In he heory of speial relaiviy, an even, suh as he launhing of he spae shule in Figure 8., is a physial happening ha ours a a erain plae and ime. In his drawing wo observers are wahing he lif-off, one sanding on he earh and one in an observer uses a referene frame ha onsiss of a se of x, y, z axes (alled a oordinae sysem) and a lok. The oordinae sysems are used o esablish where he even ours, and he loks o speify when. Eah observer is a res relaive o his own referene frame. However, he earh-based observer and he airborne observer are moving relaive o eah oher and so, also, are heir respeive referene frames. y x y x Using an earh-based referene frame, an observer sanding on he earh reords he loaion and ime of an even (he spae shule lif-off). Likewise, an observer in he airplane uses a plane-based referene frame o desribe he even.
The heory of speial relaiviy deals wih a speial kind of referene frame, alled an inerial referene frame. As Seion 4. disusses, an inerial referene frame is one in whih Newon s law of ineria is valid. Tha is, if he ne fore aing on a body is zero, he body eiher remains a res or moves a a onsan veloiy. In oher words, he aeleraion of suh a body is zero when measured in an inerial referene frame. Roaing and oherwise aeleraing referene frames are no inerial referene frames. The earh-based referene frame in Figure 8. is no quie an inerial frame beause i is subjeed o enripeal aeleraions as he earh spins on is axis and revolves around he sun. In mos siuaions, however, he effes of hese aeleraions are small, and we an negle hem. To he exen ha he earh-based referene frame is an inerial frame, so is he plane-based referene frame, beause he plane moves a a onsan veloiy relaive o he earh. The nex seion disusses why inerial referene frames are imporan in relaiviy. Einsein buil his heory of speial relaiviy on wo fundamenal assumpions or posulaes abou he way naure behaves.. The Relaiviy Posulae. The laws of physis are he same in every inerial referene frame.. The Speed-of-Ligh Posulae. The speed of ligh in a vauum, measured in any inerial referene frame, always has he same value of, no maer how fas he soure of ligh and he observer are moving relaive o eah oher. observer, using his own inerial referene frame, an make measuremens on he moion of of he eleronis on he spae shule is desribed by he laws of eleromagneism. Aording o he relaiviy posulae, any inerial referene frame is as good as any oher for expressing he laws of physis. As far as inerial referene frames are onerned, naure does no play favories. Sine he laws of physis are he same in all inerial referene frames, here is no experimen ha an disinguish beween an inerial frame ha is a res and one ha is moving a a onsan veloiy. When you are seaed on he airraf in Figure 8., for insane, i is jus as valid o say ha you are a res and he earh is moving as i is o say he onverse. I is no possible o single ou one pariular inerial referene frame as being a absolue res. Consequenly, i is meaningless o alk abou he absolue veloiy of an obje ha is, is veloiy measured relaive o a referene frame a absolue res. Thus, he earh moves relaive o he sun, whih iself moves relaive o he ener of our galaxy. And he galaxy moves relaive o oher galaxies, and so on. Aording o Einsein, only he relaive veloiy beween objes, no heir absolue veloiies, an be measured and is physially meaningful. ommon sense. For insane, Figure 8. illusraes a person sanding on he bed of a ruk ha is moving a a onsan speed of 5 m/s relaive o he ground. Now, suppose ha you son on he ruk observes he speed of ligh o be. Wha do you measure for he speed of ligh? You migh guess ha he speed of ligh would be 5 m/s. However, his guess Boh he person on he ruk and he observer on he earh measure he speed of he ligh o be, regardless of he speed of he ruk.
is inonsisen wih he speed-of-ligh posulae, whih saes ha all observers in inerial referene frames measure he speed of ligh o be nohing more, nohing less. Therefore, you mus also measure he speed of ligh o be, he same as ha measured by he person on he ruk. Aording o he speed-of-ligh posulae Sine waves, suh as waer waves and sound waves, require a medium hrough whih o propagae, i was naural for sieniss before Einsein o assume ha ligh did oo. This hypoheial medium was alled he luminiferous eher spae. Furhermore, i was believed ha ligh raveled a he speed only when measured wih respe o he eher. Aording o his view, an observer moving relaive o he eher would measure a speed for ligh ha was smaller or greaer han, depending on wheher he observer moved wih or agains he ligh, respeively. During he years 883 887, however, he Amerian sieniss A. A. Mihelson and E. W. Morley arried ou a series of famous experimens whose resuls were no onsisen wih he eher heory. Their resuls indiaed ha he speed of ligh is indeed he same in all inerial referene frames and does no depend on he moion of he observer. These experimens, and ohers, led evenually o he demise of he eher heory and he aepane of he heory of speial relaiviy. The remainder of his haper reexamines, from he viewpoin of speial relaiviy, a number of fundamenal oneps ha have been disussed in earlier hapers from he viewpoin of lassial physis. These oneps are ime, lengh, momenum, kinei energy, and he addiion of veloiies. We will see ha eah is modiof a moving obje relaive o he speed of ligh in a vauum. Figure 8.3 illusraes ha when he obje moves slowly [ is muh smaller han ( lassial version of eah onep provides an aurae desripion of realiy. However, when he obje moves so rapidly ha is an appreiable fraion of he speed of ligh [ is approximaely equal o ( )], he effes of speial relaiviy mus be onsidered. This har emphasizes ha i is he speed of a moving obje, as ompared o he speed of ligh in a vauum, ha deermines wheher he effes of speial relaiviy are measurably imporan. Alber Einsein (879 955), he auhor of he heory of speial relaiviy, is one of he mos famous sieniss of he wenieh enury. ( Hulon Arhive/Gey Images) Coneps. Time Inerval. Lengh 3. Momenum 4. Kinei Energy 5. Addiion of Veloiies << Classial Version. 0 (Seion.). L 0 (Seion.) 3. p = m (Seion 7.) 4. KE = m (Seion 6.) 5. AB = AC + CB (Seion 3.4) Relaivisi Version 0. = / (Seion 8.3). L = L 0 / (Seion 8.4) 3. p = m / (Seion 8.5) 4. KE = m / (Seion 8.6) AC + CB 5. AB = + AC CB / (Seion 8.7)
imposed by speial relaiviy. Eah of hese equaions will be disussed in laer seions of his haper. imply ha he lassial oneps of ime, lengh, momenum, kinei energy, and he addiion of veloiies, as developed by Newon and ohers, are wrong. They are jus limied o speeds ha are very small ompared o he speed of ligh. In onras, he relaivisi view of he oneps applies o all speeds beween zero and he speed of ligh. Common experiene indiaes ha ime passes jus as quikly for a person sanding on he ground as i does for an asronau in a spaeraf. In onras, speial relaiviy reveals ha he person on he ground measures ime passing more slowly for he asronau han for herself. We an see how his urious effe arises wih he help of he lok illusraed in Figure 8.4, whih uses a pulse of ligh o mark ime. A shor pulse of ligh is A ligh lok. nex o he soure. Eah ime a pulse reahes he deeor, a ik regisers on he har reorder, anoher shor pulse of ligh is emied, and he yle repeas. Thus, he ime inersoure) and an ending even (he pulse sriking he deeor). The soure and deeor are so lose o eah oher ha he wo evens an be onsidered o our a he same loaion. Suppose wo idenial loks are buil. One is kep on earh, and he oher is plaed aboard a spaeraf ha ravels a a onsan veloiy relaive o he earh. The asronau is a res wih respe o he lok on he spaeraf and, herefore, sees he ligh pulse move along he up/down pah shown in Figure 8.5a. Aording o he asronau, he ime inerval 0 required for he ligh o follow his pah is he disane D divided by he speed of ligh ; 0 D/. To he asronau, 0 is he ime inerval beween he iks of he spaeraf lok ha is, he ime inerval beween he beginning and ending evens of he lok. An earh-based observer, however, does no measure 0 as he ime inerval beween hese wo evens. Sine he spaeraf is moving, he earh-based observer sees he ligh pulse follow he diagonal pah shown in red in par b of he drawing. This pah is longer han he up/down pah seen by he asronau. Bu ligh ravels a he same speed for boh observers, in aord wih he speed-of-ligh posulae. Therefore, he earh-based observer measures a ime inerval beween he wo evens ha is greaer han he ime inerval 0 measured by he asronau. In oher words, he earh-based observer, using her own nau s lok runs slowly. This resul of speial relaiviy is known as ime dilaion. (To dilae means o expand, and he ime inerval is expanded relaive o 0.) D (a) s D s (a) The asronau measures he ime inerval 0 beween suessive iks of his ligh lok. (b) An observer on earh wahes he asronau s lok and sees he ligh pulse ravel a greaer disane beween iks han i does in par a. Consequenly, he earh-based observer measures a ime inerval beween iks ha is greaer han 0. L (b) L
The ime inerval ha he earh-based observer measures in Figure 8.5b an be deermined as follows. While he ligh pulse ravels from he soure o he deeor, he spaeraf moves a disane L o he righ, where is he speed of he spaeraf relaive o he earh. From he drawing i an be seen ha he ligh pulse ravels a oal diagonal disane of s during he ime inerval s D L D Bu he disane s is also equal o he speed of ligh imes he ime inerval s. Therefore,, so ha D Squaring his resul and solving for gives D However, D/ 0, where 0 is he ime inerval beween suessive iks of he spaeraf s lok as measured by he asronau. Wih his subsiuion, he equaion for an be expressed as Time dilaion 0 (8.) 0 proper ime inerval, whih is he inerval beween wo evens as measured by an observer who is a res wih respe o he evens and who views hem as ourring a he same plae dilaed ime inerval, whih is he inerval measured by an observer who is in moion wih respe o he evens and who views hem as ourring a differen plaes relaive speed beween he wo observers speed of ligh in a vauum For a speed ha is less han, he erm / in Equaion 8. is less han, and he dilaed ime inerval is greaer han 0. Example illusraes he exen of his ime dilaion effe. The spaeraf in Figure 8.5 is moving pas he earh a a onsan speed ha is 0.9 imes he speed of ligh. Thus, (0.9)(3.0 0 8 m/s), whih is ofen wrien as 0.9. The asronau measures he ime inerval beween suessive iks of he spaeraf lok o be 0.0 s. Wha is he ime inerval ha an earh observer measures beween iks of he asronau s lok? Sine he lok on he spaeraf is moving relaive o he earh, he earh-based observer measures a greaer ime inerval beween iks han does he asronau, who is a res relaive o he lok. The dilaed ime inerval an be deermined from he ime dilaion relaion, Equaion 8.. The dilaed ime inerval is 0.0 s 0.9.6 s
From he poin of view of he earh-based observer, he asronau is using a lok ha is running slowly, beause he earh-based observer measures a ime beween iks ha is longer (.6 s) han wha he asronau measures (.0 s). sanes exis in whih ime dilaion an reae appreiable errors if no aouned for. The Global Posiioning Sysem (GPS), for insane, uses highly aurae and sable aomi loks on board eah of 4 saellies orbiing he earh a speeds of abou 4000 m/s. These loks make i possible o measure he ime i akes for eleromagnei waves o ravel from a saellie o a ground-based GPS reeiver. From he speed of ligh and he imes measured for signals from hree or more of he saellies, i is possible o loae he posiion of he reeiver (see Seion 5.5). The sabiliy of he loks mus be beer han one par in 0 3 o ensure he posiional auray demanded of he GPS. Using Equaion 8. and he speed of he GPS saellies, we an alulae he differene beween he dilaed ime inerval and he proper ime inerval as a fraion of he proper ime inerval and ompare he resul o he sabiliy of he GPS loks: 0 0 / (4000 m/s) /(3.00 0 8 m/s). 0 0 This resul is approximaely one housand imes greaer han he GPS-lok sabiliy of one par in 0 3. Thus, if no aken ino aoun, ime dilaion would ause an error in he measured posiion of he earh-based GPS reeiver roughly equivalen o he error aused by a housand-fold degradaion in he sabiliy of he aomi loks. In Figure 8.5 boh he asronau and he person sanding on he earh are measuring even (he ligh pulse sriking he deeor). For he asronau, who is a res wih respe o he ligh lok, he wo evens our a he same loaion. (Remember, we are assuming ha he ligh soure and deeor are so lose ogeher ha hey are onsidered o be a he same plae.) Being a res wih respe o a lok is he usual or proper siuaion, so he ime inerval 0 measured by he asronau is alled he proper ime inerval. In general, he proper ime inerval 0 beween wo evens is he ime inerval measured by an observer who is a res relaive o he evens and sees hem a he same loaion in spae. On he oher hand, he earh-based observer does no see he wo evens ourring a he same loaion in spae, sine he spaeraf is in moion. The ime inerval ha he earh-based observer To undersand siuaions involving ime dilaion, i is essenial o disinguish beween 0 and a deeor. Then deermine he referene frame in whih he wo evens our a he same plae. An observer a res in his referene frame measures he proper ime inerval 0. One of he inriguing aspes of ime dilaion ours in onjunion wih spae ravel. Sine enormous disanes are involved, ravel o even he loses sar ouside our solar sysem would ake a long ime. However, as he following example shows, he ravel ime an be onsiderably less for he passengers han one migh guess. Shown here in orbi is asronau David A. Wolf as he works on he Inernaional Spae Saion during a session of exravehiular aiviy. (Couresy NASA) Alpha Cenauri, a nearby sar in our galaxy, is 4.3 ligh-years away. This means ha, as measured by a person on earh, i would ake ligh 4.3 years o reah his sar. If a roke leaves for Alpha Cenauri and ravels a a speed of 0.95 relaive o he earh, by how muh will he
passengers have aged, aording o heir own lok, when hey reah heir desinaion? Assume ha he earh and Alpha Cenauri are saionary wih respe o one anoher. The wo evens in his problem are he deparure from earh and he arrival a Alpha Cenauri. A deparure, earh is jus ouside he spaeship. Upon arrival a he desinaion, Alpha Cenauri is jus ouside. Therefore, relaive o he passengers, he wo evens our a he same plae namely, jus ouside he spaeship. Thus, he passengers measure he proper ime inerval 0 lef behind on earh, he evens our a differen plaes, so suh a person measures he dilaed ime inerval we noe ha he ime o ravel a given disane is inversely proporional o he speed. Sine i akes 4.3 years o raverse he disane beween earh and Alpha Cenauri a he speed of ligh, i would ake even longer a he slower speed of 0.95. Thus, a person on earh measures he dilaed ime inerval o be (4.3 years)/0.95 4.5 years. This value an be used wih he ime-dilaion equaion 0. 0. passengers judge heir own aging is 0 (4.5 years) 0.95.4 years Thus, he people aboard he roke will have aged by only.4 years when hey reah Alpha Cenauri, and no he 4.5 years an earhbound observer has alulaed. arried ou by J. C. Hafele and R. E. Keaing.* They ranspored very preise esium-beam aomi loks around he world on ommerial jes. Sine he speed of a je plane is onsiderably less han, he ime-dilaion effe is exremely small. However, he aomi loks were aurae o abou 0 9 s, so he effe ould be measured. The loks were in he air for 45 hours, and heir imes were ompared o referene aomi loks kep on earh. The experimenal resuls revealed ha, wihin experimenal error, he readings on he loks on he planes were differen from hose on earh by an amoun ha agreed wih he prediion of relaiviy. The behavior of subaomi pariles alled muons of ime dilaion. These pariles are reaed high in he amosphere, a aliudes of abou 0 000 m. When a res, muons exis only for abou. 0 6 s before disinegraing. Wih suh a shor lifeime, hese pariles ould never make i down o he earh s surfae, even raveling a nearly he speed of ligh. However, a large number of muons do reah he earh. The only way hey an do so is o live longer beause of ime dilaion, as Example 3 illusraes. The average lifeime of a muon a res is. 0 6 s. A muon reaed in he upper amosphere, housands of meers above sea level, ravels oward he earh a a speed of 0.998. Find, on he average, (a) how long a muon lives aording o an observer on earh, and (b) how far he muon ravels before disinegraing. The wo evens of ineres are he generaion and subsequen disinegraion of he muon. When he muon is a res, hese evens our a he same plae, so he muon s average (a res) lifeime of. 0 6 s is a proper ime inerval 0. When he muon moves a a speed 0.998 relaive o he earh, an observer on he earh measures a dilaed lifeime ha is given by Equaion 8.. The average disane x raveled by a muon, as measured by an earh observer, is equal o he muon s speed imes he dilaed ime inerval. *J. C. Hafele and R. E. Keaing, Around-he-World Aomi Cloks: Observed Relaivisi Time Gains, Siene, Vol. 77, July 4, 97, p. 68.
(a) The observer on earh measures a dilaed lifeime. Using he ime-dilaion equa- 0. 0 (b) The disane raveled by he muon before i disinegraes is. 0 6 s x (0.998)(3.00 0 8 m /s)(35 0 6 s) 0.998 35 0 6 s.0 0 4 m (8.) fae of he earh. If is lifeime were only. 0 6 s, a muon would ravel only 660 m before disinegraing and ould never reah he earh. Beause of ime dilaion, observers moving a a onsan veloiy relaive o eah oher measure differen ime inervals beween wo evens. For insane, Example in he previous seion illusraes ha a rip from earh o Alpha Cenauri a a speed of 0.95 akes 4.5 years aording o a lok on earh, bu only.4 years aording o a lok in he roke. These wo imes differ by he faor /. Sine he imes for he rip are differen, one migh ask wheher he observers measure differen disanes beween earh and Alpha Cenauri. The answer, aording o speial relaiviy, is yes. Afer all, boh he earhbased observer and he roke passenger agree ha he relaive speed beween he roke and earh is 0.95. Sine speed is disane divided by ime and he ime is differen for he wo observers, i follows ha he disanes mus also be differen, if he relaive speed is o be he same for boh individuals. Thus, he earh observer deermines he disane o Alpha Cenauri o be L 0 (0.95)(4.5 years) 4.3 ligh-years. On he oher hand, L 0 (0.95)(.4 years).3 ligh-years. The passenger, measuring he shorer ime, also measures he shorer disane. This shorening of he disane beween wo poins is one example of a phenomenon known as lengh onraion. The relaion beween he disanes measured by wo observers in relaive moion a a onsan veloiy an be obained wih he aid of Figure 8.6. Par a of he drawing shows he siuaion from he poin of view of he earh-based observer. This person measures he ime of he rip o be, he disane o be L 0, and he relaive speed of he roke o be
L 0 /. Par b of he drawing presens he poin of view of he passenger, for whom he roke is a res, and he earh and Alpha Cenauri appear o move by a a speed. The passenger deermines he disane of he rip o be L, he ime o be 0, and he relaive speed o be L/ 0. Sine he relaive speed ompued by he passenger equals ha ompued by he earh-based observer, i follows ha L/ 0 L 0 /. Using his resul and he ime-dilaion equaion, Equaion 8., we obain he following relaion beween L and L 0 : Lengh onraion L (a) L L 0 (8.) The lengh L 0 is alled he proper lengh; i is he lengh (or disane) beween wo poins as measured by an observer a res wih respe o hem. Sine is less han, he erm / is less han, and L is less han L 0. I is imporan o noe ha his lengh onraion ours only along he direion of he moion. Those dimensions ha are perpendiular o he moion are no shorened, as he nex example disusses. L (b) (a) As measured by an observer on he earh, he disane o Alpha Cenauri is L 0, and he ime required o make he rip is. (b) Aording o he passenger on he spaeraf, he earh and Alpha Cenauri move wih speed relaive o he raf. The passenger measures he disane and ime of he rip o be L and 0, respeively, boh quaniies being less han hose in par a. An asronau, using a meer sik ha is a res relaive o a ylindrial spaeraf, measures he lengh and diameer of he spaeraf o be 8 m and m, respeively. The spaeraf moves wih a onsan speed of 0.95 relaive o he earh, as in Figure 8.6. Wha are he dimensions of he spaeraf, as measured by an observer on earh? The lengh of 8 m is a proper lengh L 0, sine i is measured using a meer sik ha is a res relaive o he spaeraf. The lengh L measured by he observer on earh an be deermined from he lengh-onraion formula, Equaion 8.. On he oher hand, he diameer of he spaeraf is perpendiular o he moion, so he earh observer does no measure any hange in he diameer. The lengh L of he spaeraf, as measured by he observer on earh, is L L 0 (8 m) 0.95 6 m Boh he asronau and he observer on earh measure he same value for he diameer of he spaeraf: Diameer m. Figure 8.6a shows he size of he spaeraf as measured by he earh observer, and par b shows he size as measured by he asronau. L 0 L. When dealing wih relaivisi effes we need o disinguish arefully beween he rieria for he proper ime inerval and he proper lengh. The proper ime inerval 0 beween wo evens is he ime inerval measured by an observer who is a res relaive o he evens and sees hem ourring a he same plae. All oher moving inerial observers will measure a larger value for his ime inerval. The proper lengh L 0 of an obje is he lengh measured by an observer who is a res wih respe o he obje. All oher moving inerial observers will measure a shorer value for his lengh. The observer who measures he proper ime inerval may no be he same one who measures he proper lengh. For insane, Figure 8.6 shows ha he asronau measures he proper ime inerval 0 for he rip beween earh and Alpha Cenauri, whereas he earh-based observer measures he proper lengh (or disane) L 0 for he rip.
I should be emphasized ha he word proper in he phrases proper ime and proper lengh does no mean ha hese quaniies are he orre or preferred quaniies in any absolue sense. If his were so, he observer measuring hese quaniies would be using a preferred referene frame for making he measuremen, a siuaion ha is prohibied by he relaiviy posulae. Aording o his posulae, here is no preferred inerial referene frame. When wo observers are moving relaive o eah oher a a onsan veloiy, eah measures he oher person s lok o run more slowly han his own, and eah measures he oher person s lengh, along ha person s moion, o be onraed. Thus far we have disussed how ime inervals and disanes beween wo evens are measured by observers moving a a onsan veloiy relaive o eah oher. Speial relaiviy also alers our ideas abou momenum and energy. Reall from Seion 7. ha when wo or more objes inera, he priniple of onservaion of linear momenum applies if he sysem of objes is isolaed. This priniple saes ha he oal linear momenum of an isolaed sysem remains onsan a all imes. (An isolaed sysem is one in whih he sum of he exernal fores aing on he objes is zero.) The onservaion of linear momenum is a law of physis and, in aord wih he relaiviy posulae, is valid in all inerial referene frames. Tha is, when he oal linear momenum is onserved in one inerial referene frame, i is onserved in all inerial referene frames. As an example of momenum onservaion, suppose ha several people are wahing wo billiard balls ollide on a friionless pool able. One person is sanding nex o he pool able, and he oher is moving pas he able wih a onsan veloiy. Sine he wo balls onsiue an isolaed sysem, he relaiviy posulae requires ha boh observers momenum p of an obje o be he produ of is mass m and veloiy v. As a resul, he magniude of he lassial momenum is p m. As long as he speed of an obje is onspeed approahes he speed of ligh, an analysis of he ollision shows ha he oal linear
simply as he produ of mass and veloiy. In order o preserve he onservaion of linear ha he magniude of he relaivisi momenum Magniude of he relaivisi momenum p (8.3) The oal relaivisi momenum of an isolaed sysem is onserved in all inerial referene frames. From Equaion 8.3, we an see ha he magniudes of he relaivisi and nonrelaivisi momena differ by he same faor of / ha ours in he ime-dilaion and lengh-onraion equaions. Sine his faor is always less han and ours in he denominaor in Equaion 8.3, he relaivisi momenum is always larger han he nonrelaivisi momenum. To illusrae how he wo quaniies differ as he speed inreases, Figure 8.7 shows a plo of he raio of he momenum magniudes (relaivisi/nonrelaivisi) as a funion of. Aording o Equaion 8.3, his raio is jus / /. The graph shows ha for speeds aained by ordinary objes, suh as ars and planes, he relaivisi and nonrelaivisi momena are almos equal beause heir raio is nearly. Thus, a speeds muh less han he speed of ligh, eiher he nonrelaivisi momenum or he relaivisi momenum an be used o desribe ollisions. On he oher hand, when he speed of he obje beomes omparable o he speed of ligh, he relaivisi momenum beomes m This graph shows how he raio of he magniude of he relaivisi momenum o he magniude of he nonrelaivisi momenum inreases as he speed of an obje approahes he speed of ligh. wih he relaivisi momenum of an eleron raveling lose o he speed of ligh. The parile aeleraor a Sanford Universiy (Figure 8.8) is 3 km long and aeleraes elerons o a speed of 0.999 999 999 7, whih is very nearly equal o he speed of ligh. Find he magniude of he relaivisi momenum of an eleron ha emerges from he aeleraor, and ompare i wih he nonrelaivisi value. The magniude of he eleron s relaivisi momenum an be obained from Equaion 8.3 if we reall ha he mass of an eleron is m 9. 0 3 kg: p m This value for he magniude of he momenum agrees wih he value measured experimenally. The relaivisi momenum is greaer han he nonrelaivisi momenum by a faor of (9. 0 3 kg)(0.999 999 999 7) (0.999 999 999 7) (0.999 999 999 7) 0 7 kg m /s 4 0 4 One of he mos asonishing resuls of speial relaiviy is ha mass and energy are equivalen, in he sense ha a gain or loss of mass an be regarded equally well as a gain or loss of energy. Consider, for example, an obje of mass m raveling a a speed. Einsein showed ha he oal energy E of he moving obje is relaed o is mass and speed by he following relaion: Toal energy of an obje E m (8.4) The Sanford 3-km linear aeleraor aeleraes elerons almos o he speed of ligh. ( Bill Marsh/ Phoo Researhers, In.)
To gain some undersanding of Equaion 8.4, onsider he speial ase in whih he obje is a res. When 0 m/s, he oal energy is alled he res energy E 0, and Equaion 8.4 redues o Einsein s now-famous equaion: Res energy of an obje E 0 m (8.5) The res energy represens he energy equivalen of he mass of an obje a res. As Example 6 shows, even a small mass is equivalen o an enormous amoun of energy. A 0.046-kg golf ball is lying on he green, as Figure 8.9 illusraes. If he res energy of his ball were used o operae a 75-W ligh bulb, how long would he bulb remain li? The average power delivered o he ligh bulb is 75 W, whih means ha i uses 75 J of energy per seond. Therefore, he ime ha he bulb would remain li is equal o he oal energy used by he ligh bulb divided by he energy per seond (i.e., he average power) delivered o i. This energy omes from he res energy of he golf ball, whih is equal o is mass imes he speed of ligh squared. The daa for his problem are: Desripion Symbol Value Mass of golf ball m 0.046 kg Average power delivered o ligh bulb P 75 W Unknown Variable Time ha ligh bulb would remain li? The res energy of a golf ball is for an inredibly long ime (see Example 6). Power The average power P is equal o he energy delivered o he ligh bulb divided by he ime (see Seion 6.7 and Equaion 6.0b), or P Energy/. In his ase he energy omes from he res energy E 0 of he golf ball, so P E 0 /. Solving for he ime gives Equaion a he righ. The average power is known, and he res energy will be evaluaed in Sep.? E 0 P () Res Energy The res energy E 0 is he oal energy of he golf ball as i ress on he green. If he golf ball s mass is m, hen is res energy is E 0 m (8.5) where is he speed of ligh in a vauum. Boh m and are known, so we subsiue his expression for he res energy ino Equaion, as indiaed a he righ. E 0 P E 0 m () (8.5) Algebraially ombining he resuls of eah sep, we have E 0 P m P
The ime ha he ligh bulb would remain li is m P Expressed in years ( yr 3. 0 7 s), his ime is equivalen o (5.5 0 3 s) (0.046 kg)(3.0 0 8 m/s) yr 3. 0 7 s 75 W 5.5 0 3 s.7 0 6 yr or.7 million years! Problems 30, 3 When an obje is aeleraed from res o a speed, he obje aquires kinei energy in addiion o is res energy. The oal energy E is he sum of he res energy E 0 and he kinei energy KE, or E E 0 KE. Therefore, he kinei energy is he differene beween he obje s oal energy and is res energy. Using Equaions 8.4 and 8.5, we an wrie he kinei energy as Kinei energy of an obje KE E E 0 m (8.6) This equaion is he relaivisially orre expression for he kinei energy of an obje of mass m moving a speed. Equaion 8.6 looks nohing like he kinei energy expression inrodued in Seion 6. namely, KE m (Equaion 6.). However, for speeds muh less han he speed of ligh ( ), he relaivisi equaion for he kinei energy redues o KE m, as an be seen by using he binomial expansion* o represen he square roo erm in Equaion 8.6: Suppose ha is muh smaller han say, 0.0. The seond erm in he expansion has he value (, while he hird erm has he muh smaller value 3 ( 3.8 0 9 8 / ) / ) 5.0 0 5. The addiional erms are smaller sill, so if, we an negle 3 8 KE m whih is he familiar form for he kinei energy. However, Equaion 8.6 gives he orre kinei energy for all speeds and mus be used for speeds near he speed of ligh, as in Example 7. m An eleron (m 9.09 0 3 kg) is aeleraed from res o a speed of 0.9995 in a parile aeleraor. Deermine he eleron s (a) res energy, (b) oal energy, and () kinei energy in millions of eleron vols or MeV. (a) The eleron s res energy is E 0 m (9.09 0 3 kg)(.998 0 8 m/s) 8.87 0 4 J (8.5) *The binomial expansion saes ha ( x) n nx n(n )x /. In our ase, x / and n /.
Sine ev.60 0 9 J, he eleron s res energy is (8.87 0 4 J) (b) The oal energy of an eleron raveling a a speed of E m ev.60 0 9 J.59 0 J or 6. MeV 0.9995 is (9.09 0 3 kg)(.998 0 8 m/s) () The kinei energy is he differene beween he oal energy and he res energy: KE E E 0.59 0 J 8. 0 4 J.5 0 J or 5.7 MeV For omparison, if he kinei energy of he eleron had been alulaed from of only 0.6 MeV would have been obained. 5. 0 5 ev or 0.5 MeV 0.9995 m (8.4) (8.6), a value Sine mass and energy are equivalen, any hange in one is aompanied by a orresponding hange in he oher. For insane, life on earh is dependen on eleromagnei energy (ligh) from he sun. Beause his energy is leaving he sun (see Figure 8.0), here is a derease in he sun s mass. Example 8 illusraes how o deermine his derease. The sun emis eleromagnei energy over a broad porion of he eleromagnei sperum. These phoographs were obained using ha energy in he indiaed regions of he sperum. (Top, Mark Maren/ NASA/Phoo Researhers, In.; boom, Dr. Leon Golub/Phoo Researhers, In.) The sun radiaes eleromagnei energy a he rae of 3.9 0 6 W. (a) Wha is he hange in he sun s mass during eah seond ha i is radiaing energy? (b) The mass of he sun is.99 0 30 kg. Wha fraion of he sun s mass is los during a human lifeime of 75 years? Sine W J/s, he amoun of eleromagnei energy radiaed during eah seond is 3.9 0 6 J. Thus, during eah seond, he sun s res energy dereases by his amoun. The hange E 0 in he sun s res energy is relaed o he hange m in is mass by E 0 ( m), aording o Equaion 8.5. (a) For eah seond ha he sun radiaes energy, he hange in is mass is m Over 4 billion kilograms of mass are los by he sun during eah seond. (b) The amoun of mass los by he sun in 75 years is m (4.36 0 9 kg/s) E 0 3.9 0 6 J (3.00 0 8 m /s) 3.6 0 7 s year Alhough his is an enormous amoun of mass, i represens only a iny fraion of he sun s oal mass: m.0 0 9 kg m sun.99 0 30 kg 4.36 0 9 kg (75 years).0 0 9 kg 5.0 0 Any hange E 0 in he res energy of a sysem auses a hange in he mass of he sysem aording o E 0 ( m). I does no maer wheher he hange in energy is due o a hange in eleromagnei energy, poenial energy, hermal energy, or so on. Alhough any hange in energy gives rise o a hange in mass, in mos insanes he hange in mass is oo small o be deeed. For insane, when 486 J of hea is used o raise he emperaure of kg of waer by C, he mass hanges by only m E 0 / (486 J)/(3.00 0 8 m/s) 4.7 0 4 kg. Conepual Example 9 illusraes furher how a hange in he energy of an obje leads o an equivalen hange in is mass.
Figure 8.a shows a op view of a massless spring on a horizonal able. Iniially he spring is unsrained. Then he spring is eiher srehed or ompressed by an amoun x from is unsrained lengh, as Figure 8.b illusraes. Wha is he mass of he spring in Figure 8.b? (a) I is greaer han zero by an amoun ha is larger when he spring is srehed. (b) I is greaer han zero by an amoun ha is larger when he spring is ompressed. () I is greaer han zero by an amoun ha is he same when he spring is srehed or ompressed. (d) I remains zero. When a spring is srehed or ompressed, is elasi poenial energy hanges. As disussed in Seion 0.3, he elasi poenial energy of an ideal spring is equal o kx, where k is he spring onsan and x is he amoun of sreh or ompression. Consisen wih he heory of speial relaiviy, any hange in he oal energy of a sysem, inluding a hange in he elasi poenial energy, is equivalen o a hange in he mass of he sysem. In being srehed or ompressed by he same amoun x, he spring aquires he same amoun of elasi poenial energy ( kx ). Therefore, aording o speial relaiviy, he spring aquires he same mass regardless of wheher i is srehed or ompressed, so hese answers mus be inorre. The spring aquires elasi poenial energy in being srehed or ompressed. Speial relaiviy indiaes ha his addiional energy is equivalen o addiional mass. Sine he amoun of sreh or ompression is he same, he poenial energy is he same in eiher ase, and so is he addiional mass. Problem 6 (a) (b) x (a) This spring is unsrained and assumed o have no mass. (b) When he spring is eiher srehed or ompressed by an amoun x, i gains elasi poenial energy and, hene, mass. +x I is also possible o ransform maer iself ino oher forms of energy. For example, he posiron (see Seion 3.4) has he same mass as an eleron bu an opposie elerial harge. If hese wo pariles of maer ollide, hey are ompleely annihilaed, and a burs of high-energy eleromagnei waves is produed. Thus, maer is ransformed ino eleromagnei waves, he energy of he eleromagnei waves being equal o he oal energies of he wo olliding pariles. The medial diagnosi ehnique known as posiron emission omography or PET sanning depends on he eleromagnei energy produed when a posiron and an eleron are annihilaed (see Seion 3.6). The ransformaion of eleromagnei waves ino maer also happens. In one experimen, an exremely high-energy eleromagnei wave, alled a gamma ray (see Seion reae an eleron and a posiron. The gamma ray disappears, and he wo pariles of maer appear in is plae. Exep for piking up some momenum, he nearby nuleus remains unhanged. The proess in whih he gamma ray is ransformed ino he wo pariles is known as pair produion. I is possible o derive a useful relaion beween he oal relaivisi energy E and he relaivisi momenum p. We begin by rearranging Equaion 8.3 for he momenum, o obain Wih his subsiuion, Equaion 8.4 for he oal energy beomes Using his expression o replae / in Equaion 8.4 gives E m / m p /E or E m 4 Solving his expression for E shows ha E m p / m / p or E p m 4 p E p /E (8.7)
One of he imporan onsequenes of he heory of speial relaiviy is ha objes wih mass anno reah he speed of ligh in a vauum. Thus, he speed of ligh in a vauum represens he ulimae speed. To see ha his speed limiaion is a onsequene of speial relaiviy, onsider Equaion 8.6, whih gives he kinei energy of a moving obje. As approahes he speed of ligh, he / erm in he denominaor approahes zero. he onlusion ha objes wih mass anno aain he speed of ligh. The veloiy of an obje relaive o an observer plays a enral role in speial relaiviy, and o deermine his veloiy, i is someimes neessary o add wo or more veloreviewing some of he ideas presened here. Figure 8. illusraes a ruk moving a a onsan veloiy of TG 5 m/s relaive o an observer sanding on he ground, where he plus sign denoes a direion o he righ. Suppose someone on he ruk hrows a baseball oward he observer a a veloiy of BT 8.0 m/s relaive o he ruk. We migh onlude ha he observer on he ground would see he ball approahing a a veloiy of BG BT TG 8.0 m/s 5 m/s 3 m/s. These symbols are similar o hose used in Seion 3.4 and have he following meaning: BG BT TG veloiy of he Baseball relaive o he Ground 3 m /s veloiy of he Baseball relaive o he Truk 8.0 m/s veloiy of he Truk relaive o he Ground 5.0 m /s Alhough he resul ha BG 3m/s seems reasonable, areful measuremens would show ha i is no quie righ. Aording o speial relaiviy, he equaion is no valid for he following reason. If he veloiy of he ruk had a BG BT TG The ruk is approahing he ground-based observer a a relaive veloiy of TG 5 m/s. The veloiy of he baseball relaive o he ruk is 8.0 m/s. BT
ha he observer on he earh ould see he baseball moving faser han he speed of ligh. ligh in a vauum. For he ase in whih he ruk and ball are moving along he same sraigh line, he heory of speial relaiviy reveals ha he veloiies are relaed aording o BG BT TG BT TG 8.. For he general siuaion, he relaive veloiies are relaed by he veloiy-addiion formula: Veloiy addiion (8.8) where all he veloiies are assumed o be onsan and he symbols have he following meanings: veloiy of obje A relaive o obje B AB AB AC CB AC CB AC veloiy of obje A relaive o obje C CB veloiy of obje C relaive o obje B The ordering of he subsrips in Equaion 8.8 follows he disussion in Seion 3.4. For moion along a sraigh line, he veloiies an have eiher posiive or negaive values, depending on wheher hey are direed along he posiive or negaive direion. Furhermore, swihing he order of he subsrips hanges he sign of he veloiy, so, for example, BA AB (see Example in Chaper 3). Equaion 8.8 differs from he nonrelaivisi formula ( AB AC CB ) by he presene of he AC CB/ erm in he denominaor. This erm arises beause of he effes of ime dilaion and lengh onraion ha our in speial relaiviy. When AC and CB are small ompared o, he AC CB/ erm is small ompared o, so he veloiy-addiion formula redues o AB AC CB. However, when eiher AC or CB is omparable o, he resuls an be quie differen, as Example 0 illusraes. Imagine a hypoheial siuaion in whih he ruk in Figure 8. is moving relaive o he ground wih a veloiy of TG 0.8. A person riding on he ruk hrows a baseball a a veloiy relaive o he ruk of BT o a person sanding on he ground? 0.5. Wha is he veloiy BG of he baseball relaive The observer sanding on he ground does no see he baseball approahing a BG 0.5 0.8.3. This anno be, beause he speed of he ball would hen exeed he speed of ligh in a vauum. The veloiy-addiion formula gives he orre veloiy, whih has a magniude less han he speed of ligh. The ground-based observer sees he ball approahing wih a veloiy of BG BT TG BT TG 0.5 0.8 (0.5)(0.8) 0.93 (8.8) Example 0 disusses how he speed of a baseball is viewed by observers in differen inerial referene frames. The nex example deals wih a similar siuaion, exep ha he baseball is replaed by he ligh of a laser beam.
a beam of laser ligh. An inergalai ruiser, Figure 8.3 shows an inergalai ruiser approahing a hosile spaeraf. Boh vehiles move a a onsan veloiy. The veloiy of he ruiser relaive o he spaeraf is CS 0.7, he hosile renegades. The veloiy of he laser beam relaive o he ruiser is LC. Whih one of he following saemens orrely desribes he veloiy LS of he laser beam relaive o he renegades spaeraf and he veloiy a whih he renegades see he laser beam move away from he ruiser? (a) LS 0.7 and (b) LS 0.3 and () LS and 0.7 (d) LS and 0.3 Sine boh vehiles move a a onsan veloiy, eah onsiues an inerial referene frame. Aording o he speed-of-ligh posulae, all observers in inerial referene frames measure he speed of ligh in a vauum o be. Sine he renegades spaeraf onsiues an inerial referene frame, he veloiy of he laser beam relaive o i an only have a value of LS, aording o he speed-of-ligh posulae. The veloiy a whih he renegades see he laser beam move away from he ruiser anno be 0.7, beause hey see he ruiser moving a a veloiy of 0.7 and he laser beam moving a a veloiy of only (no.4). The renegades see he ruiser approah hem a a relaive veloiy of CS 0.7 and see he laser beam approah hem a a relaive veloiy of LS. Boh hese veloiies are measured relaive o he same inerial referene frame namely, ha of heir own spaeraf. Therefore, he renegades see he laser beam move away from he ruiser a a veloiy ha is he differene beween hese wo veloiies, or ( 0.7) 0.3. The veloiy-addiion formula, Equaion 8.8, does no apply here beause boh veloiies are measured relaive o he same inerial referene frame. Equaion 8.8 is used only when he veloiies are measured relaive o differen inerial referene frames. Problem 46 I is a sraighforward maer o show ha he veloiy-addiion formula is onsisen wih he speed-of-ligh posulae. Consider Figure 8.4, whih shows a person riding on is LT. The veloiy LG of he ligh relaive o he observer sanding on he ground is given by he veloiy-addiion formula as LG LT TG LT TG Thus, he veloiy-addiion formula indiaes ha he observer on he ground and he person on he ruk boh measure he speed of ligh o be, independen of he relaive veloiy TG beween hem. This is exaly wha he speed-of-ligh posulae saes. LT = + TG TG ( TG) ( TG) The speed of he ligh relaive o boh he ruk and he observer on he ground. TG
There are many asonishing onsequenes of speial relaiviy, wo of whih are ime dilaion and lengh onraion. Example reviews hese imporan oneps in he onex of a golf game in a hypoheial world where he speed of ligh is a lile faser han ha of a golf ar. Imagine playing golf in a world where he speed of ligh is only 3.40 m/s. Golfer A drives ing in a ar, happens o pass by jus as he ball is hi (see Figure 8.5). Golfer A sands a he ee and wahes while golfer B moves down he fairway oward he ball a a onsan speed of.80 m/s. (a) How far is he ball hi aording o a measuremen made by golfer B? (b) Aording o eah golfer, how muh ime does i ake for golfer B o reah he ball? measures he onraed lengh? Who measures he proper lengh of he drive, and who Answer Consider wo oordinae sysems, one aahed o he earh and he oher o he golf ar. The proper lengh L 0 is he disane beween wo poins as measured by an observer who is a res wih respe o hem. Sine golfer A is sanding a he ee and is a res relaive o he earh, golfer A measures he proper lengh of he drive, whih is L 0 75.0 m. Golfer B is no a res wih respe o he wo poins, however, and measures a onraed lengh for he drive ha is less han 75.0 m. Who measures he proper ime inerval, and who measures he dilaed ime inerval? Answer The proper ime inerval 0 is he ime inerval measured by an observer who is a res in his or her oordinae sysem and who views he beginning and ending evens as ourring a he same plae. The beginning even is when he ball is hi, and he ending even is when golfer B arrives a he ball. When he ball is hi, i is alongside he origin of golfer B s oordinae sysem. When golfer B arrives a he ball down he fairway, i is again alongside he origin of his oordinae sysem. Thus, golfer B measures he proper ime inerval. Golfer A, who does no see he beginning and ending evens ourring a he same plae, measures a dilaed, or longer, ime inerval. Jus as golfer A his he ball, golfer B passes by in a golf ar. Aording o speial relaiviy, eah golfer measures a differen disane for how far he ball is hi. (a) Golfer A, being a res wih respe o he beginning and ending poins, measures he proper lengh of he drive, so L 0 75.0 m. Golfer B, who is moving, measures a onraed lengh L given by Equaion 8.: L L 0 (75.0 m) (.80 m/s) (3.40 m/s) 4.5 m Thus, he moving golfer measures he lengh of he drive o be a shorened 4.5 m raher han he 75.0 m measured by he saionary golfer. (b) Aording o golfer B, he ime inerval 0 i akes o reah he ball is equal o he onraed lengh L ha he measures divided by he speed of he ground wih respe o him. The speed of he ground wih respe o golfer B is he same as he speed of he ar wih respe 0 L 4.5 m.80 m /s 5. s
Golfer A, sanding a he ee, measures a dilaed ime inerval, whih is relaed o he proper ime inerval by Equaion 8.: 0 5. s (.80 m/s) (3.40 m/s) 6.8 s In summary, golfer A measures he proper lengh (75.0 m) and a dilaed ime inerval (6.8 s), and golfer B measures a shorened lengh (4.5 m) and a proper ime inerval (5. s). Oher imporan onsequenes of speial relaiviy are he equivalene of mass and energy, and he dependene of kinei energy on he oal energy and on he res energy. Example 3 ompares hese properies for hree differen pariles. The res energy E 0 and he oal energy E of hree pariles, expressed in erms of a basi amoun of energy E 5.98 0 0 J, are lised in he able below. The speeds of hese pariles are large, in some ases approahing he speed of ligh. For eah parile, deermine is (a) mass and (b) kinei energy. Res Toal Parile Energy Energy a E E b E 4E 5E 6E Answer The res energy is he energy ha an obje has when is speed is zero. Aording o speial relaiviy, he res energy E 0 and he mass m are equivalen. The relaion beween he wo is given by Equaion 8.5 as E 0 m, where is he speed of ligh in a vauum. Thus, he res energy is direly proporional o he mass. From he able i an be seen ha pariles a and b have idenial res energies, so hey have idenial masses. Parile has is, hen a and b (a ie). Is he kinei energy KE given by he expression KE m, and wha is he ranking (larges Answer No, beause he expression KE m applies only when he speed of he obje is muh, muh less han he speed of ligh. Aording o speial relaiviy, he kinei energy is he differene beween he oal energy E and he res energy E 0, so KE E E 0. Therefore, we an examine he able and deermine he kinei energy of eah parile in erms of E. The kinei energies of pariles a, b, and are, respeively, E E E, 4E E 3E, and 6E 5E E. is b, hen a and (a ie). E 0 (a) The mass of parile a an be deermined from is res energy E 0 E (see he able), is mass is m a E 5.98 0 0 J (3.00 0 8 m /s) 6.64 0 7 kg m. Sine m b 6.64 0 7 kg As expeed, he ranking is m m a m b. and m 33. 0 7 kg
(b) Aording o Equaion 8.6, he kinei energy KE of a parile is equal o is oal energy E minus is res energy E 0 ; KE E E 0. For parile a, is oal energy is E E and is res energy is E 0 E, so is kinei energy is KE a E E E The kinei energies of pariles b and an be deermined in a similar fashion: KE b 7.9 0 0 J and As aniipaed, he ranking is KE b KE a KE. 5.98 0 0 J KE 5.98 0 0 J If you need more help wih a onep, use he Learning Aids noed nex o he disussion or equaion. Examples (Ex.) are in he ex of his haper. Go o for he following Learning Aids: Models for erain ypes of problems in he haper homework. The alulaions are arried ou ineraively. An even is a physial happening ha ours a a erain plae and ime. To reord he even an observer uses a referene frame ha onsiss of a oordinae sysem and a lok. Differen observers may use differen referene frames. The heory of speial relaiviy deals wih inerial referene frames. An inerial referene frame is one in whih Newon s law of ineria is valid. Aeleraing referene frames are no inerial referene frames. The heory of speial relaiviy is based on wo posulaes. The relaiviy posulae saes ha he laws of physis are he same in every inerial referene frame. The speed-of-ligh posulae says ha he speed of ligh in a vauum, measured in any inerial referene frame, always has he same value of, no maer how fas he soure of he ligh and he observer are moving relaive o eah oher. The proper ime inerval 0 beween wo evens is he ime inerval measured by an observer who is a res relaive o he evens and views hem ourring a he same plae. An observer who is in moion wih respe o he evens and who views hem as ourring a differen plaes measures a dilaed ime inerval. The dilaed ime inerval is greaer han he proper ime inerval, aording o he ime-dilaion equaion: 0 (8.) In his expression, who measures. is he relaive speed beween he observer who measures 0 and he observer The proper lengh L 0 beween wo poins is he lengh measured by an observer who is a res relaive o he poins. An observer moving wih a relaive speed parallel o he line beween he wo poins does no measure he proper lengh. Insead, suh an observer measures a onraed lengh L given by he lengh-onraion formula: L L 0 (8.) Lengh onraion ours only along he direion of he moion. Those dimensions ha are perpendiular o he moion are no shorened. The observer who measures he proper lengh may no be he observer who measures he proper ime inerval.
An obje of mass m, moving wih speed, has a relaivisi momenum whose magniude p is given by p (8.3) Energy and mass are equivalen. The oal energy E of an obje of mass m, moving a speed, is E m m (8.4) The res energy E 0 is he oal energy of an obje a res ( 0 m/s): E 0 m (8.5) An obje s oal energy is he sum of is res energy and is kinei energy KE, or E E 0 KE. Therefore, he kinei energy is KE E E 0 m (8.6) The relaivisi oal energy and momenum are relaed aording o E p m 4 (8.7) Objes wih mass anno aain he speed of ligh, whih is he ulimae speed for suh objes. Aording o speial relaiviy, he veloiymove along he same sraigh line, his formula is AB AC (8.8) where AB is he veloiy of obje A relaive o obje B, AC is he veloiy of obje A relaive o obje C, and CB is he veloiy of obje C relaive o obje B. The veloiies an have posiive or negaive values, depending on wheher hey are direed along he posiive or negaive direion. Furhermore, swihing he order of he subsrips hanges he sign of he veloiy, so ha, for example, BA AB. CB AC CB Noe o Insruors: are available online. However, all of he quesions are available for assignmen via an online homework managemen program suh as WileyPLUS or WebAssign. Seion 8. Evens and Inerial Referene Frames. Consider a person along wih a frame of referene in eah of he following siuaions. In whih one or more of he following siuaions is he frame of referene an inerial frame of referene? (a) The person is osillaing in simple harmoni moion a he end of a bungee ord. (b) The person is in a ar going around a irular urve a a onsan speed. () The person is in a plane ha is landing on an airraf arrier. (d) The person is in he spae shule during lif-off. (e) None of he above. Seion 8.3 The Relaiviy of Time: Time Dilaion. of he road where work is being done. Who measures he proper ime (a) A worker sanding sill on he road (b) A driver in a ar approahing a a onsan veloiy () Boh he worker and he driver (d) Neiher he worker nor he driver Seion 8.4 The Relaiviy of Lengh: Lengh Conraion 4. Two spaerafs A and B are moving relaive o eah oher a a onsan veloiy. Observers in spaeraf A see spaeraf B. Likewise, observers in spaeraf B see spaeraf A. Who sees he proper lengh of eiher spaeraf? (a) Observers in spaeraf A see he proper lengh of spaeraf B. (b) Observers in spaeraf B see he proper lengh of spaeraf A. () Observers in boh spaerafs see he proper lengh of he oher spaeraf. (d) Observers in neiher spaeraf see he proper lengh of he oher spaeraf. 6.
he baer runs a a onsan veloiy. Who measures he proper ime (a) The aher measures he proper ime, and he runner measures he proper lengh. (b) The runner measures he proper ime, and he aher measures he proper lengh. () The aher measures boh he proper ime and he proper lengh. (d) The runner measures boh he proper ime and he proper lengh. 7. To whih one or more of he following siuaions do he imedilaion and lengh-onraion equaions apply? (a) Wih respe o an inerial frame, wo observers have differen onsan aeleraions. (b) Wih respe o an inerial frame, wo observers have he same onsan aeleraion. () Wih respe o an inerial frame, wo observers are moving wih differen onsan veloiies. (d) Wih respe o an inerial frame, one observer has a onsan veloiy, and anoher observer has a onsan aeleraion. (e) All of he above. Seion 8.5 Relaivisi Momenum 0. Whih one of he following saemens abou linear momenum is rue ( p magniude of he momenum, m mass, and speed)? (a) When he magniude p m p, he linear momenum of an isolaed sysem is onserved only if he speeds of he various pars of he sysem are very / high. (b) When he magniude p p m, he linear momenum of an isolaed sysem is onserved only if he speeds of he various pars of he sysem are very high. () When he magniude p m as p, he linear momenum of an isolaed sysem is / onserved no maer wha he speeds of he various pars of he sysem are. (d) When he magniude p as p m, he linear momenum of an isolaed sysem is onserved no maer wha he speeds of he various pars of he sysem are.. Whih of he following wo expressions for he magniude p of he linear momenum an be used when he speed of an obje of mass m is very small ompared o he speed of ligh in a vauum? A. p B. p m (a) Only A (b) Only B () Neiher A nor B (d) Boh A and B Seion 8.6 The Equivalene of Mass and Energy 3. Consider he following hree possibiliies for a glass of waer a res on a kihen ouner. The emperaure of he waer is 0 C. Rank A. The waer is half liquid and half ie. B. The waer is all liquid. C. The waer is all ie. (a) C, A, B (b) B, A, C () A, C, B (d) B, C, A (e) C, B, A 5. An obje has a kinei energy KE and a poenial energy PE. I also has a res energy E 0. Whih one of he following is he orre way o express he obje s oal energy E? (a) E KE PE (b) E E 0 KE () E E 0 KE PE (d) E E 0 KE PE 7. The kinei energy of an obje of mass m is equal o is res energy. Wha is he magniude p of he obje s momenum? (a) p 3m (b) p m () p 4m (d) p m (e) p 3m Seion 8.7 m The Relaivisi Addiion of Veloiies 8. Two spaeships are raveling in he same direion. Wih respe o an inerial frame of referene, spaeship A has a speed of 0.900. Wih respe o he same inerial frame, spaeship B has a speed of 0.500. Find he speed AB of spaeship A relaive o spaeship B. Noe o Insruors: Mos of he homework problems in his haper are available for assignmen via an online homework managemen program suh as WileyPLUS or WebAssign, and hose marked wih he ion are presened in WileyPLUS using a guided uorial forma ha provides enhaned ineraiviy. See Prefae for addiional deails. Before doing any alulaions involving ime dilaion or lengh onraion, i is useful o idenify whih observer measures he proper ime inerval 0 or he proper lengh L 0. Soluion is in he Suden Soluions Manual. Soluion is available online a www.wiley.om/ollege/unell This ion represens a biomedial appliaion. Seion 8.3. The Relaiviy of Time: Time Dilaion fas is he polie ar moving relaive o he earh?. A parile known as a pion lives for a shor ime before breaking apar ino oher pariles. Suppose ha a pion is moving a a speed of 0.990, and an observer who is saionary in a laboraory measures he pion s lifeime o be 3.5 0 8 s. (a) Wha is he lifeime aording o a hypoheial person who is riding along wih he pion? (b) Aording o his hypoheial person, how far does he laboraory move before he pion breaks apar? 3. A Klingon spaeraf has a speed of 0.75 wih respe o he earh. The Klingons measure 37.0 h for he ime inerval beween wo evens on he earh. Wha value for he ime inerval would hey measure if heir ship had a speed of 0.94 wih respe o he earh? 4. Suppose ha you are raveling on board a spaeraf ha is moving wih respe o he earh a a speed of 0.975. You are breahing a a rae of 8.0 breahs per minue. As moniored on earh, wha is your breahing rae? * 5. a www.wiley.om/ollege/unell illusraes one way o model his problem. A 6.00-kg obje osillaes bak and forh a he end of a spring whose spring onsan is 76.0 N/m. An observer is raveling a a speed of.90 0 8 m/s relaive o he period of osillaion? * 6. A spaeship ravels a a onsan speed from earh o a plane orbiing anoher sar. When he spaeraf arrives, years have elapsed on earh, and 9. years have elapsed on board he ship. How far away (in meers) is he plane, aording o observers on earh?
** 7. As observed on earh, a erain ype of baerium is known o ** 7. Twins who are 9.0 years of age leave he earh and ravel o double in number every 4.0 hours. Two ulures of hese baeria are prepared, eah onsising iniially of one baerium. One ulure is lef on earh and he oher plaed on a roke ha ravels a a speed of a disan plane.0 ligh-years away. Assume ha he plane and earh are a res wih respe o eah oher. The wins depar a he same ime on differen spaeships. One win ravels a a speed of 0.866 relaive o he earh. A a ime when he earhbound ulure has 0.900, and he oher win ravels a 0.500. (a) Aording o he grown o 56 baeria, how many baeria are in he ulure on he heory of speial relaiviy, wha is he differene beween heir ages roke, aording o an earh-based observer? when hey mee again a he earlies possible ime? (b) Whih win is older? Seion 8.4 The Relaiviy of Lengh: Lengh Conraion 8. A ouris is walking a a speed of.3 m/s along a 9.0-km pah ha follows an old anal. If he speed of ligh in a vauum were 3.0 m/s, how long would he pah be, aording o he ouris? 9. How fas mus a meer sik be moving if is lengh is observed o shrink o one-half of a meer? 0. a www.wiley.om/ollege/ unell reviews he oneps ha play roles in his problem. The disane from earh o he ener of our galaxy is abou 3 000 ly ( ly ligh-year 9.47 0 5 m), as measured by an earh-based observer. A spaeship is o make his journey a a speed of 0.9990. Aording o a lok on board he spaeship, how long will i ake o make he rip? Express your answer in years ( yr 3.6 0 7 s).. A UFO sreaks aross he sky a a speed of 0.90 relaive o he earh. A person on earh deermines he lengh of he UFO o be 30 m along he direion of is moion. Wha lengh does he person measure for he UFO when i lands?. A Marian leaves Mars in a spaeship ha is heading o Venus. On he way, he spaeship passes earh wih a speed 0.80 relaive o i. Assume ha he hree planes do no move relaive o eah oher during he rip. The disane beween Mars and Venus is.0 0 m, as measured by a person on earh. (a) Wha does he Marian measure for he disane beween Mars and Venus? (b) Wha is he ime of he rip (in seonds) as measured by he Marian? 3. a www.wiley/om/ollege/unell illusraes one approah o solving his problem. A spae raveler moving a a speed of 0.70 wih respe o he earh makes a rip o a disan sar ha is saionary relaive o he earh. He measures he lengh of his rip o be 6.5 ligh-years. Wha would be he lengh of his same rip (in ligh-years) as measured by a raveler moving a a speed of 0.90 wih respe o he earh? 4. An unsable high-energy parile is reaed in he laboraory, and i moves a a speed of 0.990. Relaive o a saionary referene frame 0 3 m before disinegraing. Wha are (a) he proper disane and (b) he disane measured by a hypoheial person raveling wih he parile? Deermine he parile s () proper lifeime and (d) is dilaed lifeime. * 5. As he drawing shows, a arpener on a spae saion has onsrued a 30.0 ramp. A roke moves pas he spae saion wih a relaive speed of 0.730 in a direion parallel o side x. Wha does a person aboard he roke measure for he angle of he ramp? ** 6. An obje is made of glass and has he shape of a ube 0. m on a side, aording o an observer a res relaive o i. However, an observer moving a high speed parallel o one of he obje s edges and knowing ha he obje s mass is 3. kg deermines is densiy o be 7800 kg/m 3, whih is muh greaer han he densiy of glass. Wha is he moving observer s speed (in unis of ) relaive o he ube? x y Seion 8.5 Relaivisi Momenum 8. The speed of an ion in a parile aeleraor is doubled from 0.460 o 0.90. The iniial relaivisi momenum of he ion is 5.08 0 7 kg m/s. Deermine (a) he mass and (b) relaivisi momenum of he ion. 9. A jeliner has a mass of. 0 5 (a) Find he magniude of is momenum. (b) If he speed of ligh in a vauum had he hypoheial value of 70 m/s, wha would be he magniude of he jeliner s momenum? 0. Three pariles are lised in he able. The mass and speed of eah parile are given as muliples of he variables m and, whih have he values m.0 0 8 kg and 0.00. The speed of ligh in a vauum is 3.00 0 8 m/s. Deermine he momenum for eah parile aording o speial relaiviy. Parile Mass Speed a m b m 4 4 m. A woman is.6 m all and has a mass of 55 kg. She moves pas an observer wih he direion of he moion parallel o her heigh. The observer measures her relaivisi momenum o have a magniude of.0 0 0 kg m/s. Wha does he observer measure for her heigh?. A spaeraf has a nonrelaivisi (or lassial) momenum whose magniude is.3 0 3 kg m/s. The spaeraf moves a suh a speed ha he pilo measures he proper ime inerval beween wo evens o be one-half he dilaed ime inerval. Find he relaivisi momenum of he spaeraf. * 3. Saring from res, wo skaers push off agains eah oher on smooh level ie, where friion is negligible. One is a woman and one is a man. The woman moves away wih a veloiy of.5 m/s relaive o he ie. The mass of he woman is 54 kg, and he mass of he man is 88 kg. Assuming ha he speed of ligh is 3.0 m/s, so ha man relaive o he ie. (Hin: This problem is similar o Example 6 in Chaper 7.) Seion 8.6 The Equivalene of Mass and Energy 4. An eleron and a posiron have masses of 9. 0 3 kg. They ollide and boh vanish, wih only eleromagnei radiaion appearing afer he ollision. If eah parile is moving a a speed of 0.0 relaive o he laboraory before he ollision, deermine he energy of he eleromagnei radiaion. 5. Deermine he raio of he relaivisi kinei energy o he nonrelaivisi kinei energy ( m when a parile has a speed of (a).00 0 3 ) and (b) 0.970. 6. Review Conepual Example 9 for bakground perinen o his problem. Suppose ha he speed of ligh in a vauum were one million imes smaller han is aual value, so ha 3.00 0 m/s.