Name Solutions to Test 1 September 23, 2016

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1 Name Solutions to Test 1 September 3, 016 This test onsists of three parts. Please note that in parts II and III, you an skip one question of those offered. Possibly useful formulas: F qequb x xvt E Evpx p qrb t tvx px px ve u p p y y, y py E z z pz pz f0 f n 1 1 vos 1 1n nn1 ux v ux 1 vu uy uy 1 vu uz uz 1 vu x x x Part I: Multiple Choie [0 points] For eah question, hoose the best answer ( points eah) 1. If observer A sees light moving to the right at speed, and observer B is moving to the right at an angle, how fast will the light beam appear to be moving aording to observer B? A) Slower than B) Faster than C) Exatly at D) It depends on the speed of observer B E) It depends on the angle of observer B. If one observer stays at onstant veloity, while another one aelerates, suh that they start and end at the same plae and time, for whih observer will less proper time have passed? A) The onstant veloity one B) The aelerating one C) They will be the same D) It depends on whih one is atually moving E) There is insuffiient information, even if you know whih one is atually moving 3. If you ompress a realisti spring, how will its total energy hange? A) It dereases a lot B) It dereases a little C) It is unhanged D) It inreases a little E) It inreases a lot

2 4. To alulate the invariant mass of a olletion of objets, you would need: A) The total momentum (only) B) The total energy (only) C) The sum of the veloities of all the omponents (only) D) The total momentum and the total energy E) The total momentum and the sum of the veloities of the omponents 5. Suppose that a lok is moving at a very high veloity ompared to us, so that 4. Aording to us, in the time the lok advanes 1 s, how muh time will have passed for us? A) 3 s B) 6 s C) 1 s D) 4 s E) 4 s 6. If one observer is moving in the x-diretion ompared to the other, on whih of the following quantities will the two observers agree? A) The energy (only) B) The x-momentum (only) C) The y- and z-momentum (only) D) All three omponents of momentum, but not the energy E) All three omponents of momentum, and the energy 7. Whih of the following oordinate transformations is believed to not atually leave the laws of physis unhanged? A) Galilean boost B) Rotation C) Spae translation D) Time translation E) Atually, all of these do leave the laws of physis unhanged. If an objet is moving from our point of view diretly perpendiular to the line of sight, how will the observed frequeny differ from the natural frequeny? A) The observed frequeny will be higher To observer B) The observed frequeny will be lower C) The observed frequeny will be the same v D) It depends on the veloity of the objet E) It depends on the frequeny 9. Whih of the following formulas is the orret relativisti formula for just the kineti energy? A) m B) m C) 1 m D) mv E) mv 10. A partile that moves faster than would be a. An example that atually exists of this would be a. A) Massive partile; eletron B) Massless partile; photon C) Tahyon; neutrino D) Tahyon; there are no suh partiles E) None of the above 1

3 Part II: Short answer [0 points] Choose two of the following questions and give a short answer (1-3 sentenes) (10 points eah). 11. In H.G. Wells story The Time Mahine, the protagonist states, There is no differene between time and any of the three dimensions of spae Explain what is right and/or wrong about this statement. The story is right in the sense that time is the fourth dimension, but time isn t exatly the same as the others. For one thing, you have to use the speed of light to onvert units of time into units of distane. The other differene is that time enters the distane formula with the opposite sign, so the distane formula works out to be s x y z t 1. As we learn in relativity, simultaneity is ambiguous, and observers might even disagree on whih of two events ame first. Does this mean that the birth of your father might atually have happened after your birth, as viewed by ertain observers? Explain. Although whih of two events ours first an be ambiguous when they are spaelike separated, it is unambiguous when two events are timelike separated. Beause the separation of your parents birth from yours is large (typially 30 y) and when multiplied by the speed of light is even larger (typially 30 y ~ m), whereas the physial separation of those two events is small (less than 10 4 km = 10 7 m), the two events are timelike separated. Hene the birth of your parents is in the absolute past of your birth, or your birth is in the absolute future of your parent s birth, and all observers will agree that your parent s birth ame before your birth. dp 13. In non-relativisti physis, the two equations F ma and F are equivalent. Tell dt me whih of these (if either) is orret in relativity, and why that one is preferred. The two equations an be shown to be equivalent if p mv, but this does not apply in speial relativity, so we must rejet at least one of these two equations. If we keep the latter, then if we assume Newton s third law, F1 F, then it is easy to show that the sum of the dp momenta of two interating partiles will be onserved. So we prefer F and rejet dt F ma..

4 Part III: Calulation: [60 points] Choose three of the following four questions and perform the indiated alulations (0 points eah) 14. An ie hokey puk of diameter 3 in (7.6 m) rosses a line on a standard hokey rink of width in (5.0 m). Beause the puk is moving quikly, it looks as if the hokey puk is exatly the same diameter (in the diretion it is moving) as the line (a) What is for the hokey puk? What is the speed of the hokey puk? The hokey puk has a proper length It follows that sine L L, we have P L 7.6 m but an observed length L 5.0 m. P L P 7.6 m L 5.0 m The veloity an be found from 1 1 v, or rearranging, we have 1 1 v, so , v v v , , v m/s.3 10 m/s. (b) As viewed by an observer on the hokey puk, what is the width of the line it is rossing? The atual line has a proper width of LP 5.0 m. Sine in the hokey puk s frame of referene, the line is moving at high speed, it will look Lorentz ontrated by L LP. Sine the speed is the same as we found in part (a), the Lorentz fator will still be 1.500, so L p 5.0 m L 3.39 m () (Short essay question) At t = 0, the referee instantaneously stops the hokey puk to look at it, and sees that it is indeed no wider than the line it was rossing. But sine in the referene frame of the hokey puk, the puk is larger than the line it is rossing, how is it possible for the puk to fit in this short a spae? As viewed by an observer on the hokey puk, the two ends were not pushed down at the same time, and hene the front end was stopped first, but the bak end kept moving and was stopped a short time later. In speial relativity, there are no rigid objets, so it is inevitable that a hokey puk an be squeezed into a spae apparently too small to ontain the objet.

5 15. A muon has a mean lifetime, as viewed in its own frame, of s. The 0 probability of it lasting a proper time is given by e. Suppose a group of muons is travelling at an unknown speed (a) After how muh proper time will only 6.30% of the original muons still exist? A little algebra tells us that 0 e , 0ln , s s. 0 (b) Suppose that 6.30% of the muons still exist after travelling a distane d =.03 km. Find the time t as measured by us. We already know the proper time. Let s make things simple by assuming the muon is moving in the x-diretion. Then we an use the proper time formula, to get the time differene t : x y z t, t x x 030 m t s.9910 m/s s s.7410 s, s s 9.10 s. t, () Find the veloity of the muons in m/s. Veloity is distane over time, so x 030 m v t s.3 10 m/s.

6 16. A Lambda partile deays into a proton (m p = 93 MeV/ ) at rest and a pion (m = 140 MeV/ ) moving to the right with energy E = 14 MeV. (a) Find the initial momentum of the pion in MeV/. What is the momentum and energy of the proton? Before After p + The momentum an be found from the equation rearranged to yield 4 E p m, whih an be 4 p E m 14 MeV 140 MeV 1456 MeV, p 119 MeV, p119 MeV/. The proton is at rest, so 93 MeV and the momentum is p mu 0 E m m. (b) Find the initial energy and momentum of the Lambda. This is just a matter of onservation of momentum and energy, so p p p MeV/ 119 MeV/, p E E E 93 MeV 14 MeV 11 MeV. p () What is the mass of the Lambda m in MeV/, and its veloity as a fration of the speed of light? The mass an be found using the same equation as before, 4 6 m E p 11 MeV 119 MeV MeV, m The veloity an be found using MeV 1116 MeV, m1116 MeV/. 6 u p 119 MeV 0.106, E 11 MeV u

7 17. A nanobot, initially at rest, has a mass of kg and is being launhed to visit a nearby star. The goal is to get it to a speed of m/s (a) What is the initial and final energy (in J) of the nanobot? Beause it starts at rest, the initial energy is The final energy is E f 1 5 E m m kg.9910 m/s J. i i 5 5 m J J 5 fm J. 1v m/s m/s (b) What is the initial and final momentum (in kgm/s) of the nanobot? p f The initial momentum is zero, sine it is at rest. The final momentum is kg m/s kg m/s kg m/s mv fmv 1v m/s m/s () If the aelerator that launhes the nanobot is 30 m long, what fore is required on the nanobot? We an use the work formula, W Fd, where work is the hange in energy, so we have J J W Ef Ei F 01.7 N. d d 30 m (d) If the aeleration is ahieved with an eletri field of magnitude E = V/m, what eletrial harge must be plaed on the nanobot? The harge on the nanobot an be found from F qe, and is given by F 01.7 N q E V/m N m/v C 0.67 C.