m s () The isplcemen cn be foun from Eq -c, wih x replce by y m s y y y y 5 m 98 m s (b) The ime of fligh cn be foun from Eq -b, wih x replce by y, using isplcemen of for he isplcemen of he bll reurning o he heigh from which i ws hi m s y y, 45 s 98 m s The resul of = s is he ime for he originl isplcemen of zero (when he bll ws hi), n he resul of = 45 s is he ime o reurn o he originl isplcemen Thus he nswer is = 45 secons 37 Choose upwr o be he posiie irecion, n ke y o be he heigh from which he bll ws hrown The ccelerion is 98 m s The isplcemen upon cching he bll is, ssuming i ws cugh he sme heigh from which i ws hrown The sring spee cn be foun from Eq -b, wih x replce by y y y y y 98 m s 3 s 47 m s 5 m s 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he

12 Gincoli Physics: Principles wih Applicions, 6 h Eiion The heigh cn be clcule from Eq -c, wih finl elociy of he op of he ph 47 m s y y y y m 98 m s 38 Choose ownwr o be he posiie irecion, n ke y o be he heigh where he objec ws relese The iniil elociy is, n he ccelerion is () The spee of he objec will be gien by Eq - wih 98 m s, n so This is he equion of srigh line pssing hrough he origin wih slope of (b) The isnce fllen will be gien by equion (-b) wih, n so 98 m s 98 m s y y This is he equion of prbol, cenere on he 49 m s -xis, opening upwr spee (m/s) ime (s) isnce fllen (m) ime (s) 39 Choose ownwr o be he posiie irecion, n ke y o be he heigh where he objec ws relese The iniil elociy is 5 m s isplcemen of he pckge will be y 5 m Eq -b, wih x replce by y y, he ccelerion is 98 m s, n he The ime o rech he groun cn be foun from 5 m s 5 m y y 98 m s 98 m s 56 s, 455 s The correc ime is he posiie nswer, 56 s 4 Choose ownwr o be he posiie irecion, n ke y o be he heigh from which he objec is relese The iniil elociy is, he ccelerion is g Then we cn clcule he posiion s funcion of ime from Eq -b, wih x replce by y, s of ech secon, he posiion woul be s follows: y y ; y g ; y g 4y ; y 3 g 3 9y g A he en The isnce rele uring ech secon cn be foun by subrcing wo jcen posiion lues from he boe lis y y y ; y y 3y ; 3 y 3 y 5y 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 3

13 Chper Describing Moion: Kinemics in One Dimension We coul o his in generl y n gn y n g n n y n y n g n gn g n n g n n n g n The lue of n is lwys o, in he sequence, 3, 5, 7, 4 Choose upwr o be he posiie irecion, n ke y o be he heigh from which he bll is hrown The iniil elociy is, he ccelerion is g, n he finl locion for he roun rip is y The elociy uner hose coniions cn be foun from Eq -c, wih x replce by y y The wo resuls represen wo ifferen elociies for he sme isplcemen of The posiie sign is he iniil elociy, when he bll is moing upwrs, n he negie sign is he finl elociy, when he bll is moing ownwrs Boh of hese elociies he he sme mgniue, n so he bll hs he sme spee he en of is fligh s he beginning 4 Choose upwr o be he posiie irecion, n y o be he heigh from which he sone is hrown We he 8 m s, 98 m s, n y y m () The elociy cn be foun from Eq, -c, wih x replce by y y y y 8 m s 98 m s m 4 m s Thus he spee is 4 m s (b) The ime o rech h heigh cn be foun from equion (-b) 8 m s m y y 98 m s 98 m s s, 775 s (c) There re wo imes which he objec reches h heigh once on he wy up 775 s, n once on he wy own 9 s 43 The -cm ( mm) pple hs imeer of bou 6 mm s mesure in he phoogrph Thus ny isnces mesure from he picure nee o be muliplie by / 6 Choose he ownwr irecion o be posiie Choose y o be sem of he pple on he THIRD imge from he op of he picure I is he firs picure in which he sem of he pple is isible The elociy of he pple h posiion is no, bu i is no known eiher Cll i We will choose h he ime h poin is, n we cll he ime inerl from one picure o he nex o be T The ccelerion of he pple is g 98 m s 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 4

14 Gincoli Physics: Principles wih Applicions, 6 h Eiion The 3 r picure fer he picure (he firs one h is no oerlpping wih noher imge) hs he sem 65 mm from he origin of coorines, ime of 3T The cul posiion woul be foun by y 65 mm 6 75 mm 75 m The 6 h picure fer he picure (he nex o ls one in he picure) hs he sem 4 mm from he origin of coorines, ime of 6T The cul posiion woul be foun by y 4 mm 6 7 mm 7 m Now we he wo ses of posiion-ime, relie o he origin Boh of hose ses of posiionime mus sisfy equion Eq -b y y 75 3 T g 3 T y y 7 6 T g 6T Muliply he firs equion by, n hen subrc i from he secon equion o elimine he epenence on The resuling equion cn be sole for T 55 m 6 T 9gT 5 m 7 m 6 T 8gT flsh flsh This is equilen o T 4 s 5 m 9 gt T 4 s 9 98 m s 4 flshes per secon 44 Choose ownwr o be he posiie irecion, n y o be he heigh from which he sone is roppe Cll he locion of he op of he winow y, n he ime for he sone o fll from w relese o he op of he winow is w Since he sone is roppe from res, using Eq -b wih y subsiuing for x, we he winow is w y w y y g The locion of he boom of he w w m, n he ime for he sone o fll from relese o he boom of he winow is 8 s Since he sone is roppe from res, using Eq -b, we he y m y g 8 s Subsiuing he firs expression for w w y w ino he secon one g g w w w m 8 s 66 s Use his ime in he firs equion y g 98m s 66 s m w w 45 For he flling rock, choose ownwr o be he posiie irecion, n y o be he heigh from which he sone is roppe The iniil elociy is m s, he ccelerion is g, he isplcemen is y H, n he ime of fll is Using Eq -b wih y subsiuing for x, we he H y g For he soun we, use he consn spee equion h s H T, which cn be rerrnge 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 5

15 Chper Describing Moion: Kinemics in One Dimension o gie T H, where T 3 s is he ol ime elpse from ropping he rock o hering he s soun Inser his expression for ino he equion for H, n sole for H H g gt H g T H H gt s s s H 9H 58 H 46 m, 57 m If he lrger nswer is use in correc nswer is H 46 m 46 Choose upwr o be he posiie irecion, n y T H, negie ime of fll resuls, n so he physiclly s o be he locion of he nozzle The iniil elociy is, he ccelerion is 98 m s, he finl locion is y 5 m, n he ime of fligh is s Using Eq -b n subsiuing y for x gies he following y y y 5 m 98 m s s s 9m s 47 Choose ownwr o be he posiie irecion, n y o be he op of he cliff The iniil elociy is m s, he ccelerion is 98 m s, n he finl locion is y 7 m () Using Eq -b n subsiuing y for x, we he y y 49 m s m s 7 m 749 s, 598 s The posiie nswer is he physicl nswer: (b) Using Eq -, we he 5 s m s 98 m s 598 s 389 m s (c) The ol isnce rele will be he isnce up plus he isnce own The isnce own will be 7 m more hn he isnce up To fin he isnce up, use he fc h he spee he op of he ph will be Then using Eq -c: m s y y y y 735 m 98 m s Thus he isnce up is 735 m, he isnce own is 7735 m, n he ol isnce rele is 847 m 48 Choose upwr o be he posiie irecion, n y o be he leel from which he bll ws hrown The iniil elociy is, he insnneous elociy is 98 m s, n he locion of he winow is y 8 m () Using Eq -c n subsiuing y for x, we he y y 3m s, he ccelerion is y y 3 m s 98 m s 8 m 7 m s Choose he posiie lue becuse he iniil irecion is upwr 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 6

16 Gincoli Physics: Principles wih Applicions, 6 h Eiion (b) A he op of is ph, he elociy will be, n so we cn use he iniil elociy s foun boe, long wih Eq -c 7 m s y y y y 37 m 98 m s (c) We wn he ime elpse from hrowing (spee 3 m s ) Using Eq -, we he: 3 m s 7 m s 98 m s () We wn he ime elpse from he winow (spee 7 m s ) Using Eq -, we he: 7 m s 3 m s 98 m s 7 m s ) o reching he winow (spee 4 s 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 7 3m s ) o reching he sree (spee 4 s 49 Slighly ifferen nswers my be obine since he comes from reing he grph () The grees elociy is foun he highes poin on he grph, which is 48 s (b) The inicion of consn elociy on elociy-ime grph is slope of, which occurs from 9 s o 8 s (c) The inicion of consn ccelerion on elociy-ime grph is consn slope, which occurs from s o 38 s, gin from 65 s o 83 s, n gin from 9 s o 8 s () The mgniue of he ccelerion is grees when he mgniue of he slope is grees, which occurs from 65 s o 83 s 5 Slighly ifferen nswers my be obine since he comes from reing he grph () The insnneous elociy is gien by he slope of he ngen line o he cure A s, he slope is pproximely 3 m s 3 m s (b) A 3 s, he slope of he ngen line o he cure, n hus he insnneous elociy, is pproximely 3 m 8 m 35 s 5 s 4 m s (c) The erge elociy is gien by x 5 m x m 5 m 5 s s 5 s 3 m s () The erge elociy is gien by x 3 m x 5 m 6 m 9 m 3 s 5 s 5 s 4 m s (e) The erge elociy is gien by x 5 m x 4 m m 95 m 5 s 4 s s 95 m s 5 Slighly ifferen nswers my be obine since he comes from reing he grph () The inicion of consn elociy on posiion-ime grph is consn slope, which occurs from s o 8 s (b) The grees elociy will occur when he slope is he highes posiie lue, which occurs

17 Chper Describing Moion: Kinemics in One Dimension bou 7 s (c) The inicion of elociy on posiion-ime grph is slope of, which occurs bou from 38 s () The objec moes in boh irecions When he slope is posiie, from s o 38 s, he objec is moing in he posiie irecion When he slope is negie, from 5 s, he objec is moing in he negie irecion 38 s o 5 Slighly ifferen nswers my be obine since he comes from reing he grph We ssume h he shor, nerly horizonl porions of he grph re he imes h shifing is occurring, n hose imes re no coune s being in cerin ger () The erge ccelerion in n 4 m s 4 m s ger is gien by 5 m s 8 s 4 s The erge ccelerion in 4 h ger is gien by 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he m s 37 m s 7 s 6 s 64 m s (b) The isnce rele cn be eermine from elociy-ime grph by clculing he re beween he grph n he xis, boune by he imes uner consierion For his cse, we will pproxime he re s recngle f 44 m s 37 m s heigh 45 m s wih 7 s 6 s s Thus he isnce rele is 45 m s s 45 m 53 Slighly ifferen nswers my be obine since he comes from reing he grph We ssume h he shor, nerly horizonl porions of he grph re he imes h shifing is occurring, n hose imes re no coune s being in cerin ger 4 m s m s () The erge ccelerion in firs ger is gien by 4 m s 4 s s 37 m s 4 m s (b) The erge ccelerion in hir ger is gien by 3 m s 4 s 9 s 5 m s 44 m s (c) The erge ccelerion in fifh ger is gien by 35 m s 5 s 7 s () The erge ccelerion hrough he firs four gers is gien by 44 m s m s 6 m s 7 s s 54 Slighly ifferen nswers my be obine since he comes from reing he grph () To esime he isnce he objec rele uring he firs minue, we nee o fin he re uner he grph, from = s o = 6 s Ech "block" of he grph represens n "re" of x m s s m By couning n esiming, here re bou 75 blocks uner he s minue of he grph, n so he isnce rele uring he s minue is bou 75 m (b) For he secon minue, here re bou 5 blocks uner he grph, n so he isnce rele uring he secon minue is bou 5 m Alerniely, erge ccelerions cn be esime for rious porions of he grph, n hen he uniform ccelerion equions my be pplie For insnce, for pr (), brek he moion up ino

18 Gincoli Physics: Principles wih Applicions, 6 h Eiion wo segmens, from o 5 secons n hen from 5 o 6 secons 38 m s 4 m s () = o 5: 48 m s 5 s s o o 3m s 645 m 4 m s 5 s 48 m s 5 s 3 m 38 m s 6 s 5 s 7 m s 55 The s grph is foun by king he slope of he x s grph Boh grphs re shown here 38 m s s 7 m s s 345 m 56 () During he inerl from A o B, i is moing in he negie irecion, becuse is isplcemen is negie (b) During he inerl from A o B, i is speeing up, becuse he mgniue of is slope is incresing (chnging from less seep o more seep) (c) During he inerl from A o B, he ccelerion is negie, becuse he grph is conce ownwr, inicing h he slope is geing more negie, n hus he ccelerion is negie () During he inerl from D o E, i is moing in he posiie irecion, becuse he isplcemen is posiie (e) During he inerl from D o E, i is speeing up, becuse he mgniue of is slope is incresing (chnging from less seep o more seep) (f) During he inerl from D o E, he ccelerion is posiie, becuse he grph is conce upwr, inicing he slope is geing more posiie, n hus he ccelerion is posiie (g) During he inerl from C o D, he objec is no moing in eiher irecion The elociy n ccelerion re boh 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 9

19 Chper Describing Moion: Kinemics in One Dimension 57 () For he free-flling pr of he moion, choose ownwr o be he posiie irecion, n y o be he heigh from which he person jumpe The iniil elociy is, ccelerion is 98 m s, n he locion of he ne is y 5 m Fin he spee upon reching he ne from Eq (-c) wih x replce by y y y y 98 m s 5 m 7m s The posiie roo is selece since he person is moing ownwr For he ne-sreching pr of he moion, choose ownwr o be he posiie irecion, n y 5 m o be he heigh which he person firs concs he ne The iniil elociy is 7m s, he finl elociy is, n he locion he sreche posiion is y 6 m Fin he ccelerion from Eq (-c) wih x replce by y 7m s 5 m s y y y y m (b) For he ccelerion o be smller, in he boe equion we see h he isplcemen woul he o be lrger This mens h he ne shoul be "loosene" 58 Choose he upwr irecion o be posiie, n y o be he leel from which he objec ws hrown The iniil elociy is n he elociy he op of he ph is he op of he ph cn be foun from Eq (-c) wih x replce by y y y y y m s The heigh From his we see h he isplcemen is inersely proporionl o he ccelerion, n so if he ccelerion is reuce by fcor of 6 by going o he Moon, n he iniil elociy is unchnge, he isplcemen increses by fcor of 6 m s 59 The iniil elociy of he cr is km h 78 m s Choose x o be 36 km h locion which he ecelerion begins We he he isplcemen from Eq (-c) m s n 78 m s 3g 94 m s Fin x x x x 3m 3 m 94 m s 6 Choose ownwr o be he posiie irecion, n y o be he heigh of he brige Agen Bon hs n iniil elociy of, n ccelerion of g, n will he isplcemen of y m 5 m 5 m Fin he ime of fll from Eq -b wih x replce by y y y y 5 m If he ruck is pproching wih 98 m s 464 s 5 m s, hen he nees o jump when he ruck is isnce wy gien by 5 m s 464 s 366 m Coner his isnce ino "poles" 366 m pole 5 m 46 poles 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 3

20 Gincoli Physics: Principles wih Applicions, 6 h Eiion So he shoul jump when he ruck is bou 5 poles wy from he brige 6 () Choose ownwr o be he posiie irecion, n y o be he leel from which he cr ws roppe The iniil elociy is, he finl locion is y H, n he ccelerion is g Fin he finl elociy from Eq -c, replcing x wih y y y y y gh The spee is he mgniue of he elociy, gh (b) Soling he boe equion for he heigh, we he h m s 6 km h 667 m s, he corresponing heigh is: 36 km h H (c) For collision of H 667 m s g 98 m s 47 m 4 m m s km h 778 m s 36 km h 778 m s g 98 m s 3937 m 39 m H Thus for collision of g, he corresponing heigh is: 6 The erge spee is he isnce iie by he ime 9 km y km h km h y h 63 Use he informion for he firs 8 m o fin he ccelerion, n he informion for he full moion o fin he finl elociy For he firs segmen, he rin hs m s, 5 m s, n isplcemen of x x 8 m Fin he ccelerion from Eq -c 5 m s 736 m s x x x x 8 m Fin he spee of he rin fer i hs rele he ol isnce (ol isplcemen of x x 75 m ) using Eq -c x x x x 736 m s 75 m 3m s 64 For he moion in he ir, choose ownwr o be he posiie irecion, n y o be he heigh of he iing bor Then ier hs, (ssuming he ier oes no jump upwr or ownwr), g 98 m s, n y 4 m when reching he surfce of he wer Fin he ier s spee he wer s surfce from Eq -c, wih x replce by y y y x y y 98 m s 4 m 885 m s 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 3

21 Chper Describing Moion: Kinemics in One Dimension For he moion in he wer, gin choose own o be posiie, bu reefine y o be he surfce of he wer For his moion, 885 m s,, n y y m Fin he ccelerion from Eq -c, wih x replce by y 885 m s 96 m s m s y y y y x m The negie sign inices h he ccelerion is irece upwrs 65 This problem cn be nlyze s series of hree one-imensionl moions: he ccelerion phse, he consn spee phse, n he ecelerion phse The mximum spee of he rin is: m s 9 km h 5 m s 36 km h In he ccelerion phse, he iniil elociy is finl elociy is m s, he ccelerion is m s 5 m s Fin he elpse ime for he ccelerion phse from Eq - 5 m s 73 s cc m s Fin he isplcemen uring he ccelerion phse from Eq -b x x m s 73 s 84 m cc, n he In he ecelerion phse, he iniil elociy is 5 m s, he ccelerion is m s, n he finl elociy is m s Fin he elpse ime for he ecelerion phse from equion Eq - 5 m s ec m s 5 s Fin he isnce rele uring he ecelerion phse from Eq -b x x 5 m s 5 s m s 5 s 56 m ec The ol elpse ime n isnce rele for he ccelerion / ecelerion phses re: 7 s 5 s 35 s cc ec x x x x 84 m 56 m 44 m cc ec () If he sions re spce 8 km = 8 m pr, hen here is ol of 9 m 5 inersion segmens A rin mking he enire rip woul hus he ol of 5 iner-sion 8 m segmens n 4 sops of s ech he inermeie sions Since 44 m is rele uring ccelerion n ecelerion, 36 m of ech segmen is rele n erge spee of 5 m s The ime for h 36 m is gien by consn spee 36 m 544 s 5 m s Thus ol iner-sion segmen will ke 35 s s = 896 s Wih 5 iner-sion segmens of 896 s ech, n 4 sops of s ech, he ol ime is gien by: 5 896s 4 s 58 s 88 min 8 km 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 3

22 Gincoli Physics: Principles wih Applicions, 6 h Eiion (b) If he sions re spce 3 km =3 m pr, hen here is ol of 9 m 3 inersion segmens A rin mking he enire rip woul hus he ol of 3 iner-sion 3 m segmens n sops of s ech he inermeie sions Since 44 m is rele uring ccelerion n ecelerion, 56 m of ech segmen is rele n erge spee of 56 m 5 m s The ime for h 56 m is gien by 4 s 5 m s Thus ol iner-sion segmen will ke 35 s + 4 s = 376 s Wih 3 iner-sion segmens of 376 s ech, n sops of s ech, he ol ime is s s 453 s 75 min 3 km 66 Choose ownwr o be he posiie irecion, n y o be he sr of he pelicn s ie The pelicn hs n iniil elociy is n n ccelerion of g, n finl locion of y 6 m Fin he ol ime of he pelicn s ie from Eq -b, wih x replce by y y 6 m y y y 8 s ie 98 m s The fish cn ke esie cion if he sees he pelicn ime of 8 s ie Fin he locion of he pelicn h ime from Eq -b 98 m s 6 s 7 m y y s = 6 s ino he Thus he fish mus spo he pelicn minimum heigh from he surfce of he wer of 6 m 7 m 33 m 67 Firs consier he "uphill lie", in which he bll is being pue own he hill Choose x o be he bll's originl locion, n he irecion of he bll's rel s he posiie irecion The finl elociy of he bll is m s, he ccelerion of he bll is m s, n he isplcemen of he bll will be x x 6 m for he firs cse, n x x 8 m for he secon cse Fin he iniil elociy of he bll from Eq -c x x x x m s 6 m 49 m s m s 8 m 57 m s The rnge of ccepble elociies for he uphill lie is 49 m s o 57 m s, wih spre of 8 m/s Now consier he "ownhill lie", in which he bll is being pue up he hill Use ery similr seup for he problem, wih he bsic ifference being h he ccelerion of he bll is now 3 m s Fin he iniil elociy of he bll from Eq -c x x x x 3 m s 6 m 6 m s 3 m s 8 m 69 m s The rnge of ccepble elociies for he ownhill lie is 6 m s o 69 m s, wih spre of 9 m/s 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 33

23 Chper Describing Moion: Kinemics in One Dimension Becuse he rnge of ccepble elociies is smller for puing own he hill, more conrol in puing is necessry, n so he ownhill pu is more ifficul 68 () The rin's consn spee is 6 m s, n he locion of he empy box cr s rin funcion of ime is gien by x 6 m s The fugiie hs m s n rin rin 4 m s unil his finl spee is 8 m s The elpse ime uring ccelerion is 8 m s cc 4 m s s Le he origin be he locion of he fugiie when he srs o run The firs possibiliy o consier is, "Cn he fugiie cch he rin before he reches his mximum spee?" During he fugiie's ccelerion, his locion s funcion of ime is gien by x x 4 m s For him o cch he rin, we mus he fugiie x x 6 m s 4 m s The soluions of his re s, 3 s rin fugiie he fugiie cnno cch he cr uring his s of ccelerion Thus Now he equion of moion of he fugiie chnges Afer he s ccelerion, he runs wih consn spee of 8 m s Thus his locion is now gien (for imes s ) by he following 4 m s s 8 m s s 8 m s 8 m fugiie x So now, for he fugiie o cch he rin, we gin se he locions equl x x 6 m s 8 m s 8 m 4 s rin fugiie (b) The isnce rele o rech he box cr is gien by x 4 s 8 m s 4 s 8 m 4 m fugiie 69 Choose ownwr o be he posiie irecion, n y o be he roof from which he sones re roppe The firs sone hs n iniil elociy of n n ccelerion of g Eqs - n -b (wih x replce by y) gie he elociy n locion, respeciely, of he firs sone s funcion of ime g y y y g The secon sone hs he sme iniil coniions, bu is elpse ime 5 s, n so hs elociy n locion equions s follows g 5 s y g 5 s The secon sone reches spee of m s ime gien by m s 5 s 5 s 7 s g 98 m s The locion of he firs sone h ime is y g 98 m s 7 s 364 m The locion of he secon sone h ime is y g 5 s 98 m s 7 5 s 735 m Thus he isnce beween he wo sones is y y 364 m 735 m 9 m 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 34

24 Gincoli Physics: Principles wih Applicions, 6 h Eiion 7 To fin he erge spee for he enire rce, we mus ke he ol isnce iie by he ol ime If one lp is isnce of L, hen he ol isnce will be L The ime elpse gien 9L consn spee is gien by, so he ime for he firs 9 lps woul be, n 98 km h he ime for he ls lp woul be L, where is he erge spee for he ls lp Wrie n expression for he erge spee for he enire rce, n hen sole for ol L 9L L km h 98 km h 9 km h 98 km h 7 The iniil elociy is km h m s 8 km h 5 m s The finl elociy is 36 km h m s 75 km h 83m s The isplcemen is x x 36 km h 4 km 4 m Fin he erge ccelerion from Eq -c 83 m s 5 m s 5 m s x x x x 4 m 7 Assume h y for ech chil is he leel which he chil loses conc wih he rmpoline surfce Choose upwr o be he posiie irecion () The secon chil hs 5 m s, g 98 m s, n m s he mximum heigh posiion Fin he chil s mximum heigh from Eq -c, wih x replce by y 5 m s y y y y 76 m 3 m 98 m s (b) Since he firs chil cn bounce up o one-n--hlf imes higher hn he secon chil, he firs chil cn bounce up o heigh of 5 76 m 93m y y Eq -c is gin use o fin he iniil spee of he firs chil y y y y 98 m s 93 m 64 m s 6m s The posiie roo ws chosen since he chil ws iniilly moing upwr (c) To fin he ime h he firs chil ws in he ir, use Eq -b wih ol isplcemen of, since he chil reurns o he originl posiion y y 64 m s 98 m s s, 497 s The ime of s correspons o he ime he chil sre he jump, so he correc nswer is s 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 35

25 Chper Describing Moion: Kinemics in One Dimension 73 For he cr o pss he rin, he cr mus rel he lengh of he rin AND he isnce he rin rels The isnce he cr rels cn hus be wrien s eiher 95 km h or cr cr L km 75 km h To sole for he ime, eque hese wo expressions cr rin rin for he isnce he cr rels km 95 km h km 75 km h 55 h 33 min km h The isnce he cr rels uring his ime is 95 km h 55 h 55 km 5 km If he rin is reling he opposie irecion from he cr, hen he cr mus rel he lengh of he rin MINUS he isnce he rin rels Thus he isnce he cr rels cn be wrien s eiher 95 km h or km 75 km h To sole for he ime, eque hese wo cr cr expressions for he isnce he cr rels km 3 95 km h km 75 km h 647 h 33 s The isnce he cr rels uring his ime is 7 km h 3 95 km h 647 h 6 km 74 For he bsebll,, x x 35 m, n he finl spee of he bsebll (uring he hrowing moion ) is 44 m s The ccelerion is foun from Eq -c 44 m s 8 m s x x x x 35 m 75 () Choose upwr o be he posiie irecion, n y he groun The rocke hs, 3 m s, n m y when i runs ou of fuel Fin he elociy of he rocke when i runs ou of fuel from Eq -c, wih x replce by y y y m y y 3 m s m 8764 m s 88 m s m The posiie roo is chosen since he rocke is moing upwrs when i runs ou of fuel (b) The ime o rech he m locion cn be foun from equion (-) 8764 m s m 739 s 7 s m m m 3 m s (c) For his pr of he problem, he rocke will he n iniil elociy 8764 m s, n ccelerion of 98 m s, n finl elociy of is mximum liue The liue reche from he ou-of-fuel poin cn be foun from equion (-c) y m y m 8764 m s m mx m m m 39 m 59 m 98 m s () The ime for he "cosing" porion of he fligh cn be foun from Eq m s 894 s m cos cos 98 m s 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 36

26 Gincoli Physics: Principles wih Applicions, 6 h Eiion Thus he ol ime o rech he mximum liue is 7 s 894 s 36 s (e) For his pr of he problem, he rocke hs m s, 98 m s, n isplcemen of 6 m (i flls from heigh of 6 m o he groun) Fin he elociy upon reching he Erh from Eq -c y y (f) y y 98 m s 6 m 77 m s The negie roo ws chosen becuse he rocke is moing ownwr, which is he negie irecion The ime for he rocke o fll bck o he Erh is foun from Eq - 77 m s 8 s fll 98 m s Thus he ol ime for he enire fligh is 36 s 8 s 54 s m s 76 The spee limi is 5 km h 389 m s 36 km h () For your moion, you woul nee o rel m 6 m o ge hrough he hir ligh The ime o rel he 6 m is foun using he isnce n he consn spee 6 m 5 s 389 m s Yes, you cn mke i hrough ll hree lighs wihou sopping (b) The secon cr nees o rel 5 m before he hir ligh urns re This cr cceleres from m s o mximum of 389 m s wih m s Use Eq - o eermine he urion of h ccelerion 389 m s m s 694 s cc m s The isnce rele uring h ime is foun from Eq -b x x m s 694 s 48 m cc cc cc Since 694 sec he elpse, here re = 66 sec remining o cler he inersecion The cr rels noher 6 secons spee of 389 m/s, coering isnce of 389 m s 66 s 84 m Thus he ol isnce is 48 m + 84 m = consn spee 34 m No, he cr cnno mke i hrough ll hree lighs wihou sopping 77 Tke he origin o be he locion where he speeer psses he police cr The speeer's consn spee is speeer m s km h 333 m s, n he locion of he speeer s funcion 36 km h of ime is gien by x 333 m s The police cr hs n iniil elociy of speeer speeer speeer speeer m s n consn ccelerion of The locion of he police cr s funcion of ime police is gien by Eq -b x police police police 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 37

27 Chper Describing Moion: Kinemics in One Dimension () The posiion s ime grphs woul quliiely look like he grph shown here (b) The ime o oerke he speeer occurs when he speeer hs gone isnce of 75 m The ime is foun using he speeer's equion from boe 75 m 75 m 333 m s 5 s 3 s speeer speeer 333 m s (c) The police cr's ccelerion cn be clcule knowing h he police cr lso h gone isnce of 75 m in ime of 5 s 75 m 75 m 5 s 96 m s 3 m s p p 5 s () The spee of he police cr he oerking poin cn be foun from Eq - 96 m s 5 s 6667 m s 67 m s Noe h his is excly wice he spee of he speeer 78 Choose ownwr o be he posiie irecion, n he origin o be he roof of he builing from which he sones were roppe The firs sone hs y,, finl locion of y H (s ye unknown), n g If he ime for he firs sone o rech he groun is, hen Eq -c gies he following, replcing x wih y: y y H 98 m s The secon sone hs 5 m s, y, finl locion of y H, n g The ime for he secon sone o rech he groun is H 5 m s 98 m s s, n so Eq -c for he secon sone is () Se he wo expressions for H equl o ech oher, n sole for 98 m s 5 m s 98 m s 563 s (b) The builing heigh is gien by H g 98 m s 563 s 55 m (c) The spee of he sones is foun using Eq - #: g 98 m s 563 s 55 m s # : g 5 m s 98 m s 363 s 66 m s x Speeer Police cr 79 Choose upwr o be he posiie irecion, n he origin o be groun leel The iniil elociy of he firs sone is m s, n he ccelerion of boh sones is 98 m s The A sring locion is y H, n i kes 45 s for he sone o rech he finl locion y Use A A Eq -b (wih x replce by y) o fin lue for H A y y H m s 45 s 98 m s 45 s A H 497 m A Assume h he h floor blcony is hree imes higher boe he groun hn he 4 h floor blcony Thus he heigh of 4 h floor blcony is 497 m 66 m So for he secon sone, y 66 m, 3 B 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 38

28 Gincoli Physics: Principles wih Applicions, 6 h Eiion n i kes 45 s for he sone o rech he finl locion y Use Eq -b o fin he sring elociy, B y y 66 m 45 s 98 m s 45 s B B 8 m s 8 Choose ownwr o be he posiie irecion, n he origin o be he locion of he plne The prchuis hs, g 98 m s, n will he y y 85 m when she pulls he ripcor Eq -b, wih x replce by y, is use o fin he ime when she pulls he ripcor y y y y 85 m 98 m s 4 s The spee is foun from Eq - 98 m s 4 s 36 m s 3 m s 85 km h m 8 The spee of he coneyor bel is gien by 44 m min 5 min of burger proucion, ssuming he spcing gien is cener o cener, cn be foun s burger 44 m burgers 9 5 m min min The re 8 Choose upwr o be he posiie irecion, n he origin o be he leel where he bll ws hrown The elociy he op of he bll's ph will be, n he bll will he n ccelerion of g If he mximum heigh h he bll reches is y H, hen he relionship beween he iniil elociy n he mximum heigh cn be foun from Eq -c, wih x replce by y y y g H H g H g Bill We re ol h 5, so Bill Joe H g Bill Bill Joe Joe Joe As shown in problem 4, he spee wih which he bll ws hrown upwr is he sme s is spee on reurning o he groun From he symmery of he wo moions (boh moions he spee = op, he sme isnce rele n he sme ccelerion), he ime for he bll o rise is s Choose upwr o be he posiie irecion, n he origin o be he leel where he bll ws hrown For he bll, he op of he moion, n g Fin he iniil elociy from Eq - 98 m s s m s 84 Choose ownwr o be he posiie irecion, n he origin o be he op of he builing The bromeer hs y,, n builing, wih x replce by y y y 98 m s y g 98 m s Use Eq -b o fin he heigh of he y 3 98 m s s m 98 m s 3 s 6 m The ifference in he esimes is 6 m 5 Person Eucion, Inc, Upper Sle Rier, NJ All righs resere This meril is proece uner ll copyrigh lws s hey currenly exis No porion of his meril my be reprouce, in ny form or by ny mens, wihou permission in wriing from he 39