T T A BA T B 1 5 7

Transcription

1 Hmewrk 5. Write the fllwing equatins in matrix frm: (a) 3x5z7 4z5 3xz x z 7 5 (b) x3z 4x56z 7x89z x z 3. The transpse peratin hanges a lumn vetr t a rw vetr and visa-vera. (a) Find T T T T (b) Find T T Find the dt prdut f the fllwing vetrs: A 5 7, B 4 A BA T B Calulate the fllwing matrix prduts: 4 3 (a) (b) Cnstrut a matrix whih des a mirrr refletin in the z diretin. M z 5. Shw that the matrix u fund in Prblem 4 is its wn inverse. M z M z

2 6. Find a matrix whih rtates a three-dimensinal vetr b +9arund the z axis. We knw that suh a rtatin will leave the z rdinate f a vetr unhanged, it will make the riginal x mpnent the new mpnent, and it will make the riginal mpnent the new x mpnent. Thus: R z, R z r x x z z 7. A general rtatin abut the z axis b an angle an be shwn t be: R z ( ) s sin sin s Shw what this matrix peratin des t an arbitrar vetr r. Des this result seem rret?. R z ( ) r s sin sin s x xs sin s x sin. z z 8. Shw that the inverse f the matrix given in Prblem 7 is just a rtatin abut the z axis f. Nte that s( )=s and sin( )= sin. Then: R z ( ) R z ( ) s sin sin s s sin sin s s sin s sin s sin s sin s sin s sin 9. Find the matrix whih gives a bst in the diretin. B x v/.. (Optinal) Find a bst in a diretin f 45frm the x axis tward the axis. Use rtatin matries. We need t rtate abut the z axis b 45. The matrix fr this is given in Prblem 7 t be: R z (45).

3 Hwever, we need t reast this int fur-dimensinal spae. Sine the rtatin des nthing t the time mpnent, we have: R. Arding t Setin 6, we have: BRB x R v/ v/ v / v / v v

4 Hmewrk 5. On the grid b ding the fllwing: prvided, draw the trajetr fr a partile initiall traveling hrizntall in a gravitatinal field. Use the nn-relativisti energ-mmentum vetr as the prpagatr. Plt nl the x and diretins n the graph. Fllw these instrutins: a. The partile starts at the psitin indiated. b. The x mpnent f the prpagatr is 5 units in length. There is n x mpnent f fre, s the x mpnent f the prpagatr remains nstant.. The mpnent f the prpagatr is initiall zer. The mpnent f the fre is nstant, s the hange in the mpnent f the prpagatr remains nstant. (We take this t be ne unit dwnward with eah step. See (e) belw.) d. Draw the (new) prpagatr frm the urrent psitin. e. Draw the fre as a vetr ging dwnward b ne unit. f. Draw the new prpagatr as the vetr sum f the ld prpagatr and the fre. g. Determine the new psitin f the partile. Chse this t be /5 f the wa alng the prpagatr. (This number is arbitrar, but it shuld be small. The hie f /5 wrks well here. Wh?) h. Repeat (e) (g) as lng as u an.

5 . Nw, let s think abut what Prblem means. Keen mind the fllwing relatinships fr nn-relativisti prpagatin: x = x + ε E x = w x z m mvx E = mv mv z E px p p z E = Fx ε E F F z Let unit n the graph f Prblem t be meter fr distanes and J fr p. Als let m/s. Find the fllwing quantities (be sure t inlude units): a. is the fratin f E we add t x t get x. Hene = / 5 m/j. Nte that E has units f J and x has units f m. b. p x p x = 5 J, s p x = 5 kg m/s..e Hint: use p = J. (Where des this me frm?) Als remember E = m. J p F E mg E E ge E J g E.74 J d. t, the step size in the time dimensin. Nte that r = p, s t = E =.43 m. e. What is the mass f the partile? E = m, s m =.74 kg. f. What is the initial velit f the partile? p x mv x 5 J, v x p x 7. m/s. m g. Estimate the final velit f the partile (the last ne u are able t determine.) The final value f s abut 6 squares = 6 J. Therefre p f mv f 6 J, v f p f.5 m/s. m

6 3. If the time dimensin were t g bak int the page, desribe the appearane f the wrld line ff the partile f Prblem (still taking the prblem t be nn-relativisti). Eah time stes w = E, whih remains nstant. This means that eah suessive pint wuld be an equal distane behind the previus pint in the time dimensin. If the sale fr w is the same as fr x and, there wuld be abut seven data pints per unit displaement in w. 4. If the mtin f the partile in Prblem were relativisti. Fr sake f preisin, let the initial values be: =3, the x mpnent f the prpagatr 5 units t the right, and the hange in the mpnent f the prpagatr unit dwn. Hw wuld the prpagatr differ frm the nn-relativisti ase? Hw wuld the wrld line differ? Eah mpnent f the prpagatr wuld be times the rrespnding nn-relativisti result. T begin with, the prpagatr wuld be just the same, thugh. As the partile aelerates and the velit inreases, the mass wuld inrease, ausing the time step and spatial steps t nstantl grw lnger mpared t the nn-relativisti ase. Eah mpnent f the prpagatr wuld be times the rrespnding nn-relativisti result. T begin with, the prpagatr wuld be just the same, thugh. As the partile aelerates and the velit inreases, the mass wuld inrease, ausing the time step and spatial steps t nstantl grw lnger mpared t the nn-relativisti ase.

7 Hmewrk 5 3. The rest energ f an eletrn is.5 MeV. What is the mmentum f an eletrn traveling at % f the speed f light? At 99% f the speed f light? Hw des this mpare t the nn-relativisti results? Yu an express ur answer in units f MeV/. Nn-relativistiall p=mv=m =E / Hene the mmenta are 5. kev/ and.56 MeV/. Relativistiall p=m v=m =E / Hene the mmenta are 5. kev/ and 3.59 MeV/.. What is the kineti energ and the ttal energ f the eletrns desribed in Prblem? Cmpute the kineti energies bth relativistiall and nn-relativistiall. Relativistiall E=E. This is.5 MeV and 3.6 MeV. K=E E = 5.6 ev and 3. MeV. Nn-relativistiall K = ½mv = ½m = ½E = 5.6 MeV/ and.5 MeV/. 3. The maximum speed f a mving partile is. Is there a maximum kineti energ? Briefl explain. N, there is n maximum kineti energ beause as v apprahes, bemes infinite. Nte K = E ( -). 4. (a) If the mmentum f a partile dubles, b hw muh des its speed hange? 4 p p, E E ,,, 4 (4 4 )4 4 4, Nte that when the speed is small, the speed essentiall duble. Hwever, as the speed apprahes, the velit remains the same. p E E (b) If the speed f a partile dubles, b hw 4 muh hw des its mmentum hange? p E 4 p p 4

8 Nte that the denminatr blws uf the speed f the partile is half the speed f light. That is, the velit an duble nl if the partile is traveling slwer than half the speed f light. Near this value, the mmentum gets ver large. () If the speed f a partile dubles, b hw muh hw des its ttal energ hange? E E E 4 E E 4 E E 4 When the velit is small, the ttal energ nl hanges a small amunt beause it is dminated b the rest energ. (d) If the speed f a partile dubles, b hw muh hw des its kineti energ hange? (Nte that nn-relativistiall, its kineti energ bemes fur times larger.) K E ( )E ( ) ( ) ( ) K K 4 When the velit is small, we ma use a Talr series t apprximate the square rts: K K (4 )½ ( )½ K K ½ K 4K 5. A 5. MeV prtn (e.g., a prtn with a kineti energ f 5. MeV) strikes a arbn nuleus at rest. The prtn is deteted with a kineti energ f 35. MeV. At what angles with respet t the inident prtn s diretin d the tw partiles emerge? The rest energ f a prtn is MeV and the rest energ f the arbn nuleus is 74.8 MeV. Hint: Use nservatin f eah mpnent f the energ-mmentum fur vetr. Assume that the inident prtn mves in the z diretin and after the llisin, the prtns emerge in the x-z plane. E p E E p E p p p p sin p p p s p p sin p s

9 Starting with the energ equatin: This allws us t find the ttal energies and mmenta: Nw we ma appl the mmentum equatins: E p E E p E K p K p K K K p K p 5. MeV EKE, E p E E p 88.3 MeV, p p 79. MeV/ E p 73.3 MeV, p E 89.8 MeV, p p 74.4 MeV/ 579. MeV/ p p sin p p sin p p p p s p p s p sin p p sin p p s p p p p s p This leads t p = 47.6, and = 64.. p p sin p s p p sin p (p p p p s p ) p p p sin p p p p p p p s p p p s p p p p p p p p p p s p s p p p p p p p p p p

10 Hmewrk 5 4. Write dwn the bst that takes fur vetrs frm a frame S t a frame Smving at velit v ŷ with respet t S. L( ŷ ). Write dwn the inverse f the matrix u fund fr Prblem. Shw expliitl that this matrix is the inverse. L ( ŷ )L( ŷ ) ( ) ( ) 3. (This prblem has a lt f algebra. It is imprtant t knw hw t d the algebra, but if u understand the algebra, I dn t mind if u trust parts f m slutin rather than wrking thrugh the details. Be sure u understand the ideas, thugh.) E A partile travels in frame S with idential velities in the x and diretins and with zer velit in the z diretin. Find the fllwing matries: E E E E (a) Find a bst, L, that in ne step brings the partile t its rest frame, S. (Use Eq Yu an just write it dwn b inspetin.) L (b) Find a bst, L, in the diretin that brings the mpnent f the velit t zer in the S frame. Express the matrix in terms f and, then evaluate and.

11 L T find the bst parameters: E E E x E x x We write the last vetr beause we knw that the resultant vetr must have this general frm with the mpnent f velit zer. We ll find and later. The third line gives: /, and in turn, ½. () Find a bst, L x, in the x diretin that brings the partile t its rest frame, S. Express the matrix in terms f x and x, then evaluate x and x. First, we turn t the matrix equatin abve. Frm the first line, we have And frm the send line, x ( ) ½ (½ ) ½ x x, x

12 The matrix is a simple bst in the x diretin: L x x x x x x x (d) Find the prdut f the bsts L x L. Nte that this is nt the same as L. It is nvenient t express ur answer in terms f,, and. L x L x x x x x x x x x x x x x x x Substituting the values we have abve, L x L (e) (Optinal) We an define a rtatin matrix abut the z axis with a rtatin angle b R z s sin sin s, s ( ), sin ( ) Shw that LR z L x L.(That is, the suessive bsts are equivalent t a pure bst fllwed b a rtatin.) Tw identities ma simplif ur algebra. Prfs fllw: ( ) ( )( ) ( ) (½ ) R z L x L ( ) ( ) ( ) ( )

13 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 4 ( ( ) ) ( ( ) ) ( ) ( ) ( ) ( ) 4 ( ) ( ) 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

14 Hmewrk 5 5. A prtn strikes a arbn nuleus at rest. The initial kineti energ f the prtn is K. Assume the rest masses f the prtn, E p, and arbn, E, are knwn. (a) Find expressins fr the fllwing quantities fr the prtn befre the llisin in terms f K and E p. Ttal energ, E p E p KE p p p E p E p K E p p p E p E p (E p K) K KE p E p K E p E p E p, K KE p (b) Find a vetr that represents the energ-mmentum f the sstem prir t the llisin. In this and sueeding parts f the prblem, u ma leave ur answers in terms f an quantit that has been fund in terms f the givens. Assume that the prtn mves in the z diretin. E p E E ( ) Find the Lrentz transfrmatins that takes the fur-vetr f part (b) t the inertial frame in whih the ttal mmentum is zer. E L E W W is the ttal energ in the zer mmentum (enter f mass) frame. T find and : p i E i E i E i p i E i p i

15 (d) Find the ttal energ f the sstem in the zer mmentum frame, W, and shw that the Lrentz transfrmatin an be written L W W W W E i W E i p i p i E i p i E i p i W E i p i E i p i W, W (e) Bst eah f the initial energ-mmentum fur-vetrs int the zer mmentum frame in rder t find the ttal energ and mmentum the prtn and the arbn in this frame. E p p p W W W E p W E p p i ( E p ) W E p p i E E E E W W W W p E (f) If an elasti llisin urs, then the ttal energ-mmentum f the sstem is the same befre and after the llisin. In the zer-mmentum sstem, the energ-mmentum fur vetrs f the partiles (the prtn and arbn nuleus) after the llisin an be fund ver easil. Desribe in wrds what happens when elasti sattering urs in the zer-mmentum frame. The energ f eah partile remains the same befre and after the llisin. This means that the magnitude f the mmentum f eah partile must als be the same befre and after the llisin. In rder fr the ttal mmentum t be nserved, the tw partiles must g ff bak t bak s that the mmentum remains zer. Thus, eah partile hanges diretin, but nthing else hanges in the llisin press. (g) Assume that the sattering angle f the prtn is in the zer-mmentum frame. Find the sattering angle in the riginal lab frame. The sattering angle is measured as the angle between the riginal diretin f the prtn (the z diretin) and the sattering diretin. Fr simpliit, assume that the prtn remains in the x-z plane after the llisin. Let a ~ represent the lab quantities after the llisin. Ẽ p E p E p p p s p p sin L W p p sin Wp p sin W W W p p s L p p s E p p p s tan L sin L s L p p sin L p p s L Wp p sin E p p p s

16 . A prtn is traveling in the +z diretin at a speed f.3 8 m/s as measured b an bserver in S. Find its velit in a frame mving at a speed f.4 8 m/s in the +z diretin with respet t S. D the prblem in terms f energ-mmentum fur-vetrs. This time, we need nl t d a simple bst int the S frame: E p E p E p p p p p p p p p p p E p E p p p E p p p E p p p p p p p p p.3/3..4/3. (.3/3.)(.4/3.).56 v.68 8 m/ s 3. If an bjet is reeding frm the earth at half the speed f light, at what wavelength wuld an astrnmer view light emitted at a wavelength f 4 nm? Let the rest frame f the astrnmial bjet be S. In this frame we knw the wavelength and we knw that E=hf=h/. Let the earth, S, mve in the +z diretin with respet t S. Light frm the bjet reahing the earth is als mving in the +z diretin. Hene: E E E E hf hf ( ) ( ) ( ) E E E E 4 nm 69 nm.5.5 This means that light emitted at the ver blue end f the visible spetrum is seen at the ver red end f the spetrum.

17 Hmewrk 5 6. A bus. m lng travels at.8 8 m/s. It passes thrugh a garage with drs n the frnt and bak. The garage is 8. m lng. (a) Can bth drs f the garage be lsed at the same time with the bus inside? Answer this frm the perspetive f a persn in the garage and frm a persn n the bus. (Assume, f urse, that u an pen and lse the drs ver fast.) Is this ntraditr? =.933 and =.79. T an bserver in the garage, the bus appears t be nl 3.58 m lng and the bus an fit easil in the garage. T an bserver n the bus, the garage appears t be nl.87 m lng, s the bus annt fit inside. The seeming ntraditin arises frm the fat that simultaneit is relative. While the persn in the garage sees bth drs lse at the same time, a persn n the bus sees the frnt dr lse first and the bak dr lse later. (b) Wrk thrugh the mathematial details t prve ur argument. Define tw events and determine the spae-time rdinates f the events in bth referene frames. Let S be the garage frame and S be the bus frame. Let S mve with velit v in the +z diretin with respet t S. In S: Event : The left (bak) dr lses just after the bus enters the garage. Event : The right (frnt) dr lses simultaneusl with the left dr. Thus: x, x 8m x, x t 8m.4m z 8m 8m.9m Thus the bserver n the bus sees the frnt dr lse.4 m / = 7 ns befre the bak dr lses.. The rest energ f a partile is 35. MeV. A deas int tw -ra phtns with an average life-time f. 6 s. (The half-life is s.) (a) Can tw -ras have different energies in the rest frame f the? Explain. N, beause the must me ut bak-t-bak with the same mmenta in rder t nserve mmentum. (b) What are the energies f the tw -ras as measured in the rest frame? Eah has half the rest energ f the, 67.5 MeV. () In a labratr, a beam f is prdued. -ras frm the dea are deteted at angles f ±. with respet t

18 the beam diretin. What energ des eah -ra have? Let us all the beam diretin the z diretin. Frm the smmetr f the -ras deteted, we an nlude that in the rest frame f the, the -ras were emitted in a diretin perpendiular t z. We an all this the x diretin. We an nstrut a phtn energ-mmentum fur-vetr in the rest frame, and then bst it t the lab frame. In the lab, the beam is mving in the z diretin, s the bst must be in the z diretin. Thus, alling the lab frame the S frame: E E sin E s E E E E E Frm this, we an nlude that The energ is E = 8 MeV. tan t t t s sin.67 (d) What is the kineti energ f the in the lab frame? K = 36 MeV 35 MeV = 5 MeV. (e) What is the average distane suh a wuld travel befre deaing? The average lifetime f the is its average rest lifetime times, 3. 6 s. Nw we find the velit, and then the distane:.97 d t8.9 8 m. Hmewrk 5 7. A 3. m airplane is fling at Mah. B hw muh is its length ntrated as measured b an bserver n the grund? The velit is abut 66 m/s, s =. 6, Talr series expansin: The differene in length is then =.. Sine this desn t tell us anthing, we need t find b a ( ) / ½ m7.74 m.. The twin paradx has ften been used t refute relativit. In this paradx, tw twins are initiall n the earth

19 tgether. One twin ges ff in a rapidl mving spaeship where her lks tik slwl. After man ears, she returns and finds herself unger than her twin. Sine everthing is relative, shuldn t eah twin see the ther ne as unger? N, the traveling twin has t aelerate, s the tw twins are nt in equivalent inertial frames. Assume that the traveling twin an aelerate instantaneusl t =.8, then turn arund and return t the earth at a similar speed, and then aelerate instantaneusl t stp n the earth. If the rund trip takes ten ears as measured n the earth, b what amunt wuld the traveling twin have aged? Letting dente the time in the rest frame f the traveling twin, t, t E m.8 7 J 6 ears. 3.. kg and. kg f antimatter annihilate eah ther mpletel. Hw muh energ is prdued? Hmewrk 5 8. In assignment 5 7, we intrdued the twin paradx. Let us assume that the traveling twin experienes a nstant fre, F, as measured in the spae ship. The fre is direted awa frm the earth. (a) Cnsider the spae ship at ne instant in time s it is instantaneusl in an inertial referene frame S. Find the fre as measured n the earth, using the relatinship f de d de d w F F x F. F z In the spae ship s frame, S, the twin is at rest. Thus f. F The earth, S, mves in the z diretin with respet t S. On the earth, this bemes F F F F FF Nte that the tw s are the same. Thus the fre is the same in the earth frame as in the frame f the spae ship.

20 (b) Given that F = dp /dt, find as a funtin f time. dp F, pft, pft dt E Ft F E F E t t () (Optinal.) Find a relatinship between t and. (Hint: dt = d ) dt d F E t d dt F E t E F ln F t F t E E