3. As long as the railroad car is traveling with a constant velocity, the ball will land back in his hand.

Transcription

1 CHAPTER 36: The Seia Theor of Reaii Resonses o Quesions. No. The rain is an ineria referene frame, and he aws of hsis are he same in a ineria referene frames, so here is no eerimen ou an erform inside he rain ar o deermine if ou are moing.. The fa ha ou insinie hink ou are moing is onsisen wih he reaii rinie aied o mehanis. Een hough ou are a res reaie o he ground, when he ar ne o ou rees forward, ou are moing bakward reaie o ha ar. 3. As ong as he rairoad ar is raeing wih a onsan eoi, he ba wi and bak in his hand. 4. The reaii rinie refers on o ineria referene frames. Neiher he referene frame of he Earh nor he referene frame of he Sun is ineria. Eiher referene frame is aid, bu he aws of hsis wi no be he same in eah of he frames. 5. The sarigh woud ass a, regardess of our saeshi s seed. This is onsisen wih he seond osuae of reaii whih saes ha he seed of igh hrough em sae is indeenden of he seed of he soure or he obserer. 6. I deas wih sae-ime (someimes aed he fabri of sae-ime ) and he aua assage of ime in he referene frame, no wih he mehania workings of oks. An measuremen of ime (hearbeas or dea raes, for insane) woud be measured as sower han norma when iewed b an obserer ouside he moing referene frame. 7. Time aua asses more sow in he moing referene frames, aording o obserers ouside he moing frames.. This siuaion is an eame of he win arado aied o aren-hid insead of o wins. This migh be ossibe if he woman was raeing a high enough seeds during her ri. Time woud hae assed more sow for her and she oud hae aged ess han her son, who saed on Earh. (Noe ha he siuaions of he woman and son are no smmeri; she mus undergo aeeraion during her journe.) 9. No, ou woud no noie an hange in our hearbea, mass, heigh, or waisine, beause ou are in he ineria frame of he saeshi. Obserers on Earh, howeer, woud reor ha our hearbea is sower and our mass greaer han if ou were a res wih rese o hem. Your heigh and waisine wi deend on our orienaion wih rese o he moion. If ou are sanding u in he saeshi suh ha our heigh is erendiuar o he direion of rae, hen our heigh woud no hange bu our waisine woud shrink. If ou haened o be ing down so ha our bod is arae o he direion of moion when he Earh obserers eer hrough he eesoe, hen ou woud aear shorer bu our waisine woud no hange.. Yes. Howeer, a a seed of on 9 km/hr, / is er sma, and herefore γ is er ose o one, so he effes woud no be noieabe. 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 47

2 Phsis for Sieniss & Engineers wih Modern Phsis, 4 h Ediion Insruor Souions Manua. Lengh onraion and ime diaion woud no our. If he seed of igh were infinie, / woud be zero for a finie aues of, and herefore γ woud awas be one, resuing in and.. The effes of seia reaii, suh as ime diaion and engh onraion, woud be noieabe in our eerda aiiies beause eerda seeds woud no onger be so sma omared o he seed of igh. There woud be no absoue ime on whih we woud a agree, so i woud be more diffiu, for insane, o an o mee friends for unh a a erain ime! In addiion, 5 m/s woud be he imiing seed and nohing in he unierse woud moe faser han ha. 3. Boh he engh onraion and ime diaion formuas inude he erm. If were no he imiing seed in he unierse, hen i woud be ossibe o hae a siuaion wih >. Howeer, his woud resu in a negaie number under he square roo, whih gies an imaginar number as a resu, indiaing ha mus be he imiing seed. 4. Mr. Tomkins aears shrunk in he horizona direion, sine ha is he direion of his moion, and norma size in he eria direion, erendiuar o his direion of moion. This engh onraion is a resu of he fa ha, o he eoe on he sidewak, Mr. Tomkins is in a moing frame of referene. If he seed of igh were on mi/h, hen he amoun of onraion, whih deends on γ, woud be enough o be noieabe. Therefore, Mr. Tomkins and his bie aear er skinn. (Comare o he haer-oening figure, whih is shown from Mr. Tomkin s iewoin. In his ase, Mr. Tomkins sees himsef as norma bu a he objes moing wih rese o him are onraed.) m 5. No. The reaiisi momenum of he eeron is gien b m. A ow seeds (omared o ) his redues o he assia momenum, = m. As aroahes, γ aroahes infini so here is no uer imi o he eeron s momenum. 6. No. To aeerae a arie wih nonzero res mass u o he seed of igh woud require an infinie amoun of energ, and so is no ossibe. 7. No. E = m² does no onfi wih he rinie of onseraion of energ as ong as i is undersood ha mass is a form of energ.. Yes, mass is a form of energ so ehnia i is orre o sa ha a sring has more mass when omressed. Howeer, he hange in mass of he sring is er sma and essenia negigibe. 9. Energ an be neiher reaed nor desroed. Mass is a form of energ, and mass an be desroed when i is onered o oher forms of energ. The oa amoun of energ remains onsan.. Tehnia es, he noion ha eoiies sim add is wrong. Howeer, a eerda seeds, he reaiisi equaions redue o assia ones, so our ideas abou eoi addiion are essenia rue for eoiies ha are ow omared o he seed of igh. 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 4

3 Chaer 36 The Seia Theor of Reaii Souions o Probems. You measure he onraed engh. Find he res engh from Eq. 36-3a. 3.m 7.5m.5. We find he ifeime a res from Eq. 36-a. 6.7 m s s.7 s 3. m s 3. The numeria aues and grah were generaed in a sreadshee. The grah is shown aso. The sreadshee used for his robem an be found on he Media Manager, wih fiename PSE4_ISM_CH36.XLS, on ab Probem The measured disane is he onraed engh. Use Eq. 36-3a.. m s m s 5. The seed is deermined from he ime diaion reaionshi, Eq. 36-a..6 s.7.4 m s 4.4 s 6. The seed is deermined from he engh onraion reaionshi, Eq. 36-3a m s 7. The seed is deermined from he engh onraion reaionshi, Eq. 36-3a. Then he ime is found from he seed and he onraed disane Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 49

4 Phsis for Sieniss & Engineers wih Modern Phsis, 4 h Ediion Insruor Souions Manua 5 5 ; The seed is deermined from he engh onraion reaionshi, Eq. 36-3a The hange in engh is deermined from he engh onraion reaionshi, Eq. 36-3a. The seed is er sma omared o he seed of igh. / 3. m s m s So he eren derease is 6.97 %.. (a) The measured engh is he onraed engh. We find he res engh from Eq. 36-3a. 4.m 7.39m.76 Disanes erendiuar o he moion do no hange, so he res heigh is.35m. (b) The ime in he saeraf is he res ime, found from Eq. 36-a..s.76 3.s () To our friend, ou moed a he same reaie seed:.76. (d) She woud measure he same ime diaion: 3.s.. (a) We use Eq. 36-3a for engh onraion wih he onraed engh 99.% of he res engh (b) We use Eq. 36-a for ime diaion wih he ime as measured from a reaie moing frame.% greaer han he res ime..4. We see ha a seed of.4 resus in abou a % reaiisi effe.. (a) To an obserer on Earh,.6 is he res engh, so he ime wi be he disane diided b he seed..6 Earh 9.5 r 9.6 r.95 (b) The ime as obsered on he saeraf is shorer. Use Eq. 36-a. 9.5r r 6.r 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 43

5 Chaer 36 The Seia Theor of Reaii () To he saeraf obserer, he disane o he sar is onraed. Use Eq. 36-3a (d) To he saeraf obserer, he seed of he saeraf is heir obsered disane diided b heir obsered ime r 3. (a) In he Earh frame, he ok on he Enerrise wi run sower. Use Eq. 36-a. 5.r r (b) Now we assume he 5. ears is he ime as measured on he Enerrise. Again use Eq. 36-a. 5.r 7.4 r We find he seed of he arie in he ab frame, and use ha o find he res frame ifeime and disane. ab.m.94 m s.93 9 ab 3.4 s (a) Find he res frame ifeime from Eq. 36-a. 9 ab 3.4 s s (b) In is res frame, he arie wi rae he disane gien b is seed and he res ifeime..94 m s 6.7 s.9m This oud aso be found from he engh onraion reaionshi: ab. 5. Sine he number of aries assing er seond is redued from N o N /, a ime T mus hae eased in he aries res frame. The ime T eased in he ab frame wi be greaer, aording o Eq. 36-a. The aries moed a disane of T in he ab frame during ha ime. T T 4 T T T ; 5.94 T T 6. The dimension aong he direion of moion is onraed, and he oher wo dimensions are unhanged. Use Eq. 36-3a o find he onraed engh ; V.m. 4.m 7. The eria dimensions of he shi wi no hange, bu he horizona dimensions wi be onraed aording o Eq. 36-3a. The base wi be onraed as foows. base When a res, he ange of he sides wih rese o he base is gien b os The eria omonen of er sin sin is unhanged. The horizona 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 43

6 Phsis for Sieniss & Engineers wih Modern Phsis, 4 h Ediion Insruor Souions Manua omonen, whih is os.5 a res, wi be onraed in he same wa as he base. horizona Use he Phagorean heorem o find he engh of he eg. 4 eg horizona er In he Earh frame, he aerage ifeime of he ion wi be diaed aording o Eq. 36-a. The seed of he ion wi be he disane moed in he Earh frame imes he diaed ime. d d m s.6 s d 5m 9. We ake he osiie direion in he direion of he Enerrise. Consider he aien esse as referene frame S, and he Earh as referene frame S. The eoi of he Earh reaie o he aien esse is.6. The eoi of he Enerrise reaie o he Earh is u.9. Soe for he eoi of he Enerrise reaie o he aien esse, u, using Eq. 36-7a. u.9.6 u.65 u.6.9 We oud aso hae made he Enerrise as referene frame S, wih.9, and he eoi of he aien esse reaie o he Earh as u.6. The same answer woud resu. Choosing he wo saeraf as he wo referene frames woud aso work. Le he aien esse be referene frame S, and he Enerrise be referene frame S. Then we hae he eoi of he Earh reaie o he aien esse as u.6, and he eoi of he Earh reaie o he Enerrise as u.9. We soe for, he eoi of he Enerrise reaie o he aien esse. u u.6.9 u u.65 u uu.9.6. The Gaiean ransformaion is gien in Eq (a),, z,, z 5m 3m s 3.5s,m, 3m,m, (b),, z,, z 5m 3m s.s,m, 35m,m,. (a) The erson s oordinaes in S are found using Eq. 36-6, wih 5 m, m, z, and 3.5 s. We se. m/s. 5m. m/s 3.5 s. m/s 3. m/s m ; z z m 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 43

7 Chaer 36 The Seia Theor of Reaii (b) We reea ar (a) using he ime. s. 5m. m/s. s. m/s 3. m/s m m ; z z. We deermine he omonens of her eoi in he S frame using Eq. 36-7, where u u. m/s and. m/s. Then using rigonomer we ombine he omonens o deermine he magniude and direion. u. m/s. m/s u u /. m/s. m/s / 3. m/s.3 m/s u. m/s. m/s 3. m/s u /. m/s. m/s / 3. m/s u 7 7. m/s u u u 7.3 m/s 7. m/s.49 m/s u an an m/s u.3 m/s 3. (a) We ake he osiie direion o be he direion of moion of saeshi. Consider saeshi as referene frame S, and he Earh referene frame S. The eoi of he Earh reaie o saeshi is.6. The eoi of saeshi reaie o he Earh is u.6. Soe for he eoi of saeshi reaie o saeshi, u, using Eq. 36-7a. u.6.6 u. u.6.6 (b) Now onsider saeshi as referene frame S. The eoi of he Earh reaie o saeshi is.6. The eoi of saeshi reaie o he Earh is u.6. Soe for he eoi of saeshi reaie o saeshi, u, using Eq. 36-7a. u.6.6 u. u.6.6 As eeed, he wo reaie eoiies are he oosie of eah oher. 4. (a) The Gaiean ransformaion is gien in Eq m.9 3. m s. s 376m (b) The Lorenz ransformaion is gien in Eq Noe ha we are gien, he ok reading in frame S. 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 433

8 Phsis for Sieniss & Engineers wih Modern Phsis, 4 h Ediion Insruor Souions Manua.9 36m 6 m m s. s.9 m 5. (a) We ake he osiie direion in he direion of he firs saeshi. We hoose referene frame S as he Earh, and referene frame S as he firs saeshi. So.6. The seed of he seond saeshi reaie o he firs saeshi is u.7. We use Eq. 36-7a o soe for he seed of he seond saeshi reaie o he Earh, u. u.7.6 u.97 u.6.7 (b) The on differene is now ha u.7. u u.7.6 u The robem asks for he seed, whih woud be We assume ha he gien seed of.9 is reaie o he ane ha ou are aroahing. We ake he osiie direion in he direion ha ou are raeing. Consider our saeshi as referene frame S, and he ane as referene frame S. The eoi of he ane reaie o ou is.9. The eoi of he robe reaie o he ane is u.95. Soe for he eoi of he robe reaie o our saeshi, u, using Eq. 36-7a. u.95.9 u.34 u We se frame S as he frame a res wih he saeshi. In his frame he modue has seed u u.. Frame S is he frame ha is saionar wih rese o he Earh. The saeshi, and herefore frame S moes in he -direion wih seed.76 in his frame, or.76. We use Eq. 36-7a and 36-7b o deermine he omonens of he modue eoi in frame S. Then using rigonomer we ombine he omonens o deermine he seed and direion of rae. u u.76 ; u.533 u u/ u/ u.533 u.76 u u u ; an an 35. The eoi omonens of he arie in he S frame are u u os and u u sin. We find he omonens of he arie in he S frame from he eoi ransformaions gien in Eqs. 36-7a and 36-7b. Those ransformaions are for he S frame moing wih seed reaie o he S frame. We an find he ransformaions from he S frame o he S frame b sim hanging o and rimed o unrimed ariabes. u u u u u u u u ; u u u u 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 434

9 Chaer 36 The Seia Theor of Reaii an u u u u usin sin u u u uos os u u 9. (a) In frame S he horizona omonen of he sik engh wi be onraed, whie he eria omonen remains he same. We use he rigonomeri reaions o deermine he - and - omonens of he engh of he sik. Then using Eq. 36-3a we deermine he onraed engh of he -omonen. Fina, we use he Phagorean heorem o deermine sik engh in frame S. os ; sin ; os os sin os (b) We auae he ange from he engh omonens in he moing frame. sin an an an an an os an 3. (a) We hoose he rain as frame S and he Earh as frame S. Sine he guns fire simuaneous in S, we se hese imes equa o zero, ha is A B. To simif he robem we aso se he oaion of gunman A equa o zero in frame S when he guns were fired, A. This aes gunman B a B 55.m. Use Eq o deermine he ime ha eah gunman fired his weaon in frame S. A A A 35m/s 55. m B 4 B B.4 s Therefore, in Frame S, A fired firs. 35. m/s 3. m/s 3. m/s 4 (b) As found in ar (a), he differene in ime is.4 s. () In he Earh frame of referene, sine A fired firs, B was sruk firs. In he rain frame, A is moing awa from he bue fired oward him, and B is moing oward he bue fired oward him. Thus B wi be sruk firs in his frame as we. 3. We se frame S as he frame moing wih he obserer. Frame S is he frame in whih he wo igh bubs are a res. Frame S is moing wih eoi wih rese o frame S. We soe Eq for he ime in erms of,, and. Using he resuing equaion we deermine he ime in frame S ha eah bub is urned on, gien ha in frame S he bubs are urned on simuaneous a A B. Taking he differene in hese imes gies he ime inera as measured b he obsering moing wih eoi. = 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 435

10 Phsis for Sieniss & Engineers wih Modern Phsis, 4 h Ediion Insruor Souions Manua ; A B A A B B B A Aording o he obserer, bub B urned on firs. 3. We se u he wo frames suh ha in frame S, he firs obje is oaed a he origin and he seond obje is oaed meers from he origin, so A and B m. We se he ime when een A ourred equa o zero, so A and B. s. We hen se he oaion of he wo eens in frame S equa, and using Eq we soe for he eoi. A B m A B A A B B ;.5 m/s. s 33. From he bo s frame of referene, he oe remains a res wih rese o him. As suh, he oe wi awas remain. m ong. As he bo runs oward he barn, reaii requires ha he (reaie moing) barn onra in size, making he barn een shorer han is res engh of. m. Thus i is imossibe, in he bo s frame of referene, for he barn o be onger han he oe. So aording o he bo, he oe wi neer omee fi wihin he barn. In he frame of referene a res wih rese o he barn, i is ossibe for he oe o be shorer han he barn. We use Eq. 36-3a o auae he seed ha he bo woud hae o run for he onraed engh of he oe,, o equa he engh of he barn. A B. m. m.55 If ersons sanding a he fron and bak door of he barn were o ose boh doors ea when he oe was omee inside he barn, we woud hae wo simuaneous eens in he barn s res frame S wih he oe omee inside he barn. Le us se he ime for hese wo eens as A B. In frame S hese wo eens our a he fron and far side of he barn, or a A and B.m. Using Eq. 36-6, we auae he imes a whih he barn doors ose in he bo s frame of referene. A A A B B.55.m B. s m/s Therefore, in he bo s frame of referene he far door of he barn osed. ns before he fron door. If we mui he seed of he bo b his ime differene, we auae he disane he bo raeed beween he osing of he wo doors m/s. s 3.67 m. We use Eq. 36-3a o deermine he engh of he barn in he bo s frame of referene.. m m Subraing he disane raeed beween osing he doors from he engh of he oe, we find he engh of he barn in he bo s frame of referene.. m 3.67 m.33 m,oe barn Therefore, in he bo s frame of referene, when he fron of he oe reahed he far door i was osed. Then. ns aer, when he bak of he oe reahed he fron door, ha door was osed. In he bo s frame of referene hese wo eens are no simuaneous. 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 436

11 Chaer 36 The Seia Theor of Reaii 34. The momenum of he roon is gien b Eq m.67 kg m s m kg m s 35. (a) We omare he assia momenum o he reaiisi momenum. assia m..995 m reaiisi The assia momenum is abou.5% in error. (b) We again omare he wo momena. assia m.6. m reaiisi The assia momenum is % in error. 36. The momenum a he higher seed is o be wie he iniia momenum. We designae he iniia sae wih a subsri, and he fina sae wih a subsri f. m f f f f f.6 f m f.6.9 f f The wo momena, as measured in he frame in whih he arie was iniia a res, wi be equa o eah oher in magniude. The igher arie is designaed wih a subsri, and he heaier arie wih a subsri. m m kg.6 7 m.67 kg m We find he roon s momena using Eq m m.45 m m m ;..3333m.45. m m m.9 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 437

12 Phsis for Sieniss & Engineers wih Modern Phsis, 4 h Ediion Insruor Souions Manua (a) (b).3333m.539m %.539m 4.947m.3333m %.3333m 39. The res energ of he eeron is gien b Eq E m kg 3. m s. J 4. J 3.6 J MeV.5MeV 4. We find he oss in mass from Eq E MeV.6 J MeV m 3. m s 3.56 kg 4 kg 4. We find he mass onersion from Eq E J m 9kg 3. m s 4. We auae he mass from Eq kg.9979 m s 93. MeV 3 E m m.6 J MeV 43. Eah hoon has momenum.5 MeV/. Thus eah hoon has mass.5 MeV. Assuming he hoons hae oosie iniia direions, hen he oa momenum is, and so he rodu mass wi no be moing. Thus a of he hoon energ an be onered ino he mass of he arie. Aording, he heaies arie woud hae a mass of 3.MeV, whih is.7 kg (a) The work is he hange in kinei energ. Use Eq. 36-b. The iniia kinei energ is. W K K m fina 3.9GeV MeV.39 MeV 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher (b) The momenum of he roon is gien b Eq m 93.3MeV.99.4 MeV 4.GeV We find he energ equiaen of he mass from Eq E m 3 3. kg 3. m s 9. J We assume ha his energ is used o inrease he graiaiona oenia energ. 3 E 9. J 9 E mgh m 9. kg hg 3. m 9.m s

13 Chaer 36 The Seia Theor of Reaii 46. The work is he hange in kinei energ. Use Eq. 36-b. The iniia kinei energ is. W m ; W K K m m W W m m m The kinei energ is gien b Eq K 3 m m The oa energ of he roon is he kinei energ us he mass energ. Use Eq o find he momenum. E K m ; E m K m m K K m m 93.3MeV 95 MeV 63MeV K 95 MeV K K m K 63MeV.6GeV We find he seed in erms of. The kinei energ is gien b Eq. 36- and he momenum b Eq m s m s K m 93.3MeV MeV.67GeV.9333 m 93.3MeV MeV.44GeV We use Eq. 36- o find he seed from he kinei energ. K m m.957 K.5MeV m.5mev 5. Sine he roon was aeeraed b a oenia differene of 5 MV, is oenia energ dereased b 5 MeV, and so is kinei energ inreased from o 5 MeV. Use Eq. 36- o find he seed from he kinei energ. K m m 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 439

14 Phsis for Sieniss & Engineers wih Modern Phsis, 4 h Ediion Insruor Souions Manua.47 K 5MeV m 93.3MeV 5. We e M reresen he res mass of he new arie. The iniia energ is due o boh inoming aries, and he fina energ is he res energ of he new arie. Use Eq. 36- for he iniia energies. m E m M M m We assumed ha energ is onsered, and so here was no oss of energ in he oision. The fina kinei energ is, so a of he kinei energ was os. Kos Kiniia m m 53. Sine he eeron was aeeraed b a oenia differene of kv, is oenia energ dereased b kev, and so is kinei energ inreased from o MeV. Use Eq. 36- o find he seed from he kinei energ. K m m.3 K. MeV m.5mev 54. We use Eqs. 36- and 36-3 in order o find he mass. 4 4 E m K m K Km m m K MeV 45MeV K 45MeV The arie is mos ike a robab a meson. 4 MeV.5 kg 55. (a) Sine he kinei energ is haf he oa energ, and he oa energ is he kinei energ us he res energ, he kinei energ mus be equa o he res energ. We aso use Eq K E K m K m 3 K m m 4.66 (b) In his ase, he kinei energ is haf he res energ. 3 5 K m m Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 44

15 Chaer 36 The Seia Theor of Reaii 56. We use Eq. 36- for he kinei energ and Eq. 36- for he momenum. K m m.5 m s 93.3MeV MeV 3. m s 7.5 m s 93.3MeV m m 3. m s m 65MeV 7.5 m s Eauae wih he assia eressions MeV K m m.5 m s 3. m s 7.5 m s 93.3MeV 3. m s 34.6 MeV m m 3. m s 55MeV Cauae he eren error. K K errork K % error 65 3.% 57. (a) The kinei energ is found from Eq K m m. 4.7 kg 3. m s J.5 J (b) Use he assia eression and omare he wo resus. K m J.54 J 9.54 J kg. 3. m s.479 J % error.4% The assia aue is.4% oo ow. 5. The kinei energ of 99 GeV is used o find he seed of he roons. Sine he energ is imes he res mass, we ee he seed o be er ose o. Use Eq K m m o 7 sig. fig. K 99GeV m.93gev 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 44

16 K Phsis for Sieniss & Engineers wih Modern Phsis, 4 h Ediion Insruor Souions Manua B K m m m rq rq rq m 99GeV kg 3. m s.93gev 3 9. m.6 C 3.3T 59. B onseraion of energ, he res energ of he ameriium nueus is equa o he res energies of he oher aries us he kinei energ of he aha arie. m m m K Am N m m m K 5.5MeV u 4.56 u 4.6 u u MeV N Am 6. (a) For a arie of non-zero mass, we derie he foowing reaionshi beween kinei energ and momenum. E K m ; E m K m m K K m K K m K m 4 m 4 For he kinei energ o be osiie, we ake he osiie roo. m 4 m 4 K m m If he momenum is arge, we hae he foowing reaionshi. K m m m Thus here shoud be a inear reaionshi beween kinei energ and momenum for arge aues of momenum. If he momenum is sma, we use he binomia eansion o derie he assia reaionshi. K m m m m m m m m m Thus we ee a quadrai reaionshi for sma aues of momenum. The adjaen grah erifies hese aroimaions. (b) For a arie of zero mass, he reaionshi is m sim K. See he inuded grah. The sreadshee used for his robem an be found on he Media Manager, wih fiename PSE4_ISM_CH36.XLS, on ab Probem m 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 44

17 Chaer 36 The Seia Theor of Reaii 6. A of he energ, boh res energ and kinei energ, beomes eeromagnei energ. We use Eq Boh masses are he same. E E E m m m oa 43.6 MeV 4 MeV 6. We use Eqs. 36- and E K m ; E m K m m K K m 5.7 MeV K K m 63. (a) We assume he mass of he arie is m, and we are gien ha he eoi on has an - omonen, u. We wrie he momenum in eah frame using Eq. 36-, and we use he eoi ransformaion gien in Eq Noe ha here are hree reean eoiies: u, he eoi in referene frame S; u, he eoi in referene frame S; and, he eoi of one frame reaie o he oher frame. There is no eoi in he or z direions, in eiher frame. We resere he smbo for, and aso use Eq. 36- for energ. mu ; ; z u u u u u u u ; ; u u u z uz u u mu ; sine u ; sine u z z u Subsiue he eression for u ino he eression for. u m mu u u m u u u u u u u u u m u m u u u u u u m u m u u u u u u u 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 443

18 Phsis for Sieniss & Engineers wih Modern Phsis, 4 h Ediion Insruor Souions Manua m u mu u u u mu m m u u u E m I is obious from he firs few equaions of he robem ha and. E u u u m m m E u u u u m u m mu u u m u u u (b) We summarize hese resus, and wrie he Lorenz ransformaion from Eq. 36-6, bu soed in erms of he rimed ariabes. Tha an be easi done b inerhanged rimed and unrimed quaniies, and hanging o. E E ; ; ; E ; ; z z ; These ransformaions are idenia if we ehange wih (or Ewih ). wih, wih, z z mu z wih z, and E 64. The gaa is moing awa from he Earh, and so we use Eq. 36-5b. f f.97 f f.93 f f f.93 + f f.93 f f For soure and obserer moing owards eah oher, use Eq. 36-4b..7 f f f 95. MHz 6 MHz 3 MHz.7 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 444

19 Chaer 36 The Seia Theor of Reaii 66. We use Eq. 36-5a, and assume ha. / 67. (a) We a Eq. 36-4b o deermine he reeied/refeed frequen f. Then we a his same equaion a seond ime using he frequen f as he soure frequen o deermine he Doershifed frequen f. We subra he iniia frequen from his Doer-shifed frequen o obain he bea frequen. The bea frequen wi be muh smaer han he emied frequen when he seed is muh smaer han he seed of igh. We hen se and soe for. f f f f f f f fbea f f f f f f f 3. m/s 667 Hz Hz 7.m/s (b) We find he hange in eoi and soe for he resuing hange in bea frequen. Seing he hange in he eoi equa o km/h we soe for he hange in bea frequen. f f f f f bea bea bea f f Hz km/h m/s bea 3. m/s 3.6km/h 7Hz 6. We onsider he differene beween Doer-shifed frequenies for aoms moing dire owards he obserer and aoms moing dire awa. Use Eqs. 36-4b and 36-5b. f f f f f f We ake he seed o be he rms seed of herma moion, gien b Eq. -5. We aso assume ha he herma energ is muh ess han he res energ, and so 3 kt m. 3kT 3kT f 3kT 3kT 3kT rms m m f m m m We eauae for a gas of H aoms (no H moeues) a 55 K. Use Aendi F o find he mass. f f J K 55K 3kT.5 m 7.u.66 kg u 3. m s / bea 5 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 445

20 Phsis for Sieniss & Engineers wih Modern Phsis, 4 h Ediion Insruor Souions Manua 69. A he Norh Poe he ok is a res, whie he ok on he equaor raes he irumferene of he Earh eah da. We diide he irumferene of he Earh b he engh of he da o deermine he seed of he equaoria ok. We se he diaed ime equa o. ears and soe for he hange in res imes for he wo oks. 6 R 6.3 m 464 m/s T 4 hr 36s/hr / /,eq,eq,oe,oe,eq,oe 7. r 464 m/s 3.56 s/r 3. m/s 75 s 7. We ake he osiie direion in he direion of he moion of he seond od. Consider he firs od as referene frame S, and he saeraf as referene frame S. The eoi of he saeraf reaie o he firs od is.6. The eoi of he firs od reaie o he saeraf is u.5. Soe for he eoi of he seond od reaie o he firs od, u, using Eq. 36-7a. u u u We rea he Earh as he saionar frame, and he airane as he moing frame. The eased ime in he airane wi be diaed o he obserers on he Earh. Use Eq. 36-a. rearh rearh Earh ; ane Earh rearh rearh r Earh Earh ane 6.3 m 3km h 3.6km h. s 3. m s 6 m s 7. (a) To raeers on he saeraf, he disane o he sar is onraed, aording o Eq. 36-3a. This onraed disane is o be raeed in 4.6 ears. Use ha ime wih he onraed disane o find he seed of he saeraf. saeraf saeraf Earh Earh saeraf.69.6 saeraf Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 446

21 Chaer 36 The Seia Theor of Reaii (b) Find he eased ime aording o obserers on Earh, using Eq. 36-a. saeshi 4.6 Earh Noe ha his agrees wih he ime found from disane and seed. Earh 4.3 Earh 6.3r (a) We use Eq. 36-5a. To ge a onger waeengh han usua means ha he obje is moing awa from he Earh (b) We assume ha he quasar is moing and he Earh is saionar. Then we use Eq. 6-9b. f f We assume ha some kind of a igh signa is being ransmied from he asronau o Earh, wih a frequen of he hearbea. Tha frequen wi hen be Doer shifed, aording o Eq. 36-5b. We eress he frequenies in beas er minue. f f 6 3 f f.6 f f (a) The eoi omonens of he igh in he S frame are u and u. We ransform hose eoiies o he S frame aording o Eq (b) u u ; u u u u u an an an u u u u () In a Gaiean ransformaion, we woud hae he foowing. u u ; u u ; u ; an 76. We ake he osiie direion as he direion of moion of roke A. Consider roke A as referene frame S, and he Earh as referene frame S. The eoi of he Earh reaie o roke A is.65. The eoi of roke B reaie o he Earh is u.5. Soe for he eoi of roke B reaie o roke A, u, using Eq. 36-7a. u.5.65 u.45 u.65.5 Noe ha a Gaiean anasis woud hae resued in u.. 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 447

22 Phsis for Sieniss & Engineers wih Modern Phsis, 4 h Ediion Insruor Souions Manua 77. (a) We find he seed from Eq K m m 4, m 4, 4, 3. m s.77m s 4, 4, (b) The ube wi be onraed in he res frame of he eeron, aording o Eq. 36-3a m.m 4, 7. The eerosai fore roides he radia aeeraion. We soe ha reaionshi for he seed of he eeron. e meeron Feerosai Fenriea 4 r r e.99 N m C C 3 4 meeronr 9. kg.53 m Beause his is muh ess han., he eeron is no reaiisi. 6. m s The minimum energ required woud be he energ o rodue he air wih no kinei energ, so he oa energ is heir res energ. The boh hae he same mass. Use Eq E m.5mev. MeV.64 J. The waage imes he ime is he energ required. We use Eq. 36- o auae he mass. 7 P 75W 3.6 s g 5 E P m m.6 g 3. m s kg. Use Eqs. 36-3, 36-, and 36-. E m E m 4 4 de 4 / m m d E E m /. The kinei energ aaiabe omes from he derease in res energ. K mn m me m MeV 93.7MeV.5MeV.79MeV 3. (a) We find he rae of mass oss from Eq E m E m m E 6 4 J s 3. m s kg s 4 kg s 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 44

23 Chaer 36 The Seia Theor of Reaii (b) Find he ime from he mass of he Sun and he rae deermined in ar (a). 4 m 5.9 kg Earh m 4.44 kg s 3.56 s () We find he ime for he Sun o ose a of is mass a his same rae. 3 m.99 kg Sun m 4.44 kg s 3.56 s 4. Use Eq. 36- for he momenum o find he mass. m m 3.7 kg m s.4 m s 3 9. kg 3. m s m.4 m s This arie has he mass of an eeron, and a negaie harge, so i mus be an eeron. 5. The oa binding energ is he energ required o roide he inrease in res energ. E m m m +e n He 93.5MeV u.73u.67 u 4.6 u.3 MeV 6. The momenum is gien b Eq. 36-, and he energ is gien b Eq. 36- and Eq m E P m E m m 4 7. (a) The magniudes of he momena are equa. We use Eq m m 93.3MeV.95 m 5356 MeV GeV 5.36GeV.6 kg m s.6 J GeV 3. m s GeV (b) Beause he roons are moing in oosie direions, he eor sum of he momena is. () In he referene frame of one roon, he aboraor is moing a.95. The oher roon is moing a.95 reaie o he aboraor. We find he seed of one roon reaie o he oher, and hen find he momenum of he moing roon in he res frame of he oher roon b using ha reaie eoi. u u.9999 u Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 449

24 Phsis for Sieniss & Engineers wih Modern Phsis, 4 h Ediion Insruor Souions Manua MeV mu m u.95 mu 6MeV u u GeV 6.GeV kg m s.6 J GeV 3. m s GeV. We find he oss in mass from Eq E 44 J m 5.3 kg 3. m s Two moes of waer has a mass of 5.3 kg 3 36 kg 3 36 kg. Find he erenage of mass os % 9. Use Eq. 36- for kinei energ, and Eq. 36- for res energ. K m m Enerrise onered m onered 9 7 m Enerrise 6 kg 3 kg. 9. We se he kinei energ of he saeraf equa o he res energ of an unknown mass. Use Eqs. 36- and 36-. K m m shi m m m shi shi. kg 7. kg From he Earh s oin of iew, he disane is 35 and he seed is.7. Tha daa is used o auae he ime from he Earh frame, and hen Eq. 36-a is used o auae he ime in he saeshi frame. d 35 5 ; We assume one arie is moing in he negaie direion in he aboraor frame, and he oher arie is moing in he osiie direion. We onsider he arie moing in he negaie direion as referene frame S, and he aboraor as referene frame S. The eoi of he aboraor reaie o he negaie-moing arie is.5, and he eoi of he osiiemoing arie reaie o he aboraor frame is u.5. Soe for he eoi of he osiiemoing arie reaie o he negaie-moing arie, u. u.5.5 u.97 u Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 45

25 Chaer 36 The Seia Theor of Reaii 9. We onsider he moion from he referene frame of he saeshi. The assengers wi see he ri disane onraed, as gien b Eq. 36-3a. The wi measure heir seed o be ha onraed disane diided b he ear of rae ime (as measured on he shi). Use ha seed o find he work done (he kinei energ of he shi). W K m m kg 3. m s.6 J 93. The kinei energ is gien b Eq K m m.9 4,5 kg 3. m s 5.3 J We omare his wih annua U.S. energ onsumion: The saeshi s kinei energ is oer 5 imes as grea. 5.3 J J The i meson deas a res, and so he momenum of he muon and he neurino mus eah hae he same magniude (and oosie direions). The neurino has no res mass, and he oa energ mus be onsered. We ombine hese reaionshis using Eq / ; E m 4 / 4 / E E E m m m 4 / 4 m m m m Soe for he momenum. 4 4 m m m m m Wrie he kinei energ of he muon using Eqs. 36- and K E m ; E E E m K m m m m m m m m m m m m m m m m m m m m m m m m m m m m m 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 45

26 Phsis for Sieniss & Engineers wih Modern Phsis, 4 h Ediion Insruor Souions Manua 95. (a) The reaie seed an be auaed in eiher frame, and wi be he same aue in boh frames. The ime as measured on he Earh wi be onger han he ime measured on he saeshi, as gien b Eq. 36-a. Earh saeshi saeshi ; Earh Earh Earh Earh Earh Earh saeshi Earh saeshi Earh Earh saesh i (b) The disane as measured b he saeshi wi be onraed. Earh saeshi saeshi.5 saeshi Earh Earh saeshi Earh This is he same disane as found using he engh onraion reaionshi. 96. (a) To obserers on he shi, he eriod is non-reaiisi. Use Eq. 4-7b. T m.kg.939s k 4. N m (b) The osiaing mass is a ok. Aording o obserers on Earh, oks on he saeraf run sow. T.939s TEarh.5s.9 Earh 97. We use he Lorenz ransformaions o derie he resu. ; 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 45

27 Chaer 36 The Seia Theor of Reaii 9. We assume ha he ef edge of he gass is een wih oin A when he fash of igh is emied. There is no oss of generai wih ha assumion. We do he auaions in he frame of referene in whih oins A and B are a res, and he gass is hen moing o he righ wih seed. If he gass is no moing, we woud hae his no moion resu. disane in gass disane in auum d gass auum seed in gass seed in auum d d nd d nd d n d n If he inde of refraion is n, hen he gass wi hae no effe on he igh, and he ime woud sim be he disane diided b he seed of igh. disane in gass disane in auum d d d d n gass auum seed in gass seed in auum Now, e us onsider he robem from a reaiisi oin of iew. The seed of igh in he gass wi be he reaiisi sum of he seed of igh in saionar gass, n, and he seed of he gass,, b Eq. 36-7a. We define o simif furher eressions. n n n n igh in gass n n n n n n The onraed widh of he gass, from he Earh frame of referene, is gien b Eq. 36-3a. d d d moing gass gass d We assume he igh eners he bok when he ef edge of he bok is a oin A, and wrie sime equaions for he disaemen of he eading edge of he igh, and he eading edge of he bok. Se hem equa and soe for he ime when he igh eis he righ edge of he bok. d igh igh ; righ ; in gass n edge d d n igh righ gass gass gass edge n n d n Where is he fron edge of he bok when he igh emerges? Use gass eression for he eading edge of he igh, or he eading edge of he bok. d n d igh igh gass in gass n n n d d d n d n dn d righ gass edge n n n n wih eiher The ar of he ah ha is ef, d, n wi be raeed a seed b he igh. We eress ha ime, and hen find he oa ime. d n auum 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 453

28 K ( 7 J) Phsis for Sieniss & Engineers wih Modern Phsis, 4 h Ediion Insruor Souions Manua oa gass auum gass d d n n d n n n n d We hek his for he aroriae imiing ases. Case : n d n d oa This resu was eeed, beause he seed of he igh woud awas be. Case : n d n d n d oa This resu was obained earier in he souion. Case 3: n d oa n This resu was eeed, beause hen here is no seed hange in he gass. 99. The sreadshee used for his robem an be found on he Media Manager, wih fiename PSE4_ISM_CH36.XLS, on ab Probem Cassia Reaiisi /. (a) We use Eq Sine here is moion in wo dimensions, we hae ˆ d d d F Fj ; m ; F F m d d d Use he omonen equaions o obain eressions for and. m m m m m F F F m F m m m F m F Subsiue he eression for ino he eression for.. 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 454

29 / Chaer 36 The Seia Theor of Reaii F 4 m F m F m m m m F m m F m F 4 m m F m F m F m F / m F Use he eression for o soe for. m F F m F m F m F m F / F F m F m F F m F m F F m F The negaie sign omes from aking he negaie square roo of he reious equaion. We know ha he arie is moing down. (b) See he grah. We are oing and. The sreadshee used for his robem an be found on he Media Manager, wih fiename PSE4_ISM_CH36.XLS, on ab Probem (- ) ( s) () The ah is no araboi, beause he is no onsan. Een hough here is no fore in he - direion, as he ne seed of he arie inreases, inreases. Thus mus derease as ime eases in order for o sa onsan. 9 Pearson Eduaion, In., Uer Sadde Rier, NJ. A righs resered. This maeria is roeed under a origh aws as he urren eis. No orion of his maeria ma be rerodued, in an form or b an means, wihou ermission in wriing from he ubisher. 455