Physics 73 PS 2

University of the Philippines College of Science

Physics 73 Second Long Problem Set Second Semester, AY 2015–2016

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Second Long Problem Set

Second Semester, AY 2015–2016

Physics 73

Instructions: Choose the letter of the best answer to each of the following questions. To change your answer, cross (X) out the shaded letter and supply your new answer with your signature beside the item. Any form of cheating in examinations or any act of dishonesty in relation to studies, such as plagiarism, shall be subject to disciplinary action. USEFUL CONSTANTS Speed of light in vacuum: c = 3 × 108 m/s Planck’s constant: h = 6.626 × 10−34 J · s USEFUL CONVERSION FACTOR 1 eV = 1.602 × 10−19 J Hour 17 1. Frame Master. Which of the following is an inertial reference frame? A. Freely falling debris B. Powered spacecraft C. Space station near a massive planet D. Plane on earth’s atmosphere 2. Natural Momentum. What is 5.00 × 10−21 kg-m/s of momentum in natural units? A. 7.81 × 10−39 kg B. 4.39 × 10−3 eV C. 1.67 × 10−29 kg D. 3.12 × 10−2 eV 3. Survive. Which of the following ideas from Newtonian dynamics survived after Einstein’s postulates? A. simulteneity B. causality C. absolute time D. velocity transformation

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Second Long Problem Set

Second Semester, AY 2015–2016

Physics 73

Hour 18 4. Spam. In a lab frame, events A and B occur at the same time at locations xA = 2 m and xB = 11 m, respectively. If the same events occur at times t0A = 1 m and t0B = 8 m in a rocket frame, what is the space separation of the two events in the rocket frame? A. 5.60 m

C. 11.4 m

B. 32.0 m

D. 130 m

5. Relative speed. An observer in a lab frame observes two events that occur at the same location but separated in time by 6.00 s. If the same events are separated in time by 9.00 s in a rocket frame, what is the speed of the rocket frame relative to the lab frame? A. 0.67

C. 0.35

B. 0.41

D. 0.75

6. Traveller. Consider a space traveller leaving Earth bound for Alpha Centauri, on a spaceship that is moving with velocity 0.6. Alpha Centauri is 4.2 light-years away from Earth. What is the proper time between the event that the space traveller has left Earth and the event that the space traveller has reached Alpha Centauri? A. 2.52 y B. 4.20 y C. 5.60 y D. 11.67 y 7. Extreme. An event occurs at (t, x) = (5 m, 3 m) in some lab frame. At what speed of the rocket frame will this event have the largest time separation with respect to the common origin of the inertial frames? A. 3/4 B. 4/5 C. 3/5 D. 1

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Second Long Problem Set

Second Semester, AY 2015–2016

Physics 73

Hour 19 8. Affected. Consider events A–D and O in the spacetime diagram. Which of the following statement(s) is (are) TRUE? I. Event O can affect event A by particle emission. II. Event C could have been affected by event D by light-ray emission. III. Event D can affect event O by particle emission. A. I and II only

C. I and III only

B. II only

D. III only

9. e-World Line. Which of the following represents a world line of an electron through spacetime?

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Second Long Problem Set

Second Semester, AY 2015–2016

Physics 73

10. Longest. Which of the following particle worldlines have the greatest proper time?

Hour 20 11. Velocity. Disintegration of a particle occurs at (4.0 m, –3.0 m) in the lab frame. If this event is observed at (7.0 m, 2.0 m) in the rocket frame, what is the velocity of the rocket frame relative to the lab frame? A. +0.59 B. −0.59 C. +0.85 D. −0.85 12. Missing Link. In frame S, event O occurs at (0.00 m, 2.50 m). This same event is also observed in frame S 0 (which moves at a constant velocity of β 0 > 0 relative to frame S) and frame S 00 (which moves at a constant velocity of β 00 = +0.68 relative to frame S 0 ) upon coincidence of the origins of the three frames. If event O occurs at time t00 = 4.00 m in frame S 00 , what are the coordinates of event O in frame S 0 ? A. (9.81 m, 10.12 m) B. (9.81 m, 1.07 m) C. (10.12 m, 9.81 m) D. (1.07 m, 9.81 m)

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Second Long Problem Set

Second Semester, AY 2015–2016

Physics 73

13. Nyan Cat and Doge. Nyan Cat moves with speed 0.800 relative to Doge. Doge observes an event occurring at coordinates (1.50 m, 0.900 m). What are the coordinates according to Nyan Cat? A. (−0.500 m, 1.30 m) B. (1.30 m, −0.500 m) C. (3.70 m, 3.50 m) D. (3.50 m, 3.70 m) 14. Relative. Two astronauts, one at a lab frame and another on a rocket frame moving at some velocity with respect to the lab frame, both possessing identical metersticks. The astronaut in the lab frame observes that the meterstick on the rocket frame is a bit shorter than the one in lab frame. What will the astronaut in the rocket frame observe about the length of the lab frame meterstick compared to the rocket frame meterstick? A. Shorter B. Longer C. Shorter if approaching, longer if receding D. Shorter if receding, longer if approaching Hour 21 15. Javelin Throw. You threw a meterstick such that it moves at a constant speed of 0.500 with respect to you. How long is the stick as measured in your reference frame? A. 1.00 m B. 1.15 m C. 0.866 m D. 0.750 m 16. TGIF. Hobbes, while at rest, is holding a 10- m long tree branch. Calvin, who is moving, measures the branch’s length to be 4 m only. What is Calvin’s speed with respect to Hobbes? A. 0.9 B. 2.7 C. 1.1 D. 0.8

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Second Long Problem Set

Second Semester, AY 2015–2016

Physics 73

17. Bad blood. Taylor heared two consecutive explosions at a time interval of 3.0 m. If Taylor is moving at 0.75 relative to Selena, what would be the time interval between the two explosions according to Selena? A. 4.5m B. 2.0m C. 1.5m D. 3.0m 18. Marvin the relativistic metal man. Marvin running at 0.62 moves across the length of a big rectangular metal sheet with length 100 m and width 70 m. What is the area of the rectangular metal sheet according to Marvin? A. 7000 m2 B. 5492 m2 C. 4309 m2 D. 8922 m2 Hour 22 19. Same position or same time? Consider the t-x axes of an inertial observer O, and the events plotted on the diagram shown. Which of the following is correct regarding the marked events?

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Second Long Problem Set

Second Semester, AY 2015–2016

Physics 73

A. There exists an inertial observer moving to the right such that E and G are simultaneous. B. There exists an inertial observer moving to the left such that F and G occur at the same position. C. There exists an inertial observer moving to the left such that E and F occur at the same position. D. There exists an inertial observer moving to the right such that G and H are simultaneous. 20. Spacetime diagram. Two spaceships, each of proper length L, pass near each other heading in opposite directions. Events P, Q, R, and S are defined as follow.

Which of the following correctly depicts the two-observer spacetime diagram of the problem?

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Second Long Problem Set

Second Semester, AY 2015–2016

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Physics 73

Second Long Problem Set

Second Semester, AY 2015–2016

Physics 73

21. Esme’s Pets. Consider the given two observer spacetime diagram, where (t0 , x0 ) denote rocket frame coordinates and (t, x) denote lab frame coordinates. The rocket frame is moving with respect to the lab frame. Which of the following events does the rocket frame observe to happen at the same place as event P?

A. ELK B. CAT C. DOG D. FOX

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Second Long Problem Set

Second Semester, AY 2015–2016

Physics 73

22. Ti-prime, Ex-prime. Which of the following shows the correct time and space axes of a rocket moving at β = 1/3 to the left relative to the lab frame, assuming that their origins coincide?

Hour 23 23. At Last I See It. Rapunzel is at rest in a rocket which is moving with a constant velocity 0.50 according to a lab frame. If Rapunzel measures a particle which has travelled a distance of 25. m during a time 25. m, what is the speed of the particle as measured by the lab frame? A. 0.40

C. 0.80

B. 0.60

D. 1.0

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Second Long Problem Set

Second Semester, AY 2015–2016

Physics 73

24. θ-θ0 ,. Shown in the table below are the rapidity measurements by the lab and rocket frames for a particle moving at constant velocity. What is the velocity of the rocket frame as measured by the lab frame? Reference Frame Rapidity Lab 6.52 Rocket 5.15 A. −0.879 B. 0.879 C. −0.999 D. 0.999 25. Doge chase cate. Two dogs, Pluto and Scooby are chasing a cat, Hello Kitty. If Pluto sees Hello Kitty running away from him at 0.50 and Scooby running towards him at 0.25, how fast is Hello Kitty according to Scooby? (Assume all motion are in one line.) A. 0.67 B. 0.29 C. 0.86 D. 0.75 Hour 24 26. E! A 3.1 × 10−27 kg-particle is travelling at 0.5. What is its energy? A. 1.2 × 10−27 kg B. 5.7 × 10−27 kg C. 3.6 × 10−27 kg D. 9.8 × 10−27 kg 27. Super bird. A super bird flying at relativistic speed is measured to have a total energy of 30 MeV and a momentum of 13 MeV. What is the speed of the bird? A. 0.48 B. 0.43 C. 0.39 D. 0.52

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Second Long Problem Set

Second Semester, AY 2015–2016

Physics 73

28. Photon momentum. What is the momentum of a photon with frequency 3.41 × 1016 Hz moving to the right? A. 2.51 × 10−34 kg B. 2.26 × 10−17 kg C. 7.53 × 10−26 kg D. 1.75 × 10−33 kg Hour 25 29. Doppler Frequency. A photon with frequency 2.00 MHz is emitted at an angle of 60.0◦ with respect to the x-axis of a lab frame. What is the measured frequency of this photon in a rocket frame moving at a velocity 0.50 with respect to the lab frame? A. 2.89 MHz

C. 1.34 MHz

B. 1.73 MHz

D. 2.00 MHz

30. Run to you. Your friend holding a monochromatic light of wavelength 670 nm is running towards you at a speed of 0.45. What will be your perceived wavelength of the light? A. 1088 nm B. 972 nm C. 369 nm D. 413 nm 31. What Rocket? A particle was observed by the lab frame to have a total energy of 7 kg and momentum of 5 kg. A rocket frame measures the total energy of this particle to be zero. What is the velocity of this rocket frame relative to the lab frame? A. 2/3

C. 1/4

B. 5/7

D. No such rocket frame exists

32. Energy. Consider a particle moving with velocity β = 0.8 along the +x-direction relative to an inertial observer O. The measured energy of the particle is 15.00 MeV, relative to O. Another inertial observer O’ is moving with some unknown velocity β 0 relative to O. If the energy of the particle as measured by O’ is 41.00 MeV, what is β 0 ? A. -0.2

C. 0.6

B. 0.4

D. -0.8

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Second Long Problem Set

Second Semester, AY 2015–2016

Physics 73

33. Momenergy. According to a lab frame, the momentum and energy of a particle is measured as 7.0 kg and 25 kg respectively. Relative to the lab frame, a rocket frame moves with a velocity 0.60. What are the momentum and energy of the particle as measured by the rocket frame? A. E = 26 kg, p = 10. kg B. E = 26 kg, p = −10. kg C. E = 30 kg, p = −18 kg D. E = 30 kg, p = 18 kg Hour 26 34. April 29. The energy and momentum of a particle in the lab frame is 4.0 kg and −2.0 kg, respectively. In a rocket frame where the particle has energy of 9.0 kg, what is its magnitude of momentum? A. 3.0 kg B. 5.0 kg C. 8.3 kg D. 6.9 kg 35. When photons collide. Two photons with energies E and 2E in the lab frame are on a head-on collision. What is the total mass associated with this system? √ A. 2 2E B. 3E C. zero √ D. E 2 36. Don’t rest yet! A particle with kinetic energy K moves with speed 0.600. What is the particle’s rest mass? A. 4K

C. 35 K

B. 23 K

D. 45 K

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Second Long Problem Set

Second Semester, AY 2015–2016

Physics 73

37. System Mass. Consider a particle of mass 2m and total energy 4m non-interacting with a photon with energy 6m shown in the √ figure. What is the mass of this system? A. 2 23m B. 52m √ C. 2 13m D. 2m Hour 27 38. Bang Bang. A particle with rest mass m and kinetic energy 3m collides with a stationary particle with the same rest mass, forming a single moving particle. What is the rest mass of the resulting particle? √ A. 5m √ B. 10m √ C. 15m √ D. 20m 39. Positronium Gaming. A positronium of mass m and initial energy E = 2m decays into two photons that move in opposite directions. What is the momentum of the photon that moves in the same direction as the initial particle? A. m √ B. (1 + 23 )m √

C. (1 − √

D.

3 2 )m

3 2 m

40. To p or not to p. A particle of mass M decays from rest into two particles that move in opposite directions. One particle has mass m and the other particle is massless. What is the velocity of particle m? A. 1 m2 B. 2 M + m2 M −m C. M +m M 2 − m2 D. 2 M + m2 Page 15