Chapter 1. Problem Solutions

Transcription

1 Chapter Prblem Slutins If the speed f light were smaller than it is, wuld relatiisti phenmena be mre r less nspiuus than they are nw? All else being the same, inluding the rates f the hemial reatins that gern ur brains and bdies, relatiisiti phenmena wuld be mre nspiuus if the speed f light were smaller If we uld attain the abslute speeds btainable t us in the unierse as it is, but with the speed f light being smaller, we wuld be able t me at speeds that wuld rrespnd t larger fratins f the speed f light, and in suh instanes relatiisti effets wuld be mre nspiuus 3 An athlete has learned enugh physis t knw that if he measures frm the earth a time interal n a ming spaeraft, what he finds will be greater than what smebdy n the spaeraft wuld measure He therefre prpses t set a wrld rerd fr the 00-m dash by haing his time taken by an bserer n a ming spaeraft Is this a gd idea? Een if the judges wuld allw it, the bserers in the ming spaeship wuld measure a lnger time, sine they wuld see the runners being timed by lks that appear t run slwly mpared t the ship's lks Atually, when the effets f length ntratin are inluded (disussed in Setin 4 and Appendix ), the runner's speed may be greater than, less than, r the same as that measured by an bserer n the grund Inha Uniersity Department f Physis

2 5 Tw bserers, A n earth and B in a spaeraft whse speed is 00 x 0 8 ms, bth set their wathes t the same time when the ship is abreast f the earth (a) Hw muh time must elapse by A's rekning befre the wathes differ by 00 s? (b) T A, B's wath seems t run slw T B, des A's wath seem t run fast, run slw, r keep the same time as his wn wath? Nte that the nnrelatiisti apprximatin is nt alid, as 3 (a) See Example In Equatin (3), with t representing bth the time measured by A and the time as measured in A's frame fr the lk in B's frame t adane by t, we need t t0 t t s t 3 frm whih t 393 s (b) A ming lk always seems t run slwer In this prblem, the time t is the time that bserer A measures as the time that B's lk takes t rerd a time hange f t Inha Uniersity Department f Physis

3 7 Hw fast must a spaeraft trael relatie t the earth fr eah day n the spaeraft t rrespnd t d n the earth? Frm Equatin (3), fr the time t n the earth t rrespnd t twie the time t 0 elapsed n the ship s lk, 3, s 60 0 relating three signifiant figures 9 A ertain partile has a lifetime f 00 x0-7 s when measured at rest Hw far des it g befre deaying if its speed is 099 when it is reated? The lifetime f the partile is t 0, and the distane the partile will trael is, frm Equatin (3), t t ( 0 99) 8 ms, ( 0 99)( ms)( 00 0 s) t tw signifiant figures 0 m Inha Uniersity Department f Physis

4 A galaxy in the nstellatin Ursa Majr is reeding frm the earth at 5,000 kms If ne f the harateristi waelengths f the light the galaxy emits is 550 nm, what is the rrespnding waelength measured by astrnmers n the earth? See Example 3; fr the intermediate alulatins, nte that λ λ, + where the sign nentin fr is that f Equatin (8), whih psitie fr an apprahing sure and negatie fr a reeding sure Fr this prblem, s that 50 0 kms ms , λ λ + ( 550 nm) nm Inha Uniersity Department f Physis

5 3 A spaeraft reeding frm the earth emits radi waes at a nstant frequeny f 0 9 Hz If the reeier n earth an measure frequenies t the nearest hertz, at what spaeraft speed an the differene between the relatiisti and lassial Dppler effets be deteted? Fr the lassial effet, assume the earth is statinary This prblem may be dne in seeral ways, all f whih need t use the fat that when the frequenies due t the lassial and relatiisti effets are fund, thse frequenies, while differing by Hz, will bth be suffiiently lse t 0 9 Hz s that uld be used fr an apprximatin t either In Equatin (4), we hae 0 and V -u, where u is the speed f the spaeraft, ming away frm the earth (V < 0) In Equatin (6), we hae u (r -u in Equatin (8)) The lassial and relatiisti frequenies, and r respetiely, are ( u ) ( u ) 0, r + ( u ) + ( u ) + ( u ) The last expressin fr, is mtiated by the deriatin f Equatin (6), whih essentially inrprates the lassial result (unting the number f tiks), and allws expressin f the rati r ( u ) Inha Uniersity Department f Physis

6 Use f the abe frms fr the frequenies allws the alulatin f the rati Hz 9 r ( u ) 0 + ( u ) 9 0 Hz Attempts t sle this equatin exatly are nt likely t be met with suess, and een numerial slutins wuld require a higher preisin than is mmnly aailable Hweer, regnizing that the numeratr ( u ) is f the frm that an be apprximated using the methds utlined at the beginning f this hapter, we an use ( u ) ( )( u ) The denminatr will be indistinguishable frm at lw speed, with the result u 9 0, whih is sled fr u ms 34kms Inha Uniersity Department f Physis

7 Inha Uniersity Department f Physis 5 If the angle between the diretin f mtin f a light sure f frequeny and the diretin frm it t an bserer is 0, the frequeny the bserer finds is gien by where is the relatie speed f the sure Shw that this frmula inludes Eqs (5) t (7) as speial ases The transerse Dppler effet rrespnds t a diretin f mtin f the light sure that is perpendiular t the diretin frm it t the bserer; the angle θ ±π (r ±90 ), s s θ 0, and whih is Equatin (5) Fr a reeding sure, θ π (r 80 ), and s θ The gien expressin bemes,, + + whih is Equatin (8) Fr an apprahing sure, θ 0, s θ, and the gien expressin bemes, + whih is Equatin (8) θ )s (

8 7 An astrnaut whse height n the earth is exatly 6 ft is lying parallel t the axis f a spaeraft ming at 090 relatie t the earth What is his height as measured by an bserer in the same spaeraft? By an bserer n the earth? The astrnaut s prper length (height) is 6 ft, and this is what any bserer in the spaeraft will measure Frm Equatin (9), an bserer n the earth wuld measure L L ( 6 ft) ( 0 90) 6 ft 9 Hw muh time des a meter stik ming at 000 relatie t an bserer take t pass the bserer? The meter stik is parallel t its diretin f mtin The time will be the length as measured by the bserer diided by the speed, r t 8 L L 00 m 0 00 ( ) ( ) ( 0 00)( ms) s Inha Uniersity Department f Physis

9 A spaeraft antenna is at an angle f 0 relatie t the axis f the spaeraft If the spaeraft mes away frm the earth at a speed f 070, what is the angle f the antenna as seen frm the earth? If the antenna has a length L' as measured by an bserer n the spaeraft (L' is nt either L r L O in Equatin (9)), the prjetin f the antenna nt the spaeraft will hae a length L's(0 ), and the prjetin nt an axis perpendiular t the spaeraft's axis will hae a length L'sin(0 ) T an bserer n the earth, the length in the diretin f the spaeraft's axis will be ntrated as desribed by Equatin (9), while the length perpendiular t the spaeraft's mtin will appear unhanged The angle as seen frm the earth will then be L sin( 0 ) artan L s( 0 ) artan tan( 0 ) ( 0 70) 4 The generalizatin f the abe is that if the angle is 00 as measured by an bserer n the spaeraft, an bserer n the earth wuld measure an angle θ gien by tanθ tanθ Inha Uniersity Department f Physis

10 3 A wman leaes the earth in a spaeraft that makes a rund trip t the nearest star, 4 lightyears distant, at a speed f 09 The age differene will be the differene in the times that eah measures the rund trip t take, r 4 yr ( ) ( 0 9 ) 5 yr L t All definitins are arbitrary, but sme are mre useful than thers What is the bjetin t defining linear mmentum as p m instead f the mre mpliated p gm? It is nenient t maintain the relatinship frm Newtnian mehanis, in that a fre n an bjet hanges the bjet's mmentum; symblially, F dpdt shuld still be alid In the absene f fres, mmentum shuld be nsered in any inertial frame, and the nsered quantity is p -γm, nt m 7 Dynamite liberates abut 54 x 0 6 Jkg when it expldes What fratin f its ttal energy ntent is this? Fr a gien mass M, the rati f the mass liberated t the mass energy is 6 M ( Jkg) M ( ms) Inha Uniersity Department f Physis

11 9 At what speed des the kineti energy f a partile equal its rest energy? If the kineti energy K E m, then E m and Equatin (3) redues t (γ in the ntatin f Setin 7) Sling fr, ms 3 An eletrn has a kineti energy f 000 MeV Find its speed arding t lassial and relatiisti mehanis Classially, Inha Uniersity 9 K 0 00 MeV 60 0 JeV ms kg m e Relatiistially, sling Equatin (3) fr as a funtin f K, m e E m me e + K + K ( m e ) Department f Physis

12 With K(m e ) (000 MeV)(05 MeV) 00005, ms 64 0 ms + ( 0 00)( 0 5) The tw speeds are mparable, but nt the same; fr larger alues f the rati f the kineti and rest energies, larger disrepanies wuld be fund 33 A partile has a kineti energy 0 times its rest energy Find the speed f the partile in terms f Using Equatin () in Equatin (3) and sling fr, With E E, that is, E E + 0E, E E Inha Uniersity Department f Physis

13 Inha Uniersity Department f Physis 35 Hw muh wrk (in MeV) must be dne t inrease the speed f an eletrn frm x 0 8 ms t 4 X 0 8 ms? The differene in energies will be, frm Equatin (3), MeV MeV 0 ) ( ) ( ) ( m e 37 Pre that ½γm, des nt equal the kineti energy f a partile ming at relatiisti speeds Using the expressin in Equatin (0) fr the kineti energy, the rati f the tw quantities is K m γ γ γ

14 39 An alternatie deriatin f the mass-energy frmula E O m, als gien by Einstein, is based n the priniple that the latin f the enter f mass (CM) f an islated system annt be hanged by any press that urs inside the system Figure 7 shws a rigid bx f length L that rests n a fritinless surfae; the mass M f the bx is equally diided between its tw ends A burst f eletrmagneti radiatin f energy E is emitted by ne end f the bx Arding t lassial physis, the radiatin has the mmentum p E, and when it is emitted, the bx reils with the speed E 0 M s that the ttal mmentum f the system remains zer After a time t L the radiatin reahes the ther end f the bx and is absrbed there, whih brings the bx t a stp after haing med the distane S If the CM f the bx is t remain in its riginal plae, the radiatin must hae transferred mass frm ne end t the ther Shw that this amunt f mass is m E O Measured frm the riginal enter f the bx, s that the riginal psitin f the enter f mass is 0, the final psitin f the enter f mass is M L M L m + S + m S 0 Expanding the prduts and aneling similar terms [(M)(L), ms], the result MS ml is btained The distane 5 is the prdut t, where, as shwn in the prblem statement, EM (apprximate in the nnrelatiisti limit M >> El ) and t L Then, MS M E L E Inha Uniersity m L L M Department f Physis

15 4 In its wn frame f referene, a prtn takes 5 min t rss the Milky Way galaxy, whih is abut 0 5 light-years in diameter (a) What is the apprximate energy f the prtn in eletrnlts? (b) Abut hw lng wuld the prtn take t rss the galaxy as measured by an bserer in the galaxy's referene frame? T rss the galaxy in a matter f minutes, the prtn must be highly relatiisti, with (but <, f urse) The energy f the prtn will be E E γ, where E O is the prtn's rest energy and γ Hweer, γ, frm Equatin (9), is the same as the rati L O L, where L is the diameter f the galaxy in the prtn's frame f referene, and fr the highly-relatiisti prtn L t, where t is the time in the prtn's frame that it takes t rss the galaxy Cmbining, E L L 9 0 ly 7 E E E γ ( 0 ev) ( 3 0 syr ) 0 L t ( 300 s) 43 Find the mmentum (in MeV) f an eletrn whse speed is 0600 Taking magnitudes in Equatin (6), m p e ( 0 5 MeV )( ) MeV ( 0 600) 5 9 ev Inha Uniersity Department f Physis

16 45 Find the mmentum f an eletrn whse kineti energy equals its rest energy f 5 kev When the kineti energy f an eletrn is equal t its rest energy, the ttal energy is twie the rest energy, and Equatin (4) bemes 4m 4 e 4 m 4 e 4 + p, r p 3( m ) 3( 5 kev ) 94 GeV The result f Prblem -9 uld be used diretly; γ, ( p m3 e, as abe e ), 3 and Equatin (7) gies 47 Find the speed and mmentum (in GeV) f a prtn whse ttal energy is 3500 GeV Sling Equatin (3) fr the speed in terms f the rest energy E O and the ttal energy E, ( E E) ( ) numerially 888 x 0 8 ms (The result f Prblem -3 des nt gie an answer aurate t three signifiant figures) The alue f the speed may be substituted int Equatin (6) (r the result f Prblem -46), r Equatin (4) may be sled fr the magnitude f the mmentum, p ( E ) ( E ) ( GeV ) ( GeV ) 3 37 GeV Inha Uniersity Department f Physis

17 49 A partile has a kineti energy f 6 MeV and a mmentum f 335 MeV Find its mass (in MeV ) and speed (as a fratin f ) Frm E m + K and Equatin (4), 4 ( m + K ) m + p Expanding the binmial, anelling the m 4 term, and sling fr m, ( p) K ( 335MeV) ( 6 MeV) m 874 MeV K ( 6 MeV) The partile's speed may be fund any number f ways; a ery nenient result is that f Prblem -46, giing p p 335 MeV 0 36 E m + K 874MeV + 6 MeV Inha Uniersity Department f Physis

18 5 An bserer detets tw explsins, ne that urs near her at a ertain time and anther that urs 00 ms later 00 km away Anther bserer finds that the tw explsins ur at the, same plae What time interal separates the explsins t the send bserer? The gien bseratin that the tw explsins ur at the same plae t the send bserer means that x' 0 in Equatin (4), and s the send bserer is ming at a speed 5 x 00 0 m ms t s with respet t the first bserer Inserting this int Equatin (44), x x t ( x t) t t t t t ( x t) x t ( ms) ( 00 ms) 97 ms 8 (998 0 ms) (Fr this alulatin, the apprximatin is ( x t) ( x t ) alid t three signifiant figures) An equally alid methd, and a gd heek, is t nte that when the relatie speed f the bserers (500 x 0 7 ms) has been determined, the time interal that the send bserer measures shuld be that gien by Equatin (3) (but be areful f whih time it t, whih is t) Algebraially and numerially, the different methds gie the same result Inha Uniersity 7 Department f Physis

19 53 A spaeraft ming in the +x diretin reeies a light signal frm a sure in the xy plane In the referene frame f the fixed stars, the speed f the spaeraft is and the signal arries at an angle θ t the axis f the spaeraft (a) With the help f the Lrentz transfrmatin find the angle θ ' at whih the signal arries in the referene frame f the spaeraft (b) What wuld yu nlude frm this result abut the iew f the stars frm a prthle n the side f the spaeraft? (a) A nenient hie fr the rigins f bth the unprimed and primed rdinate systems is the pint, in bth spae and time, where the ship reeies the signal Then, in the unprimed frame (gien here as the frame f the fixed stars, ne f whih may be the sure), the signal was sent at a time t -r, where r is the distane frm the sure t the plae where the ship reeies the signal, and the minus sign merely indiates that the signal was sent befre it was reeied Take the diretin f the ship's mtin (assumed parallel t its axis) t be the psitie x-diretin, s that in the frame f the fixed stars (the unprimed frame), the signal arries at an angle 0 with respet t the psitie x-diretin In the unprimed frame, x r s θ and y r sin θ Frm Equatin (4), x t r sθ ( r ) sθ + ( ) x r, and y y r sin θ Then, Inha Uniersity Department f Physis

20 tanθ 55 A man n the mn sees tw spaeraft, A and B, ming tward him frm ppsite diretins at the respetie speeds f 0800 and 0900 (a) What des a man n A measure fr the speed with whih he is apprahing the mn? Fr the speed with whih he is apprahing B? (b) What des a man n B measure fr the speed with whih he is apprahing the mn? Fr the speed with whih he is apprahing A? (a) If the man n the mn sees A apprahing with speed 0800, then the bserer n A will see the man in the mn apprahing with speed 0800 The relatie elities will hae ppsite diretins, but the relatie speeds will be the same The speed with whih B is seen t apprah A, t an bserer in A, is then V V x x V + ( 0 800)( 0 900) Inha Uniersity y x (sθ sinθ + ( )), and sinθ θ artan sθ + ( ) (b) Frm the frm f the result f part (a), it an be seen that the numeratr f the term in square brakets is less than sinθ, and the denminatr is greater than sθ, and s tan θ and θ < θ when 0 Lking ut f a prthle, the sures, inluding the stars, will appear t be in the diretins lse t the diretin f the ship s mtin than they wuld fr a ship with 0 As, θ 0, and all stars appear t be almst n the ship s axis(farther frward in the field f iew) x Department f Physis

21 (b) Similarly, the bserer n B will see the man n the mn apprahing with speed 0900, and the apparent speed f A, t an bserer n B, will be ( 0 900)( 0 800) (Nte that Equatin (49) is unhanged if V x and are interhanged) S (mn) S B V x O A Inha Uniersity Department f Physis