Q39.2 An ellipsoid. The dimension in the direction of motion would be measured to be scrunched in.

Transcription

1 9 Rltiity CHAPTER UTNE 9 Th Prinil o Gliln Rltiity 9 Th Mihlson-Morly Exrimnt 9 Einstin s Prinil o Rltiity 94 Consquns o th Sil Thory o Rltiity 95 Th orntz Trnsormtion Equtions 96 Th orntz Vloity Trnsormtion Equtions 97 Rltiisti inr Momntum nd th Rltiisti orm o Nwton s ws 9 Rltiisti Enrgy 99 Mss nd Enrgy 90 Th Gnrl Thory o Rltiity ANSWERS T QUESTNS Q9 Th sd o light nd th sd o thir rlti motion Q9 An llisoid Th dimnsion in th dirtion o motion would b msurd to b srunhd in Q9 No Th rinil o rltiity imlis tht nothing n trl str thn th sd o light in uum, whih is 00 Mm/s Th ltron would mit light in onil shok w o Crnko rdition Q94 Th lok in orbit runs slowr No, thy r not synhronizd Although thy both tik t th sm rt tr rturn, tim dirn hs dlod btwn th two loks Q95 Suos rilrod trin is moing st you n wy to msur its lngth is this: You mrk th trks t th owthr orming th ront o th moing ngin t 9:00:00 AM, whil your ssistnt mrks th trks t th bk o th boos t th sm tim Thn you ind th distn btwn th mrks on th trks with t msur You nd your ssistnt must mk th mrks simultnously in your rm o rrn, or othrwis th motion o th trin would mk its lngth dirnt rom th distn btwn mrks Q96 () Yours dos (b) () His dos th loity o rlti motion is onstnt, both obsrrs h qully lid iws Q97 Gt Mr Tomkins book by Gorg Gmow or wondrul itionl xlortion o this qustion Driing hom in hurry, you ush on th gs dl not to inrs your sd by ry muh, but rthr to mk th bloks gt shortr Big Dolr shits in w rqunis mk rd lights look grn s you roh thm nd mk r horns nd r rdios uslss High-sd trnsorttion is ry xnsi, rquiring hug ul urhss And it is dngrous, s sding r n knok down building Hing hd brkst t hom, you rturn hungry or lunh, but you ind you h missd dinnr Thr is i-dy dly in trnsmission whn you wth th lymis in Austrli on li tlision t tks ninty-i yrs or sunlight to rh Erth W nnot s th Milky Wy; th irbll o th Big Bng surrounds us t th distn o Rigl or Dnb Q9 Nothing hysilly unusul An obsrr riding on th lok dos not think tht you r rlly strng, ithr 4

2 44 Rltiity Q99 By urd lin This n b sn in th middl o Sdo s world-lin in igur 9, whr h turns round nd bgins his tri hom Q90 Aording to γm u, doubling th sd u will mk th momntum o n obt inrs by th tor N M u 4u Q P Q9 As th obt rohs th sd o light, its kinti nrgy grows without limit t would tk n ininit instmnt o work to lrt th obt to th sd o light Q9 Thr is no ur limit on th momntum o n ltron As mor nrgy E is d into th obt without limit, its sd rohs th sd o light nd its momntum rohs E Q9 Rll tht whn sring o or onstnt k is omrssd or strthd rom its rlxd osition distn x, it stors lsti otntil nrgy U kx Aording to th sil thory o rltiity, ny hng in th totl nrgy o th systm is quilnt to hng in th mss o th systm Thror, th mss o omrssd or strthd sring is grtr thn th mss o rlxd sring by n mount U Th rtionl hng is tyilly unobsrbly smll or mhnil sring Q94 You s no hng in your rltion t ny sd you n ttin You nnot ttin th sd o light, or tht would tk n ininit mount o nrgy Q95 Qusr light mos t thr hundrd million mtrs r sond, ust lik th light rom irly t rst Q96 A hoton trnsorts nrgy Th rltiisti quiln o mss nd nrgy mns tht is nough to gi it momntum Q97 Any hysil thory must gr with xrimntl msurmnts within som domin Nwtonin mhnis grs with xrimnt or obts moing slowly omrd to th sd o light Rltiisti mhnis grs with xrimnt or obts t ll sds Thus th two thoris must nd do gr with h othr or ordinry nonrltiisti obts Both sttmnts gin in th qustion r ormlly orrt, but th irst is lumsily hrsd t sms to suggst tht rltiisti mhnis lis only to st-moing obts Q9 Th oint o intrstion mos to th right To stt th roblm risly, lt us ssum tht h o th two rds mos towrd th othr rlll to th long dimnsion o th itur, with loity o mgnitud Th oint o intrstion mos to th right t sd otφ, whr φ is th tnφ smll ngl btwn th rds As φ rohs zro, otφ rohs ininity Thus th oint o intrstion n mo with sd str thn i is suiintly lrg nd φ suiintly smll or xml, tk 500 m s nd φ you r worrid bout holding th rds stdy nough to b sur o th ngl, ut th dg o on rd long ur so tht th ngl will nssrily b suiintly smll t som l long th dg t us ssum th sinning lshlight is t th ntr o grin ltor, orming irulr srn o rdius R Th linr sd o th sot on th srn is gin by ω R, whr ω is th ngulr sd o rottion o th lshlight With suiintly lrg ω nd R, th sd o th sot moing on th srn n xd ontinud on nxt g

3 Chtr 9 45 Nithr o ths xmls iolts th rinil o rltiity Both ss r dsribing oint o intrstion: in th irst s, th intrstion o two rds nd in th sond s, th intrstion o light bm with srn A oint o intrstion is not md o mttr so it hs no mss, nd hn no nrgy A bug momntrily t th intrstion oint ould yl, tk bit out o on rd, or rlt th light Non o ths tions would rsult in ommunition rhing nothr bug so soon s th intrstion oint rhs him Th sond bug would h to wit or sound or light to trl ross th distn btwn th irst bug nd himsl, to gt th mssg As hild, th uthor usd n Ertor st to build surluminl sd gnrtor using th intrsting-rds mthod Cn you gt isibl dot to run ross omutr srn str thn light? Wnt s it gin? Q99 n this s, both th rltiisti nd Gliln trtmnts would yild th sm rsult: it is tht th xrimntlly obsrd sd o on r with rst to th othr is th sum o th sds o th rs Q90 Th hottr obt hs mor nrgy r molul thn th oolr on Th quiln o nrgy nd mss rdits tht h molul o th hottr obt will, on rg, h lrgr mss thn thos in th oolr obt This imlis tht gin th sm nt lid or, th oolr obt would h lrgr lrtion thn th hottr obt would xrin n ontrolld xrimnt, th dirn will likly b too smll to noti Q9 Sil rltiity dsribs inrtil rrn rms: tht is, rrn rms tht r not lrting Gnrl rltiity dsribs ll rrn rms Q9 Th downstirs lok runs mor slowly bus it is losr to th Erth nd hn in strongr grittionl ild thn th ustirs lok Q9 Th nts noti tht thy h strongr sns o bing ushd outwrd whn thy ntur losr to th rim o th mrry-go-round thy wish, thy n ll this th t o strongr grittionl ild rodud by som mss onntrtion towrd th dg o th disk An nt nmd Albrt igurs out tht th strong grittionl ild mks msuring rods ontrt whn thy r nr th rim o th disk H shows tht this t risly ounts or th disrny SUTNS T PRBEMS Stion 9 Th Prinil o Gliln Rltiity P9 n th rst rm, b gb g b gb g b g b g m + m 000 kg 0 0 m s kg 0 m s kg m s i i i m + m 000 kg kg kg m s Sin i, 49 ms 000 kg kg n th moing rm, ths loitis r ll rdud by +00 m/s ms ms 0 0 ms i i 0 ms + 00 ms 00 ms i i ur initil momntum is thn 49 ms ms 49 ms i m i + m i 000 kg 0 0 m s kg 0 0 m s kg m s nd our inl momntum is b b gb g b gb g 000 kg kg 500 kg 49 m s kg m s b g b gb g b b g g g 4

4 46 Rltiity P9 () m s T (b) 0 0 m s T B B T B () m s P9 Th irst obsrr wths som obt lrt undr lid ors Cll th instntnous loity o th obt Th sond obsrr hs onstnt loity rlti to th irst, nd msurs th obt to h loity d d Th sond obsrr msurs n lrtion o dt dt This is th sm s tht msurd by th irst obsrr n this nonrltiisti s, thy msur th sm ors s wll Thus, th sond obsrr lso onirms tht m P94 Th lbortory obsrr nots Nwton s sond lw to hold: m (whr th subsrit imlis th msurmnt ws md in th lbortory rm o rrn) Th obsrr in th lrting rm msurs th lrtion o th mss s (whr th subsrit imlis th msurmnt ws md in th lrting rm o rrn, nd th rimd lrtion trm is th lrtion o th lrtd rm with rst to th lbortory rm o rrn) Nwton s sond lw hld or th lrting rm, tht obsrr would thn ind lid th rltion m or m (sin nd th mss is unhngd in h) But, instd, th lrting rm obsrr will ind tht m m whih is not Nwton s sond lw Stion 9 Stion 9 Stion 94 Th Mihlson-Morly Exrimnt Einstin s Prinil o Rltiity Consquns o th Sil Thory o Rltiity P95 H G K J P96 t Tking whr 00 m gis t bg t H G t K J N M P Q t so H G t K J or t t H G K J P

5 P97 () γ bg 0500 Th tim intrl btwn ulss s msurd by th Erth obsrr is t t 60 0 s H G γ K J 094 s 750 Thus, th Erth obsrr rords uls rt o 60 0 smin 64 9 min 094 s Chtr 9 47 (b) At rlti sd 0990, th rltiisti tor γ inrss to 709 nd th uls rt rordd by th Erth obsrr drss to 0 6 min Tht is, th li sn o th stronut (rkond by th durtion o th totl numbr o his hrtbts) is muh longr s msurd by n Erth lok thn by lok bord th s hil *P9 () Th 0 nd th 0 ly r msurd in th Erth rm, x 0 ly 0 ly so in this rm, t 5 0 yr 0 0 ly yr (b) () W s lok on th mtoroid moing, so w do not msur ror tim; tht lok msurs ror tim t t γ t : t 5 0 yr γ 5 0 yr yr yr - Mthod on: W msur th 0 ly on stik sttionry in our rm, so it is ror lngth Th tourist msurs it to b ontrtd to 0 ly 0 ly 0 ly γ Mthod two: Th tourist ss th Erth rohing t 0 b0 ly yr gb5 yr g 0 ly Not only do distns nd tims dir btwn Erth nd mtoroid rrn rms, but within th Erth rm rnt distns dir rom tul distns As w h intrrtd it, th 0-lightyr tul distn rom th Erth to th mtoroid t th tim o disory must b lultd rsult, dirnt rom th distn msurd dirtly Bus o th init mximum sd o inormtion trnsr, th stronomr ss th mtoroid s it ws yrs riously, whn it ws muh rthr wy Cll its rnt distn d Th tim d rquird or light to rh us rom th nwly-isibl mtoroid is th lookbk tim t Th stronomr lults tht th mtoroid hs rohd to b 0 ly wy s it mod with onstnt loity throughout th lookbk tim W n work bkwrds to ronstrut hr lultion: 0 d d 0 ly + 0 t 0 ly + 0 d 0ly d 00 ly Thus in trms o dirt obsrtion, th mtoroid w s ors 00 ly in only 5 yrs Suh n rnt surluminl loity is tully obsrd or som ts o mtril mnting rom qusrs, bus thy hn to b ointd nrly towrd th Erth w n wth nts unold on th mtoroid, w s thm slowd by rltiisti tim diltion, but lso grtly sdd u by th Dolr t

6 4 Rltiity P99 t γ t t H so t t t G J nd t t t K K J 00 0 m 000 km h 77 ms 600 s thn nd t s 54 0 s 54 ns 9 t b g 6 P90 or 0990,γ 709 () Th muon s litim s msurd in th Erth s rst rm is t 460 km 0990 nd th litim msurd in th muon s rst rm is t t γ 709 P Q m m s P µ s (b) H G K J γ m m P9 Th sshi is msurd by th Erth obsrr to b lngth-ontrtd to or Also, th ontrtd lngth is rltd to th tim rquird to ss orhd by: t or t t Equting ths two xrssions gis t or + t Using th gin lus: 00 m nd t s 5 4 this boms 4 0 m m giing 000

7 Chtr 9 49 P9 () Th sshi is msurd by Erth obsrrs to b o lngth, whr nd t t H G Soling or, t + K nd t J t + 00 m (b) Th tnks mo nonrltiistilly, so w h 75 s 400 ms () or th dt in roblm, 00 m 6 0 ms s + 00 m 00 m in grmnt with roblm or th dt in rt (b), 00 m 0 ms 75 s + 00 m in grmnt with rt (b) m m m ms P9 W ind Coor s sd: Soling, Thn th tim riod o on orbit, GMm m r r N M Q + P Q 6 R+ h π GM P R+ h π T s b g 7 0 ms T s µ s 0 ms () Th tim dirn or orbits is t t γ t T t t + 7 km s P 9 µ s (b) or on orbit, t t 7 µ s Th rss rort is urt to on digit P94 γ 0 so 040

8 440 Rltiity P95 () Sin your shi is idntil to his, nd you r t rst with rst to your own shi, its lngth is 0 0 m (b) His shi is in motion rlti to you, so you msur its lngth ontrtd to 9 0 m () W h rom whih 9 0 m m nd 0 *P96 n th Erth rm, Sdo s tri lsts or tim x 0 0 ly t 05 yr 0950 ly yr Sdo s g dns only by th ror tim intrl t t 05 yr 095 γ yr during his tri Similrly or Goslo, x t 0 0 ly yr 0750 ly yr Whil Sdo hs lndd on Plnt X nd is witing or his brothr, h gs by 0 0 ly 0 0 ly 564 yr 0750 ly yr 0950 ly yr b g Thn Goslo nds u oldr by 7 64 yr yr yr 5 45 yr P97 () t γ t t 5 0 yr bg yr b g b gb g (b) d t yr ly yr 0 yr 4 7 ly () Th stronuts s Erth lying out th bk window t 0 700: b g b gb g d t yr ly yr 5 0 yr 0 5 ly (d) Mission ontrol gts signls or 0 yr whil th bttry is orting, nd thn or 47 yrs tr th bttry stos owring th trnsmittr, 47 ly wy: 0 yr yr 5 7 yr

9 Chtr 9 44 P9 Th orbitl sd o th Erth is s dsribd by m : Gm m r Gm N m kg 99 0 kg S 9 0 r m Th mximum rquny rid by th xtrtrrstrils is m r S E E 0 4 ms + 6 obs sour Hz Th minimum rquny rid is 6 obs sour Hz ms 00 0 ms 9 0 ms 00 0 ms ms 00 0 ms ms 00 0 ms Th dirn, whih lts thm igur out th sd o our lnt, is b g 0 6 Hz 0 4 Hz P99 () t b th rquny s sn by th r Thus, nd, i is th rquny o th rltd w, Combining gis sour + sour + + (b) Using th bo rsult, Hz Hz sour b sourg b + sourg sour whih gis sour Th bt rquny is thn bt sour λ b g b g b g 9 00 ms 00 0 Hz () bt 00 0 ms (d) 00 0 λ sour bt λ so 9 ms 00 m Hz 00 ms 000Hz 00 khz 0000 m b g bt λ 5 Hz 0000 m ms 0 mih

10 44 Rltiity P90 () Whn th sour mos wy rom n obsrr, th obsrd rquny is s obs sour + s whr s sour Whn s <<, th binomil xnsion gis H G K J + H G K J s s s s s + s s So, obs sour K J Th obsrd wlngth is ound rom λ λ s Sin, λ obs obs obs sour : λsour λsour λ obs sour g s λ λobs λ λ λ λ λ sour (b) sour nm λ K J 0 0 K J λ 97 nm b s s s s K J K J K J *P9 or th light s obsrd + obs λ obs sour + λ sour + λ sour 650 nm λ obs 50 nm ms Stion 95 Th orntz Trnsormtion Equtions *P9 t Suznn b ixd in rrn rom S nd s th two light-mission nts with oordints x 0, t 0, x 0, t µ s t Mrk b ixd in rrn rm S nd gi th nts oordint x 0, t 0, 9 µ s t () Thn w h t γ t x K J b 9 µ s µ s γb g g 6 0 ms (b) x x t s 55 0 m

11 P9 γ W r lso gin: 00 m, nd θ 0 0 (both msurd in rrn rm moing rlti to th rod) Thus, x os θ 00 m m nd y sin θ 00 m m x is ror lngth, rltd to x by x x γ Thror, x 0 0x 7 m nd y y 00 m (ngths rndiulr to th motion r unhngd) Chtr 9 44 G P9 x y () + b g gis 7 4 m (b) θ tn y x gis θ 0 *P94 Einstin s rsoning bout lightning striking th nds o trin shows tht th moing obsrr ss th nt towrd whih sh is moing, nt B, s ourring irst Th S-rm oordints o th nts w my tk s (x 0, y 0, z 0, t 0) nd (x 00 m, y 0, z 0, t 0) Thn th oordints in S r gin by th orntz trnsormtion Ent A is t ( x 0, y 0, z 0, t 0) Th tim o nt B is 7 s 0 K J 0 m t γ t x 0 00 m K J m s Th tim lsing bor A ours is 444 ns P95 () rom th orntz trnsormtion, th srtions btwn th blu-light nd rd-light nts r dsribd by γ 0 γ x x t 00 m s ms γ (b) Agin rom th orntz trnsormtion, x γxt: K J m s 50 0 ms 00 0 ms x 00 m 50 0 m s 00 0 s x 497 m 9 () t γ t x : t 00 0 s t 0 s ms ms 00 m

12 444 Rltiity Stion 96 Th orntz Vloity Trnsormtion Equtions P96 u x Entrris loity Klingon loity rom Eqution 96 ux u x u x G P96 ux P97 u x u x *P9 t rm S b th Erth rm o rrn Thn 07 Th omonnts o th loity o th irst srt r ux 06 os nd uy As msurd rom th S rm o th sond srt, ux u x u uy uy γ u x x Th mgnitud o u is nd its dirtion is t tn 05 6 bo th x -xis G P97 Stion 97 Rltiisti inr Momntum nd th Rltiisti orm o Nwton s ws P99 () γ mu; or n ltron moing t 000 0, γ bug b0000 g b g Thus, kg m s kg m s (b) ollowing th sm sts s usd in rt (), w ind t 0500, γ 5 nd 5 0 kg m s () At 0900, γ 9 nd kg m s

13 mu P90 Using th rltiisti orm, bug Chtr γ mu b g w ind th dirn rom th lssil momntum, mu: γ mu mu γ mu () Th dirn is 00% whn bγ gmu 0000 γ mu : γ 0990 thus 0990 H G u K J, nd u 04 (b) Th dirn is 00% whn bγ gmu 000 γ mu : γ 0900 thus 0900 H G u K J nd u 046 b g u b g u P9 mu γ mu mu γ : γ mu mu u b g mu 90 0 ms mu 00 0 ms + H G u K J H G u K J 4 mu P9 bug u mu boms m whih gis: u + or m u + nd u m + P9 Rltiisti momntum o th systm o rgmnts must b onsrd or totl momntum to b zro tr s it ws bor, w must h, with subsrit rrring to th hir rgmnt, nd subsrit to th lightr, 50 0 kg or γ mu γ mu or Proding to sol, w ind 67 0 b u kg u 7 g u u u nd u 05 kg

14 446 Rltiity Stion 9 Rltiisti Enrgy P94 E γ γ m b g or n ltron, m 05 MV () E m H (b) E m G J P95 W K Ki or W () W J K K G m J H G d i b i g d i b i g m MV 45 MV K J m kg m s (b) W 57 0 J W kg m s W 0 9 J P96 Th rltiisti kinti nrgy o n obt o mss m nd sd u is K r u or u 000, Kr J m m 0000 Th lssil qution K mu gis K m m H G K dirnt by 075% or still smllr sds th grmnt will b still bttr m

15 Chtr P97 E γ m m or γ Thus, u H G K J or γ u Th momntum is thn mu m γ m H G H G 9 MV 6 0 MV H G K J 5 P9 () Using th lssil qution, K m 7 0 kg 06 0 m s 4 0 J (b) Using th rltiisti qution, K K M K J b g m H G K J b g Pb g H G K J H G K J 70 kg 99 0 m s 4 0 J Whn <<, th binomil sris xnsion gis Thus, + H G K J H G K J nd th rltiisti xrssion or kinti nrgy boms K m m H G K J Tht is, in th limit o sds muh smllr thn th sd o light, th rltiisti nd lssil xrssions yild th sm rsults 7 P99 () ER m 67 0 kg 99 0 m s 50 0 J 9 MV 0 (b) E γ m 50 0 b g J J 00 0 MV () K E m 4 0 J 50 0 J 0 J 07 0 MV P940 Th rltiisti dnsity is ER m m m V V V u 00 g b g m 4 g m

16 44 Rltiity P94 W must onsr both nrgy nd rltiisti momntum o th systm o rgmnts With subsrit rrring to th 0 6 rtil nd subsrit to th 0 97 rtil, γ 0 nd γ Consrtion o nrgy gis E + E E totl whih is γ m + γ m m totl 7 or 0 m + 6 m 4 0 kg 7 This rdus to: m + 09m 66 0 kg () Sin th inl momntum o th systm must qul zro, gis γ mu γ mu or 0 0 6m m whih boms m 5 m () G P94 Soling () nd () simultnously, m 4 0 kg nd m 5 0 kg *P94 Enrgy onsrtion: 0 400kg 900 kg M kg M 900 kg 0 5 M Momntum onsrtion: kg M () Diiding gis (b) Now by substitution 0 kg M 75 0 kg P94 E γ m E γ m γ mu bγ mug u u γ bγ g γ m H K E m mu m m u m QED b g P944 () q V K γ m Thus, γ b g u q V + m rom whih u 00 b g (b) K γ m q V 60 0 C 50 0 J C J

17 Chtr P945 () E γ m 0 0 GV with m 05 MV or ltrons Thus, V γ V (b) γ b g u rom whih u () u H G K J γ 9 0 m m 767 m nrgy J *P946 () P t 00 0 s W (b) Th kinti nrgy o on ltron with is H K 4 bγ gm 9 0 kg G J 0 m s J 57 0 Thn w rquir 00 J J J 7 N J J N P947 Consring totl momntum o th dying rtil systm, 0 bor dy γm u γ 07m u µ µ tr dy b g Consrtion o mss-nrgy or th systm gis Eµ + E Eπ: γ mµ + mπ γb07mg+ 7m Substituting rom th momntum qution bo, γ 07m + γ 07m u 7m u or γ + b g b g 7 + u u K J : u 070 µ b g µ b g : K µ 0 70 Thn, K γ m γ 07 m K µ 40 MV Also, E Eπ Eµ : E m π γ m µ 7 07γ m E E 7 b MV 9 6 MV g MV

18 450 Rltiity *P94 t obsrr A hold th unrimd rrn rm, with u u rst in th rimd rm with u 0 u u () Thn u u b gb g 4 sd u nd u t obsrr B b t 4 4 u 4 (b) n th unrimd rm th obts, h o mss m, togthr h nrgy m γ m + γ m 0 m 075 n th rimd rm th nrgy is 457 m 0 m m m tims grtr s msurd by obsrr B 457 m, grtr by Stion 99 Mss nd Enrgy P949 t 0-kg lg b run u lgol 7 m high W ut into it nrgy mgh 0 kg 9 m s 7m 0J So w ut into it xtr mss m E or rtionl inrs o kg kg 0 0 J 5 ms ~ kg P950 E J Also, th mss-nrgy rltion sys tht E m Thror, m E J 00 0 m s 0 No, mss loss o this mgnitud (out o totl o 900 g) ould not b dttd P95 m E t P Js 00 yr 6 0 syr 00 0 ms kg 9 7 b g 04 kg 9 b g P95 m E m T V T ρ 00 kg m m 46 J kg C 00 C 00 0 ms m 67 0 kg

19 6 W de dm dm P95 P 77 0 dt dt dt 6 Thus, dm dt 77 0 Js 00 0 ms kg s Chtr 9 45 P954 m 0 MV E γ 0 MV Stion 90 Th Gnrl Thory o Rltiity *P955 () or th stllit m (b) GM T r E 4π r r 66 0 : GM E m r m r m r π r T H G K J N m kg 4 00 s kg 4π 7 m π r π 66 0 T 4 00 s b g 7 m 7 0 ms () (d) Th smll rtionl drs in rquny rid is qul in mgnitud to th rtionl inrs in riod o th moing osilltor du to tim diltion: rtionl hng in γ b g K J 4 0 Th orbit ltitud is lrg omrd to th rdius o th Erth, so w must us GMEm U g r U g Nm kg m Nm kg m kg 66 0 m kg 67 0 m Jkgm U 7 g m s m 0 ms 0 0 ()

20 45 Rltiity Additionl Problms 6 P956 () drth trth γ tstro so 00 0 yr 0 0 yr (b) 50 0 H G K J K 5 so () J J kwh P957 () m b gb g 0 yr 7 7 h 0 H G K J $ 7 0 H G K J k W s H G 600 K J 0 J s MV bγ gm so γ 0 0 yr kg 0 m s J t 0 yr 5 t 0 yr ~ 0 s 0 γ 0 (b) d t ~0 km 5 P95 () Whn K K, m γ m γ n this s, m b g 05 MV, m nd γ Substituting, but m γ γ + + m γ u 9 MV b g b g Thror, u γ 006 (b) Whn γ mu γ mu or γ u Thus, γ u 0 5 MV MV γ mu m b g MV MV 4 nd u u 4 H G K J whih yilds u km s 4

21 Chtr 9 45 P959 () Sin Mry is in th sm rrn rm, S, s Td, sh msurs th bll to h th sm sd Td obsrs, nmly u x m (b) t u m s x 0600 () 0 0 m 44 0 m Sin 0600 nd u x 000, th loity Jim msurs or th bll is u x u + x + u x (d) Jim msurs th bll nd Mry to b initilly srtd by 44 0 m Mry s motion t 0600 nd th bll s motion t 05 nibbl into this distn rom both nds Th g loss t th rt , so th bll nd thr mt tr tim 44 0 m t m s 4 0 *P960 () Th hrgd bttry stors nrgy E P t 0 J s 50min 60s min 600J P96 (b) m m so its mss xss is m E 4 m m kg kg MV 7 4 MV 00% 0 7% 4 97 MV s s b g b g 600J 0 m s too smll to msur kg *P96 Th nrgy o th irst rgmnt is gin by E + m 75 MV + 00 MV E 0 MV or th sond, E 00 MV + 50 MV E 50 MV () Enrgy is onsrd, so th unstbl obt hd E 45 MV Eh omonnt o momntum is onsrd, so th originl obt mod with 00 x + y H G 75 MV K J + H G MV K J Thn or it 45 MV 75 MV + 00 MV + m 65 MV m (b) Now E γ m gis 45 MV 65 MV 0654, 059

22 454 Rltiity P96 () Tk th sshi s th rimd rm, moing towrd th right t (b) Thn ux + 000, nd u x u x + + u b x g b g : 0 00 ly ly γ 0946 () (d) Th lins obsr th 060-ly distn losing bus th rob nibbls into it rom on nd t 000 nd th Erth rdus it t th othr nd t ly Thus, tim K u m : K yr kg 00 0 m s K J P964 n this s, th ror tim is T 0 (th tim msurd by th studnts on lok t rst rlti to thm) Th diltd tim msurd by th rossor is: t γt 0 whr t T+ t Hr T is th tim sh wits bor snding signl nd t is th tim rquird or th signl to rh th studnts Thus, w h: T + tγ T0 () To dtrmin th trl tim t, rliz tht th distn th studnts will h mod byond th rossor bor th signl rhs thm is: d T+ t Th tim rquird or th signl to trl this distn is: Soling or t gis: Substituting this into qution () yilds: or Thn T T 0 bg T bg + b g b g T 0 0 b g + d t T t H G K J + b g b g bt g bg T t T + γ T T γ T0 0

23 Chtr P965 ook t th sitution rom th instrutors iwoint sin thy r t rst rlti to th lok, nd hn msur th ror tim Th Erth mos with loity 00 rlti to th instrutors whil th studnts mo with loity u 0600 rlti to Erth Using th loity ddition qution, th loity o th studnts rlti to th instrutors (nd hn th lok) is: + u u + u (studnts rlti to lok) () With ror tim intrl o t 50 0 min, th tim intrl msurd by th studnts is: t γ with γ t Thus, th studnts msur th xm to lst T min 760 minuts (b) Th durtion o th xm s msurd by obsrrs on Erth is: t γ with γ t 0 0 so T min 5 minuts P966 Th nrgy whih rris in on yr is E P t 79 0 Js 6 0 s J Thus, m E J 00 0 m s kg P967 Th obsrr msurs th ror lngth o th tunnl, 500 m, but msurs th trin ontrtd to lngth 00 m 0950 m shortr thn th tunnl by 50 0 m so it is omltly within th tunnl P96 th nrgy rquird to rmo mss m rom th sur is qul to its rst nrgy m, thn nd GMsm m R R g g GMs N m kg 99 0 kg 00 0 ms R g 47 0 m 47 km

24 456 Rltiity P969 () At ny sd, th momntum o th rtil is gin by γ mu mu bug Sin qe d : qe dt So nd d dt mu u u qe m qe m du dt du u + mu dt u + u u du qe u dt m u du K J dt (b) qe or u smll omrd to, th rltiisti xrssion rdus to th lssil As u m rohs, th lrtion rohs zro, so tht th obt n nr rh th sd o light () u t du z u tz 0 0 t z x udt qe 0 t z 0 qe m dt tdt m + q E t u m H qet + q E t x m + q E t m qe K P970 () An obsrr t rst rlti to th mirror ss th light trl distn D dx, whr x t S is th distn th shi mos towrd th mirror in tim t S Sin this obsrr grs tht th sd o light is, th tim or it to trl distn D is t S D d t S d t S + (b) Th obsrr in th rokt msurs lngth-ontrtd initil distn to th mirror o d nd th mirror moing towrd th shi t sd Thus, h msurs th distn th light t trls s D byg whr y is th distn th mirror mos towrd th shi bor th light rlts rom it This obsrr lso msurs th sd o light to b, so th tim or it to trl distn D is: D t d t d so + t + or t d +

25 Chtr *P97 Tk th two olliding rotons s th systm E K+ m E m 4 E + m 0 n th inl stt, E K + M 4 : E + M By nrgy onsrtion, E + E E, so E + EE + E E m + K+ m m + m + M By onsrtion o momntum, Thn Km M Km + 4m +4m m K M m + m By ontrst, or olliding bms w h n th originl stt, E K+ m E K+ m n th inl stt, E M 4 G P97 E + E E : K+ m + K+ m M M K m + m K J *P97 Consrtion o momntum γ mu: mu mu M mu + u u u Consrtion o nrgy γ m : m m M + u u 4m u u u To strt soling w n diid: 4 Thn M u 4 M m 4 u u 4m M u 4 u b g Not tht whn <<, this rdus to M 4m, in grmnt with th lssil rsult

26 45 Rltiity 0 0x 0y x y P97 () + nd + Th motion is in th x dirtion: sinθ Thus, or y 0y (b) tnθ γ tnθ0 x 0x b g y x 0y 0 0 b g x H G K J H G 0 0 osθ 0 K J H G 0 K J 0 M H G K J + H G K J 0 0 os θ sin θ0 0 os θ0 os θ 0 P974 (b) Considr hrmit who lis on n stroid hlwy btwn th Sun nd Tu Cti, sttionry with rst to both Just s our sshi is ssing him, h lso ss th blst ws rom both xlosions Judging both strs to b sttionry, this obsrr onluds tht th two strs blw u simultnously () W in th sshi moing st th hrmit do not lult th xlosions to b simultnous W msur th distn w h trld rom th Sun s H G K J b g ly ly W s th Sun lying wy rom us t 000 whil th light rom th Sun rohs t 00 Thus, th g btwn th Sun nd its blst w hs ond t 0, nd th tim w lult to h lsd sin th Sun xlodd is 60 ly 00 yr 0 W s Tu Cti s moing towrd us t 000, whil its light rohs t 00, only 000 str W msur th g btwn tht str nd its blst w s 60 ly nd growing t 000 W lult tht it must h bn oning or 60 ly 0 yr 000 nd onlud tht Tu Cti xlodd 60 yrs bor th Sun

27 P975 Sin th totl momntum is zro bor dy, it is nssry tht tr th dy Eγ 4 0 kv nulus hoton Also, or th roiling nulus, E m + with m 9 Thus, m + K 4 0 kv + m K or + m + + H G K J + H G K J K So kv kv m m m nd P976 Tk m 00 kg 4 0 kv V K 9 m 5 0 V 60 0 J 5 GV 4 0 K J H G kv K J + m 0 Th lssil kinti nrgy is K mu m nd th tul kinti nrgy is u K J Kr J K 0990 Kr, whn u K J 0990 K (Binomil Thorm) V u u H G K J H G K J 6 J r m H G 6 u K J H G b g b g u Similrly, K 0950 K whn u 057 nd K 0500 K whn u 076 r r, yilding u 05 Chtr J K J J bug G P976

28 460 Rltiity ANSWERS T EVEN PRBEMS P9 () 60 0 m s ; (b) 0 0 m s; () 44 7 m s P944 () 0 0; (b) 4 00 J P94 s th solution P946 () 00 TW; (b) ltrons P P94 () 0960; (b) 5 tims grtr s msurd by B P9 () 50 yr; (b) 50 yr; () 0 ly P950 0 kg, not dttbl P90 () µ s; (b) Th moon ss th lnt sur moing 649 m u towrd it P kg P9 () t + () s th solution P ; (b) 400 m s; P954 0 MV P956 () 0 0 ; (b) J; () $7 0 0 P yr, Goslo is oldr P95 () 0 0 6; (b) P9 khz P960 () kg ; (b) 60 0 P90 () s th solution; (b) P9 () 0 94; (b) 55 km P94 B ourrd 444 ns bor A P P9 09 t 6 bo th x -xis P90 () 0 4 ; (b) 0 46 P9 s th solution P94 () 05 MV; (b) 45MV P96 () 65 MV P964 s th solution P kg P96 47 km d P970 () + ; (b) d P97 M m 4 u u ; (b) P96 s th solution P974 () Tu Cti xlodd 60 yr bor th Sun; (b) thy xlodd simultnously P9 () 4 GJ ; (b) 4 GJ P940 4 g m P94 () 0467; (b) 75 0 kg P976 s th solution, 0 5, 0 57, 0 76