1. Show that the form of Newton’s second law is invariant under the Galilean transformation. Get solution
1q. Michelson used the motion of the Earth around the sun to try to determine the effects of the ether. Can you think of a more convenient experiment with a higher speed that Michelson might have used in the 1880s? What about today? Get solution
2. Show that the definition of linear momentum, p = mv, has the same form p’= mv’ under a Galilean transformation. Get solution
2q. If you wanted to set out today to find the effects of the ether, what experimental apparatus would you want to use? Would a laser be included? Why? Get solution
3. Show that the equation for t2 in Section 2.2 expresses the time required for the light to travel to the mirror D and back in Figure 2.2. In this case the light is traveling perpendicular to the supposed direction of the ether. In what direction must the light travel to be reflected by the mirror if the light must pass through the ether? Get solution
3q. For what reasons would Michelson and Morley repeat their experiment on top of a mountain? Why would they perform the experiment in summer and winter? Get solution
4. A swimmer wants to swim straight across a river with current flowing at a speed of v1 = 0.350 m/s. If the swimmer swims in still water with speed v2 = 1.25 m/s, at what angle should the swimmer point upstream from the shore, and at what speed will the swimmer swim across the river? Get solution
4q. Does the fact that Maxwell’s equations do not need to be modified because of the special theory of relativity, whereas Newton’s laws of motion do, mean that Maxwell’s work is somehow greater or more significant than Newton’s? Explain. Get solution
5. Show that the time difference Δt’ given by Equation (2.4) is correct when the Michelson interferometer is rotated by 90°. ... Get solution
5q. The special theory of relativity has what effect on measurements done today? (a) None whatsoever, because any correction would be negligible. (b) We need to consider the effects of relativity when objects move close to the speed of light. (c) We should always make a correction for relativity because Newton’s laws are basically wrong. (d) It doesn’t matter, because we can’t make measurements where relativity would matter. Get solution
6. In the 1887 experiment by Michelson and Morley, the length of each arm was 11 m. The experimental limit for the fringe shift was 0.005 fringes. If sodium light was used with the interferometer (λ=589 nm), what upper limit did the null experiment place on the speed of the Earth through the expected ether? Get solution
6q. Why did it take so long to discover the theory of relativity? Why didn’t Newton figure it out? Get solution
7. Show that if length is contracted by the factor ...in the direction of motion, then the result in Equation (2.3) will have the factor needed to make Δt = 0 as needed by Michelson and Morley. ... Get solution
7q. Can you think of a way you can make yourself older than those born on your same birthday? Get solution
8. Explain why Einstein argued that the constancy of the speed of light (postulate 2) actually follows from the principle of relativity (postulate 1). Get solution
8q. Will metersticks manufactured on Earth work correctly on spaceships moving at high speed? Explain. Get solution
9. Prove that the constancy of the speed of light (postulate 2) is inconsistent with the Galilean transformation. Get solution
9q. Devise a system for you and three colleagues, at rest with you, to synchronize your clocks if your clocks are too large to move and are separated by hundreds of miles. Get solution
10. Use the spherical wavefronts of Equations (2.9) to derive the Lorentz transformation given in Equations (2.17). Supply all the steps. ... ... Get solution
11. Show that both Equations (2.17) and (2.18) reduce to the Galilean transformation when v c. ... ... Get solution
11q. Can you think of an experiment to verify length contraction directly? Explain. Get solution
12. Determine the ratio β = v/c for the following: (a) A car traveling 95 km/h. (b) A commercial jet airliner traveling 240 m/s. (c) A supersonic airplane traveling at Mach 2.3 (Mach number = v/vsound). (d) The space station, traveling 27,000 km/h. (e) An electron traveling 25 cm in 2 ns. (f) A proton traveling across a nucleus (10-14 m) in 0.35 × 10-22 s. Get solution
12q. Would it be easier to perform the muon decay experiment in the space station orbiting above Earth and then compare with the number of muons on Earth? Explain. Get solution
13. Two events occur in an inertial system K as follows: Event 1: x1 = a, t1 = 2a/c, y1 = 0, z1 = 0 Event 2: x2 = 2a, t2 = 3a/ 2c, y2 = 0, z2 = 0 In what frame K’ will these events appear to occur at the same time? Describe the motion of system K’. Get solution
13q. On a spacetime diagram, can events above t = 0 but not in the shaded area in Figure 2.25 affect the future? Explain. ... Get solution
14. Is there a frame K’ in which the two events described in Problem 13 occur at the same place? Explain. Event 1: x1 = a, t1 = 2a/c, y1 = 0, z1 = 0 Event 2: x2 = 2a, t2 = 3a/ 2c, y2 = 0, z2 = 0 Get solution
14q. Why don’t we also include the spatial coordinate z when drawing the light cone? Get solution
15. Find the relativistic factor γ for each of the parts of Problem 12. Determine the ratio β = v/c for the following: (a) A car traveling 95 km/h. (b) A commercial jet airliner traveling 240 m/s. (c) A supersonic airplane traveling at Mach 2.3 (Mach number = v/vsound). (d) The space station, traveling 27,000 km/h. (e) An electron traveling 25 cm in 2 ns. (f) A proton traveling across a nucleus (10-14 m) in 0.35 × 10-22 s. Get solution
15q. What would be a suitable name for events connected by ...s2 = 0? Get solution
16. An event occurs in system K’ at x’ = 2 m, y’ = 3.5 m, z’= 3.5 m, and t’= 0. System K’ and K have their axes coincident at t = t’= 0, and system K’ travels along the x axis of system K with a speed 0.8c. What are the coordinates of the event in system K? Get solution
16q. Is the relativistic Doppler effect valid only for light waves? Can you think of another situation in which it might be valid? Get solution
17. A light signal is sent from the origin of a system K at t = 0 to the point x = 3 m, y = 5 m, z = 10 m. (a) At what time t is the signal received? (b) Find (x’, y’, z’, t’) for the receipt of the signal in a frame K’ that is moving along the x axis of K at a speed of 0.8c. (c) From your results in (b) verify that the light traveled with a speed c as measured in the K’ frame. Get solution
17q. In Figure 2.22, why can a real worldline not have a slope less than one? ... Get solution
18. Show that the experiment depicted in Figure 2.11 and discussed in the text leads directly to the derivation of length contraction. ... Get solution
18q. Explain how in the twin paradox, we might arrange to compare clocks at the beginning and end of Mary’s journey and not have to worry about acceleration effects. Get solution
19. A rocket ship carrying passengers blasts off to go from New York to Los Angeles, a distance of about 5000 km. (a) How fast must the rocket ship go to have its own length shortened by 1%? (b) Ignore effects of general relativity and determine how much time the rocket ship’s clock and the ground-based clocks differ when the rocket ship arrives in Los Angeles. Get solution
19q. In each of the following pairs, which is the more massive: a relaxed or compressed spring, a charged or uncharged capacitor, or a piston-cylinder when closed or open? Get solution
20. Astronomers discover a planet orbiting around a star similar to our sun that is 20 lightyears away. How fast must a rocket ship go if the round trip is to take no longer than 40 years in time for the astronauts aboard? How much time will the trip take as measured on Earth? Get solution
20q. In the fission of 235U, the masses of the final products are less than the mass of 235U. Does this make sense? What happens to the mass? Get solution
21. Particle physicists use particle track detectors to determine the lifetime of short-lived particles. A muon has a mean lifetime of 2.2 μs and makes a track 9.5 cm long before decaying into an electron and two neutrinos. What was the speed of the muon? Get solution
21q. In the fusion of deuterium and tritium nuclei to produce a thermonuclear reaction, where does the kinetic energy that is produced come from? Get solution
22. The Apollo astronauts returned from the moon under the Earth’s gravitational force and reached speeds of almost 25,000 mi/h with respect to Earth. Assuming (incorrectly) they had this speed for the entire trip from the moon to Earth, what was the time difference for the trip between their clocks and clocks on Earth? Get solution
22q. Mary, the astronaut, wants to travel to the star system Alpha Centauri, which is 4.3 lightyears away. She wants to leave on her 30th birthday, travel to Alpha Centauri but not stop, and return in time for her wedding to Vladimir on her 35th birthday. What is most likely to happen? (a) Vladimir is a lucky man, because he will marry Mary after she completes her journey. (b) Mary will have to hustle to get in her wedding gown, and the wedding is likely to be watched by billions of people. (c) It is a certainty that Mary will not reach Alpha Centauri if she wants to marry Vladimir as scheduled. (d) Mary does reach Alpha Centauri before her 35th birthday and sends a radio message to Vladimir from Alpha Centauri that she will be back on time. Vladimir is relieved to receive the message before the wedding date. Get solution
23. A clock in a spaceship is observed to run at a speed of only 3/5 that of a similar clock at rest on Earth. How fast is the spaceship moving? Get solution
23q. A salesman driving a very fast car was arrested for driving through a traffic light while it was red, according to a policeman parked near the traffic light. The salesman said that the light was actually green to him, because it was Doppler shifted. Is he likely to be found innocent? Explain. Get solution
24. A spaceship of length 40 m at rest is observed to be 20 m long when in motion. How fast is it moving? Get solution
25. The Concorde traveled 8000 km between two places in North America and Europe at an average speed of 375 m/s. What is the total difference in time between two similar atomic clocks, one on the airplane and one at rest on Earth during a one-way trip? Consider only time dilation and ignore other effects such as Earth’s rotation. Get solution
26. mechanism on Earth used to shoot down geosynchronous satellites that house laser-based weapons is fi nally perfected and propels golf balls at 0.94c. (Geosynchronous satellites are placed 3.58 x 104 km above the surface of the Earth.) (a) What is the distance from the Earth to the satellite, as measured by a detector placed inside the golf ball? (b) How much time will it take the golf ball to make the journey to the satellite in the Earth’s frame? How much time will it take in the golf ball’s frame? Get solution
27. Two events occur in an inertial system K at the same time but 4 km apart. What is the time difference measured in a system K’ moving parallel to these two events when the distance separation of the events is measured to be 5 km in K’? Get solution
28. Imagine that in another universe the speed of light is only 100 m/s. (a) A person traveling along an interstate highway at 120 km/h ages at what fraction of the rate of a person at rest? (b) This traveler passes by a meterstick at rest on the highway. How long does the meterstick appear? Get solution
29. In another universe where the speed of light is only 100 m/s, an airplane that is 40 m long at rest and flies at 300 km/h will appear to be how long to an observer at rest? Get solution
30. Two systems K and K_ synchronize their clocks at t = t’ = 0 when their origins are aligned as system K’ passes by system K along the x axis at relative speed 0.8c. At time t = 3 ns, Frank (in system K) shoots a proton gun having proton speeds of 0.98c along his x axis. The protons leave the gun at x = 1 m and arrive at a target 120 m away. Determine the event coordinates (x, t) of the gun fi ring and of the protons arriving as measured by observers in both systems K and K’. Get solution
31. A spaceship is moving at a speed of 0.84c away from an observer at rest. A boy in the spaceship shoots a proton gun with protons having a speed of 0.62c. What is the speed of the protons measured by the observer at rest when the gun is shot (a) away from the observer and (b) toward the observer? Get solution
32. A proton and an antiproton are moving toward each other in a head-on collision. If each has a speed of 0.8c with respect to the collision point, how fast are they moving with respect to each other? Get solution
33. Imagine the speed of light in another universe to be only 100 m/s. Two cars are traveling along an interstate highway in opposite directions. Person 1 is traveling 110 km/h, and person 2 is traveling 140 km/h. How fast does person 1 measure person 2 to be traveling? How fast does person 2 measure person 1 to be traveling? Get solution
34. In the Fizeau experiment described in Example 2.5, suppose that the water is flowing at a speed of 5 m/s. Find the difference in the speeds of two beams of light, one traveling in the same direction as the water and the other in the opposite direction. Use n = 1.33 for water. Get solution
35. Three galaxies are aligned along an axis in the order A, B, C. An observer in galaxy B is in the middle and observes that galaxies A and C are moving in opposite directions away from him, both with speeds 0.60c. What is the speed of galaxies B and C as observed by someone in galaxy A? Get solution
36. Consider the gedanken experiment discussed in Section 2.6 in which a giant floodlight stationed 400 km above the Earth’s surface shines its light across the moon’s surface. How fast does the light fl ash across the moon? Get solution
37. A group of scientists decide to repeat the muon decay experiment at the Mauna Kea telescope site in Hawaii, which is 4205 m above sea level. They count 104 muons during a certain time period. Repeat the calculation of Section 2.7 and find the classical and relativistic number of muons expected at sea level. Why did they decide to count as many as 104 muons instead of only 103? Get solution
38. Consider a reference system placed at the U.S. Naval Observatory in Washington, D.C. Two planes take off from Washington Dulles Airport, one going eastward and one going westward, both carrying a cesium atomic clock. The distance around the Earth at 39° latitude (Washington, D.C.) is 31,000 km, and Washington rotates about the Earth’s axis at a speed of 360 m/s. Calculate the predicted differences between the clock left at the observatory and the two clocks in the airplanes (each traveling at 300 m/s) when the airplanes return to Washington. Include the rotation of the Earth but no general relativistic effects. Compare with the predictions given in the text. Get solution
39. Derive the results in Table 2.1 for the frequencies f’ and f” . During what time period do Frank and Mary receive these frequencies? Get solution
40. Derive the results in Table 2.1 for the time of the total trip and the total number of signals sent in the frame of both twins. Show your work. ... Get solution
41. Use the Lorentz transformation to prove that s2 = s’2. Get solution
42. Prove that for a timelike interval, two events can never be considered to occur simultaneously. Get solution
43. Prove that for a spacelike interval, two events cannot occur at the same place in space. Get solution
44. Given two events, (x1, t1) and (x2, t2), use a spacetime diagram to find the speed of a frame of reference in which the two events occur simultaneously. What values may Δs2 have in this case? Get solution
46. Consider a fixed and a moving system with their clocks synchronized and their origins aligned at t = t’= 0. (a) Draw on a spacetime diagram in the fixed system a line expressing all the events occurring at t’ = 0. (b) Draw on this diagram a line expressing all the events occurring at x’ = 0. (c) Draw all the worldlines for light that pass through t = t’ = 0. (d) Are the x’ and ct’ axes perpendicular? Explain. Get solution
47. Use the results of the two previous problems to show that events simultaneous in one system are not simultaneous in another system moving with respect to the first. Consider a spacetime diagram with x, ct and x’, ct’ axes drawn such that the origins coincide and the clocks were synchronized at t = t’ = 0. Then consider events 1 and 2 that occur simultaneously in the fixed system. Are they simultaneous in the moving system? Get solution
48. An astronaut is said to have tried to get out of a traffic violation for running a red light (λ= 650 nm) by telling the judge that the light appeared green (λ = 540 nm) to her as she passed by in her high-powered transport. If this is true, how fast was the astronaut going? Get solution
49. Derive Equation (2.32) for the case where the source is fixed but the receiver approaches it with velocity v. ... Get solution
50. Do the complete derivation for Equation (2.33) when the source and receiver are receding with relative velocity v. ... Get solution
51. A spacecraft traveling out of the solar system at a speed of 0.95c sends back information at a rate of 1400 kHz. At what rate do we receive the information? Get solution
52. Three radio-equipped plumbing vans are broadcasting on the same frequency f 0. Van 1 is moving east of van 2 with speed v, van 2 is fixed, and van 3 is moving west of van 2 with speed v. What is the frequency of each van as received by the others? Get solution
53. Three radio-equipped plumbing vans are broadcasting on the same frequency f 0. Van 1 is moving north of van 2 with speed v, van 2 is fixed, and van 3 is moving west of van 2 with speed v. What frequency does van 3 hear from van 2; from van 1? Get solution
54. A spaceship moves radially away from Earth with acceleration 29.4 m/s2 (about 3g). How much time does it take for the sodium streetlamps (λ = 589 nm) on Earth to be invisible (with a powerful telescope) to the human eye of the astronauts? The range of visible wavelengths is about 400 to 700 nm. Get solution
55. Newton’s second law is given by .... If the force is always perpendicular to the velocity, show that ... ...where ...is the acceleration. Get solution
56. Use the result of the previous problem to show that the radius of a particle’s circular path having charge q traveling with speed v in a magnetic field perpendicular to the particle’s path is r = p/qB. What happens to the radius as the speed increases as in a cyclotron? Get solution
57. Newton’s second law is given by ....If the force is always parallel to the velocity, show that .... Get solution
58. Find the force necessary to give a proton an acceleration of 1019 m/s2 when the proton has a velocity (along the same direction as the force) of (a) 0.01c, (b) 0.1c, (c) 0.9c, and (d) 0.99c. Get solution
59. A particle having a speed of 0.92c has a momentum of 10-16 kg . m/s. What is its mass? Get solution
60. A particle initially has a speed of 0.5c. At what speed does its momentum increase by (a) 1%, (b) 10%, (c) 100%? Get solution
61. The Bevatron accelerator at the Lawrence Berkeley Laboratory accelerated protons to a kinetic energy of 6.3 GeV. What magnetic field was necessary to keep the protons traveling in a circle of 15.2 m? (See Problem 56.) Use the result of the previous problem to show that the radius of a particle’s circular path having charge q traveling with speed v in a magnetic field perpendicular to the particle’s path is r = p/qB. What happens to the radius as the speed increases as in a cyclotron? Get solution
62. Show that linear momentum is conserved in Example 2.9 as measured by Mary. Get solution
63. Show that ...does not give the correct kinetic energy. Get solution
64. How much ice must melt at 0°C in order to gain 2 g of mass? Where does this mass come from? The heat of fusion for water is 334 J/g. Get solution
65. Physicists at the Stanford Linear Accelerator Center (SLAC) bombarded 9-GeV electrons head-on with 3.1-GeV positrons to create B mesons and anti-B mesons. What speeds did the electron and positron have when they collided? Get solution
66. The Tevatron accelerator at the Fermi National Accelerator Laboratory (Fermilab) outside Chicago boosts protons to 1 TeV (1000 GeV) in fi ve stages (the numbers given in parentheses represent the total kinetic energy at the end of each stage): Cockcroft- Walton (750 keV), Linac (400 MeV), Booster (8 GeV), Main ring or injector (150 GeV), and fi nally the Tevatron itself (1 TeV). What is the speed of the proton at the end of each stage? Get solution
67. Calculate the momentum, kinetic energy, and total energy of an electron traveling at a speed of (a) 0.020c, (b) 0.20c, and (c) 0.90c. Get solution
68. The total energy of a body is found to be twice its rest energy. How fast is it moving with respect to the observer? Get solution
69. A system is devised to exert a constant force of 8 N on an 80-kg body of mass initially at rest. The force pushes the mass horizontally on a frictionless table. How far does the body have to be pushed to increase its mass-energy by 25%? Get solution
70. What is the speed of a proton when its kinetic energy is equal to twice its rest energy? Get solution
71. What is the speed of an electron when its kinetic energy is (a) 10% of its rest energy, (b) equal to the rest energy, and (c) 10 times the rest energy? Get solution
72. Derive the following equation: ... Get solution
73. Prove that .... This is a useful relation to fi nd the velocity of a highly energetic particle. Get solution
74. A good rule of thumb is to use relativistic equations whenever the kinetic energies determined classically and relativistically differ by more than 1%. Find the speeds when this occurs for (a) electrons and (b) protons. Get solution
75. How much mass-energy (in joules) is contained in a peanut weighing 0.1 ounce? How much mass-energy do you gain by eating 10 ounces of peanuts? Compare this with the food energy content of peanuts, about 100 kcal per ounce. Get solution
76. Calculate the energy needed to accelerate a spaceship of mass 10,000 kg to a speed of 0.3c for intergalactic space exploration. Compare this with a projected annual energy usage on Earth of 1021 J. Get solution
77. Derive Equation (2.58) for the relativistic kinetic energy and show all the steps, especially the integration by parts. ... Get solution
78. A test automobile of mass 1000 kg moving at high speed crashes into a wall. The average temperature of the car is measured to rise by 0.5°C after the wreck. What is the change in mass of the car? Where does this change in mass come from? (Assume the average specific heat of the automobile is close to that of steel, 0.11 cal . g-1 . °C-1.) Get solution
79. A helium nucleus has a mass of 4.001505 u. What is its binding energy? Get solution
80. A free neutron is an unstable particle and beta decays into a proton with the emission of an electron. How much kinetic energy is available in the decay? Get solution
81. The Large Hadron Collider at Europe’s CERN facility is designed to produce 7.0 TeV (that is, 7.0 × 1012 eV) protons. Calculate the speed, momentum, and total energy of the protons. Get solution
82. What is the kinetic energy of (a) an electron having a momentum of 40 GeV/c? (b) a proton having a momentum of 40 GeV/c ? Get solution
83. A muon has a mass of 106 MeV/c 2. Calculate the speed, momentum, and total energy of a 200-MeV muon. Get solution
84. The reaction 2H + 2H ...n + 3He (where n is a neutron) is one of the reactions useful for producing energy through nuclear fusion. (a) Assume the deuterium nuclei (2H) are at rest and use the atomic mass units of the masses in Appendix 8 to calculate the mass-energy imbalance in this reaction. (Note: You can use atomic masses for this calculation, because the electron masses cancel out.) This amount of energy is given up when this nuclear reaction occurs. (b) What percentage of the initial rest energy is given up? Get solution
85. The reaction 2H + 3H ...n + 4He is one of the reactions useful for producing energy through nuclear fusion. (a) Assume the deuterium (2H) and tritium (3H) nuclei are at rest and use the atomic mass units of the masses in Appendix 8 to calculate the mass-energy imbalance in this reaction. This amount of energy is given up when this nuclear reaction occurs. (b) What percentage of the initial rest energy is given up? Get solution
86. Instead of one positive charge outside a conducting wire, as was discussed in Section 2.14 and shown in Figure 2.34, consider a second conducting wire parallel to the first one. Both wires have positive and negative charges, and the wires are electrically neutral. Assume that in both wires the positive charges travel to the right and negative charges to the left. (a) Consider an inertial frame moving with the negative charges of wire 1. Show that the second wire is attracted to the first wire in this frame. (b) Now consider an inertial frame moving with the positive charges of the second wire. Show that the first wire is attracted to the second. (c) Use this argument to show that electrical and magnetic forces are relative. ... Get solution
87g. An ... particle has rest energy 1672 MeV and mean lifetime 8.2 × 10-11 s. It is created and decays in a particle track detector and leaves a track 24 mm long. What is the total energy of the ...particle? Get solution
88g. Show that the following form of Newton’s second law satisfies the Lorentz transformation. Assume the force is parallel to the velocity. ... Get solution
89g. Use the results listed in Table 2.1 to find (a) the number of signals Frank receives at the rate f ‘ and the time at which Frank detects Mary’s turnaround, and (b) the number of signals Mary receives at the rate f ‘ and her clock reading when she turns around. (c) From Frank’s perspective, find the time for the remainder of the trip (after he detects Mary’s turnaround), the number of signals he receives at the rate f “, the total number of signals he receives, and Mary’s age, based on that total number of signals. (d) From Mary’s perspective, find the time for the remainder of the trip (after her turnaround), the number of signals she receives at the rate f “, the total number of signals she receives, and Frank’s age, based on that total number of signals. ... Get solution
90g. For the twins Frank and Mary described in Section 2.8, consider Mary’s one-way trip at a speed of 0.8c to the star system 8 lightyears from Earth. Compute the spacetime interval s in the fixed frame and s’ in the moving frame, and compare the results. Get solution
91g. Frank and Mary are twins. Mary jumps on a spaceship and goes to the star system Alpha Centauri (4.30 lightyears away) and returns. She travels at a speed of 0.8c with respect to Earth and emits a radio signal every week. Frank also sends out a radio signal to Mary once a week. (a) How many signals does Mary receive from Frank before she turns around? (b) At what time does the frequency of signals Frank receives suddenly change? How many signals has he received at this time? (c) How many signals do Frank and Mary receive for the entire trip? (d) How much time does the trip take according to Frank and to Mary? (e) How much time does each twin say the other twin will measure for the trip? Do the answers agree with those for (d)? Get solution
92g. A police radar gun operates at a frequency of 10.5 GHz. The officer, sitting in a patrol car at rest by the highway, directs the radar beam toward a speeding car traveling 80 mph directly away from the patrol car. What is the frequency shift of the reflected beam, relative to the original radar beam? Get solution
93g. A spaceship moving 0.80c direction away from Earth fi res a missile that the spaceship measures to be moving at 0.80c perpendicular to the ship’s direction of travel. Find the velocity components and speed of the missile as measured by Earth. Get solution
94g. An electron has a total energy that is 250 times its rest energy. Determine its (a) kinetic energy, (b) speed, and (c) momentum. Get solution
95g. A proton moves with a speed of 0.90c. Find the speed of an electron that has (a) the same momentum as the proton, and (b) the same kinetic energy. Get solution
96g. A high-speed K0 meson is traveling at a speed of 0.90c when it decays into a π+and a π- meson. What are the greatest and least speeds that the mesons may have? Get solution
97g. Frank and Mary are twins, and Mary wants to travel to our nearest star system, Alpha Centauri (4.30 lightyears away). Mary leaves on her 30th birthday and intends to return to Earth on her 52nd birthday. (a) Assuming her spaceship returns from Alpha Centauri without stopping, how fast must her spaceship travel? (b) How old will Frank be when she returns? Get solution
98g. The International Space Federation constructs a new spaceship that can travel at a speed of 0.995c. Mary, the astronaut, boards the spaceship to travel to Barnard’s star, which is the second nearest star to our solar system after Alpha Centauri and is 5.98 lightyears away. After reaching Barnard’s star, the spaceship travels slowly around the star system for three years doing research before returning back to Earth. (a) How much time does her journey take? (b) How much older is her twin Frank than Mary when she returns? Get solution
99g. A powerful laser on Earth rotates its laser beam in a circle at a frequency of 0.030 Hz. (a) How fast does the spot that the laser makes on the moon move across the moon’s landscape? (b) With what rotation frequency should the laser rotate if the laser spot moves across the moon’s landscape at speed c ? Get solution
100g. The Lockheed SR-71 Blackbird may be the fastest non-research airplane ever built; it traveled at 2200 miles/ hour (983 m/s) and was in operation from 1966 to 1990. Its length is 32.74 m. (a) By what percentage would it appear to be length contracted while in flight? (b) How much time difference would occur on an atomic clock in the plane compared to a similar clock on Earth during a flight of the Blackbird over its range of 3200 km? Get solution
101g. A spaceship is coming directly toward you while you are in the International Space Station. You are told that the spaceship is shining sodium light (with an intense yellow doublet of wavelengths 588.9950 and 589.5924 nm). You have an apparatus that can resolve two closely spaced wavelengths if the difference is Δ λ Get solution
102g. Quasars are among the most distant objects in the universe and are moving away from us at very high speeds, as discussed in Chapter 16. Astrophysicists use the redshift parameter z to determine the redshift of such rapidly moving objects. The parameter z is determined by observing a wavelength λ’ of a known spectral line of wavelength λsource on Earth; z = Δ λ’/ λsource= (λ’ - λsource)/ λsource Find the speed of two quasars having z values of 1.9 and 4.9. Get solution
103g. One possible decay mode of the neutral kaon is K0 ... π0 + π0. The rest energies of the K0 and π0 are 498 MeV and 135 MeV, respectively. The kaon is initially at rest when it decays. (a) How much energy is released in the decay? (b) What are the momentum and relative directions of the two neutral pions (π0)? Get solution
104g. The sun radiates energy at a rate of 3.9 × 1026 W. (a) At what rate is the sun losing mass? (b) At that rate, how much time would it take to exhaust the sun’s fuel supply? The sun’s mass is 2.0 × 1030 kg, and you may assume that the reaction producing the energy is about 0.7% efficient. Compare your answer with the sun’s expected remaining lifetime, about 5 Gy. Get solution
105g. One way astrophysicists have identified “extrasolar” planets orbiting distant stars is by observing redshifts or blueshifts in the star’s spectrum due to the fact that the star and planet each revolve around their common center of mass. (See Scientifi c American, August 2010, p. 41.) Consider a star the size of our sun (mass _ 1.99 _ 1030 kg), with a planet the size of Jupiter (1.90 × 1027 kg) in a circular orbit of radius 7.79 × 1011 m and a period of 11.9 years. (a) Find the speed of the star revolving around the system’s center of mass. (b) Assume that Earth is in the planet’s orbital plane, so that at one point in its orbit the star is moving directly toward Earth, and at the opposite point it moves directly away from Earth. How much is 550-nm light redshifted and blueshifted at those two extreme points? Get solution
106g. Small differences in the wavelengths in the sun’s spectrum are detected when measurements are taken from different parts of the sun’s disk. Specifi cally, measurements of the 656-nm line in hydrogen taken from opposite sides on the sun’s equator—one side approaching Earth and the other receding—differ from each other by 0.0090 nm. Use this information to fi nd the rotational period of the sun’s equator. Express your answer in days. (The sun’s equatorial radius is 6.96 × 108 m.) Get solution
1q. Michelson used the motion of the Earth around the sun to try to determine the effects of the ether. Can you think of a more convenient experiment with a higher speed that Michelson might have used in the 1880s? What about today? Get solution
2. Show that the definition of linear momentum, p = mv, has the same form p’= mv’ under a Galilean transformation. Get solution
2q. If you wanted to set out today to find the effects of the ether, what experimental apparatus would you want to use? Would a laser be included? Why? Get solution
3. Show that the equation for t2 in Section 2.2 expresses the time required for the light to travel to the mirror D and back in Figure 2.2. In this case the light is traveling perpendicular to the supposed direction of the ether. In what direction must the light travel to be reflected by the mirror if the light must pass through the ether? Get solution
3q. For what reasons would Michelson and Morley repeat their experiment on top of a mountain? Why would they perform the experiment in summer and winter? Get solution
4. A swimmer wants to swim straight across a river with current flowing at a speed of v1 = 0.350 m/s. If the swimmer swims in still water with speed v2 = 1.25 m/s, at what angle should the swimmer point upstream from the shore, and at what speed will the swimmer swim across the river? Get solution
4q. Does the fact that Maxwell’s equations do not need to be modified because of the special theory of relativity, whereas Newton’s laws of motion do, mean that Maxwell’s work is somehow greater or more significant than Newton’s? Explain. Get solution
5. Show that the time difference Δt’ given by Equation (2.4) is correct when the Michelson interferometer is rotated by 90°. ... Get solution
5q. The special theory of relativity has what effect on measurements done today? (a) None whatsoever, because any correction would be negligible. (b) We need to consider the effects of relativity when objects move close to the speed of light. (c) We should always make a correction for relativity because Newton’s laws are basically wrong. (d) It doesn’t matter, because we can’t make measurements where relativity would matter. Get solution
6. In the 1887 experiment by Michelson and Morley, the length of each arm was 11 m. The experimental limit for the fringe shift was 0.005 fringes. If sodium light was used with the interferometer (λ=589 nm), what upper limit did the null experiment place on the speed of the Earth through the expected ether? Get solution
6q. Why did it take so long to discover the theory of relativity? Why didn’t Newton figure it out? Get solution
7. Show that if length is contracted by the factor ...in the direction of motion, then the result in Equation (2.3) will have the factor needed to make Δt = 0 as needed by Michelson and Morley. ... Get solution
7q. Can you think of a way you can make yourself older than those born on your same birthday? Get solution
8. Explain why Einstein argued that the constancy of the speed of light (postulate 2) actually follows from the principle of relativity (postulate 1). Get solution
8q. Will metersticks manufactured on Earth work correctly on spaceships moving at high speed? Explain. Get solution
9. Prove that the constancy of the speed of light (postulate 2) is inconsistent with the Galilean transformation. Get solution
9q. Devise a system for you and three colleagues, at rest with you, to synchronize your clocks if your clocks are too large to move and are separated by hundreds of miles. Get solution
10. Use the spherical wavefronts of Equations (2.9) to derive the Lorentz transformation given in Equations (2.17). Supply all the steps. ... ... Get solution
11. Show that both Equations (2.17) and (2.18) reduce to the Galilean transformation when v c. ... ... Get solution
11q. Can you think of an experiment to verify length contraction directly? Explain. Get solution
12. Determine the ratio β = v/c for the following: (a) A car traveling 95 km/h. (b) A commercial jet airliner traveling 240 m/s. (c) A supersonic airplane traveling at Mach 2.3 (Mach number = v/vsound). (d) The space station, traveling 27,000 km/h. (e) An electron traveling 25 cm in 2 ns. (f) A proton traveling across a nucleus (10-14 m) in 0.35 × 10-22 s. Get solution
12q. Would it be easier to perform the muon decay experiment in the space station orbiting above Earth and then compare with the number of muons on Earth? Explain. Get solution
13. Two events occur in an inertial system K as follows: Event 1: x1 = a, t1 = 2a/c, y1 = 0, z1 = 0 Event 2: x2 = 2a, t2 = 3a/ 2c, y2 = 0, z2 = 0 In what frame K’ will these events appear to occur at the same time? Describe the motion of system K’. Get solution
13q. On a spacetime diagram, can events above t = 0 but not in the shaded area in Figure 2.25 affect the future? Explain. ... Get solution
14. Is there a frame K’ in which the two events described in Problem 13 occur at the same place? Explain. Event 1: x1 = a, t1 = 2a/c, y1 = 0, z1 = 0 Event 2: x2 = 2a, t2 = 3a/ 2c, y2 = 0, z2 = 0 Get solution
14q. Why don’t we also include the spatial coordinate z when drawing the light cone? Get solution
15. Find the relativistic factor γ for each of the parts of Problem 12. Determine the ratio β = v/c for the following: (a) A car traveling 95 km/h. (b) A commercial jet airliner traveling 240 m/s. (c) A supersonic airplane traveling at Mach 2.3 (Mach number = v/vsound). (d) The space station, traveling 27,000 km/h. (e) An electron traveling 25 cm in 2 ns. (f) A proton traveling across a nucleus (10-14 m) in 0.35 × 10-22 s. Get solution
15q. What would be a suitable name for events connected by ...s2 = 0? Get solution
16. An event occurs in system K’ at x’ = 2 m, y’ = 3.5 m, z’= 3.5 m, and t’= 0. System K’ and K have their axes coincident at t = t’= 0, and system K’ travels along the x axis of system K with a speed 0.8c. What are the coordinates of the event in system K? Get solution
16q. Is the relativistic Doppler effect valid only for light waves? Can you think of another situation in which it might be valid? Get solution
17. A light signal is sent from the origin of a system K at t = 0 to the point x = 3 m, y = 5 m, z = 10 m. (a) At what time t is the signal received? (b) Find (x’, y’, z’, t’) for the receipt of the signal in a frame K’ that is moving along the x axis of K at a speed of 0.8c. (c) From your results in (b) verify that the light traveled with a speed c as measured in the K’ frame. Get solution
17q. In Figure 2.22, why can a real worldline not have a slope less than one? ... Get solution
18. Show that the experiment depicted in Figure 2.11 and discussed in the text leads directly to the derivation of length contraction. ... Get solution
18q. Explain how in the twin paradox, we might arrange to compare clocks at the beginning and end of Mary’s journey and not have to worry about acceleration effects. Get solution
19. A rocket ship carrying passengers blasts off to go from New York to Los Angeles, a distance of about 5000 km. (a) How fast must the rocket ship go to have its own length shortened by 1%? (b) Ignore effects of general relativity and determine how much time the rocket ship’s clock and the ground-based clocks differ when the rocket ship arrives in Los Angeles. Get solution
19q. In each of the following pairs, which is the more massive: a relaxed or compressed spring, a charged or uncharged capacitor, or a piston-cylinder when closed or open? Get solution
20. Astronomers discover a planet orbiting around a star similar to our sun that is 20 lightyears away. How fast must a rocket ship go if the round trip is to take no longer than 40 years in time for the astronauts aboard? How much time will the trip take as measured on Earth? Get solution
20q. In the fission of 235U, the masses of the final products are less than the mass of 235U. Does this make sense? What happens to the mass? Get solution
21. Particle physicists use particle track detectors to determine the lifetime of short-lived particles. A muon has a mean lifetime of 2.2 μs and makes a track 9.5 cm long before decaying into an electron and two neutrinos. What was the speed of the muon? Get solution
21q. In the fusion of deuterium and tritium nuclei to produce a thermonuclear reaction, where does the kinetic energy that is produced come from? Get solution
22. The Apollo astronauts returned from the moon under the Earth’s gravitational force and reached speeds of almost 25,000 mi/h with respect to Earth. Assuming (incorrectly) they had this speed for the entire trip from the moon to Earth, what was the time difference for the trip between their clocks and clocks on Earth? Get solution
22q. Mary, the astronaut, wants to travel to the star system Alpha Centauri, which is 4.3 lightyears away. She wants to leave on her 30th birthday, travel to Alpha Centauri but not stop, and return in time for her wedding to Vladimir on her 35th birthday. What is most likely to happen? (a) Vladimir is a lucky man, because he will marry Mary after she completes her journey. (b) Mary will have to hustle to get in her wedding gown, and the wedding is likely to be watched by billions of people. (c) It is a certainty that Mary will not reach Alpha Centauri if she wants to marry Vladimir as scheduled. (d) Mary does reach Alpha Centauri before her 35th birthday and sends a radio message to Vladimir from Alpha Centauri that she will be back on time. Vladimir is relieved to receive the message before the wedding date. Get solution
23. A clock in a spaceship is observed to run at a speed of only 3/5 that of a similar clock at rest on Earth. How fast is the spaceship moving? Get solution
23q. A salesman driving a very fast car was arrested for driving through a traffic light while it was red, according to a policeman parked near the traffic light. The salesman said that the light was actually green to him, because it was Doppler shifted. Is he likely to be found innocent? Explain. Get solution
24. A spaceship of length 40 m at rest is observed to be 20 m long when in motion. How fast is it moving? Get solution
25. The Concorde traveled 8000 km between two places in North America and Europe at an average speed of 375 m/s. What is the total difference in time between two similar atomic clocks, one on the airplane and one at rest on Earth during a one-way trip? Consider only time dilation and ignore other effects such as Earth’s rotation. Get solution
26. mechanism on Earth used to shoot down geosynchronous satellites that house laser-based weapons is fi nally perfected and propels golf balls at 0.94c. (Geosynchronous satellites are placed 3.58 x 104 km above the surface of the Earth.) (a) What is the distance from the Earth to the satellite, as measured by a detector placed inside the golf ball? (b) How much time will it take the golf ball to make the journey to the satellite in the Earth’s frame? How much time will it take in the golf ball’s frame? Get solution
27. Two events occur in an inertial system K at the same time but 4 km apart. What is the time difference measured in a system K’ moving parallel to these two events when the distance separation of the events is measured to be 5 km in K’? Get solution
28. Imagine that in another universe the speed of light is only 100 m/s. (a) A person traveling along an interstate highway at 120 km/h ages at what fraction of the rate of a person at rest? (b) This traveler passes by a meterstick at rest on the highway. How long does the meterstick appear? Get solution
29. In another universe where the speed of light is only 100 m/s, an airplane that is 40 m long at rest and flies at 300 km/h will appear to be how long to an observer at rest? Get solution
30. Two systems K and K_ synchronize their clocks at t = t’ = 0 when their origins are aligned as system K’ passes by system K along the x axis at relative speed 0.8c. At time t = 3 ns, Frank (in system K) shoots a proton gun having proton speeds of 0.98c along his x axis. The protons leave the gun at x = 1 m and arrive at a target 120 m away. Determine the event coordinates (x, t) of the gun fi ring and of the protons arriving as measured by observers in both systems K and K’. Get solution
31. A spaceship is moving at a speed of 0.84c away from an observer at rest. A boy in the spaceship shoots a proton gun with protons having a speed of 0.62c. What is the speed of the protons measured by the observer at rest when the gun is shot (a) away from the observer and (b) toward the observer? Get solution
32. A proton and an antiproton are moving toward each other in a head-on collision. If each has a speed of 0.8c with respect to the collision point, how fast are they moving with respect to each other? Get solution
33. Imagine the speed of light in another universe to be only 100 m/s. Two cars are traveling along an interstate highway in opposite directions. Person 1 is traveling 110 km/h, and person 2 is traveling 140 km/h. How fast does person 1 measure person 2 to be traveling? How fast does person 2 measure person 1 to be traveling? Get solution
34. In the Fizeau experiment described in Example 2.5, suppose that the water is flowing at a speed of 5 m/s. Find the difference in the speeds of two beams of light, one traveling in the same direction as the water and the other in the opposite direction. Use n = 1.33 for water. Get solution
35. Three galaxies are aligned along an axis in the order A, B, C. An observer in galaxy B is in the middle and observes that galaxies A and C are moving in opposite directions away from him, both with speeds 0.60c. What is the speed of galaxies B and C as observed by someone in galaxy A? Get solution
36. Consider the gedanken experiment discussed in Section 2.6 in which a giant floodlight stationed 400 km above the Earth’s surface shines its light across the moon’s surface. How fast does the light fl ash across the moon? Get solution
37. A group of scientists decide to repeat the muon decay experiment at the Mauna Kea telescope site in Hawaii, which is 4205 m above sea level. They count 104 muons during a certain time period. Repeat the calculation of Section 2.7 and find the classical and relativistic number of muons expected at sea level. Why did they decide to count as many as 104 muons instead of only 103? Get solution
38. Consider a reference system placed at the U.S. Naval Observatory in Washington, D.C. Two planes take off from Washington Dulles Airport, one going eastward and one going westward, both carrying a cesium atomic clock. The distance around the Earth at 39° latitude (Washington, D.C.) is 31,000 km, and Washington rotates about the Earth’s axis at a speed of 360 m/s. Calculate the predicted differences between the clock left at the observatory and the two clocks in the airplanes (each traveling at 300 m/s) when the airplanes return to Washington. Include the rotation of the Earth but no general relativistic effects. Compare with the predictions given in the text. Get solution
39. Derive the results in Table 2.1 for the frequencies f’ and f” . During what time period do Frank and Mary receive these frequencies? Get solution
40. Derive the results in Table 2.1 for the time of the total trip and the total number of signals sent in the frame of both twins. Show your work. ... Get solution
41. Use the Lorentz transformation to prove that s2 = s’2. Get solution
42. Prove that for a timelike interval, two events can never be considered to occur simultaneously. Get solution
43. Prove that for a spacelike interval, two events cannot occur at the same place in space. Get solution
44. Given two events, (x1, t1) and (x2, t2), use a spacetime diagram to find the speed of a frame of reference in which the two events occur simultaneously. What values may Δs2 have in this case? Get solution
46. Consider a fixed and a moving system with their clocks synchronized and their origins aligned at t = t’= 0. (a) Draw on a spacetime diagram in the fixed system a line expressing all the events occurring at t’ = 0. (b) Draw on this diagram a line expressing all the events occurring at x’ = 0. (c) Draw all the worldlines for light that pass through t = t’ = 0. (d) Are the x’ and ct’ axes perpendicular? Explain. Get solution
47. Use the results of the two previous problems to show that events simultaneous in one system are not simultaneous in another system moving with respect to the first. Consider a spacetime diagram with x, ct and x’, ct’ axes drawn such that the origins coincide and the clocks were synchronized at t = t’ = 0. Then consider events 1 and 2 that occur simultaneously in the fixed system. Are they simultaneous in the moving system? Get solution
48. An astronaut is said to have tried to get out of a traffic violation for running a red light (λ= 650 nm) by telling the judge that the light appeared green (λ = 540 nm) to her as she passed by in her high-powered transport. If this is true, how fast was the astronaut going? Get solution
49. Derive Equation (2.32) for the case where the source is fixed but the receiver approaches it with velocity v. ... Get solution
50. Do the complete derivation for Equation (2.33) when the source and receiver are receding with relative velocity v. ... Get solution
51. A spacecraft traveling out of the solar system at a speed of 0.95c sends back information at a rate of 1400 kHz. At what rate do we receive the information? Get solution
52. Three radio-equipped plumbing vans are broadcasting on the same frequency f 0. Van 1 is moving east of van 2 with speed v, van 2 is fixed, and van 3 is moving west of van 2 with speed v. What is the frequency of each van as received by the others? Get solution
53. Three radio-equipped plumbing vans are broadcasting on the same frequency f 0. Van 1 is moving north of van 2 with speed v, van 2 is fixed, and van 3 is moving west of van 2 with speed v. What frequency does van 3 hear from van 2; from van 1? Get solution
54. A spaceship moves radially away from Earth with acceleration 29.4 m/s2 (about 3g). How much time does it take for the sodium streetlamps (λ = 589 nm) on Earth to be invisible (with a powerful telescope) to the human eye of the astronauts? The range of visible wavelengths is about 400 to 700 nm. Get solution
55. Newton’s second law is given by .... If the force is always perpendicular to the velocity, show that ... ...where ...is the acceleration. Get solution
56. Use the result of the previous problem to show that the radius of a particle’s circular path having charge q traveling with speed v in a magnetic field perpendicular to the particle’s path is r = p/qB. What happens to the radius as the speed increases as in a cyclotron? Get solution
57. Newton’s second law is given by ....If the force is always parallel to the velocity, show that .... Get solution
58. Find the force necessary to give a proton an acceleration of 1019 m/s2 when the proton has a velocity (along the same direction as the force) of (a) 0.01c, (b) 0.1c, (c) 0.9c, and (d) 0.99c. Get solution
59. A particle having a speed of 0.92c has a momentum of 10-16 kg . m/s. What is its mass? Get solution
60. A particle initially has a speed of 0.5c. At what speed does its momentum increase by (a) 1%, (b) 10%, (c) 100%? Get solution
61. The Bevatron accelerator at the Lawrence Berkeley Laboratory accelerated protons to a kinetic energy of 6.3 GeV. What magnetic field was necessary to keep the protons traveling in a circle of 15.2 m? (See Problem 56.) Use the result of the previous problem to show that the radius of a particle’s circular path having charge q traveling with speed v in a magnetic field perpendicular to the particle’s path is r = p/qB. What happens to the radius as the speed increases as in a cyclotron? Get solution
62. Show that linear momentum is conserved in Example 2.9 as measured by Mary. Get solution
63. Show that ...does not give the correct kinetic energy. Get solution
64. How much ice must melt at 0°C in order to gain 2 g of mass? Where does this mass come from? The heat of fusion for water is 334 J/g. Get solution
65. Physicists at the Stanford Linear Accelerator Center (SLAC) bombarded 9-GeV electrons head-on with 3.1-GeV positrons to create B mesons and anti-B mesons. What speeds did the electron and positron have when they collided? Get solution
66. The Tevatron accelerator at the Fermi National Accelerator Laboratory (Fermilab) outside Chicago boosts protons to 1 TeV (1000 GeV) in fi ve stages (the numbers given in parentheses represent the total kinetic energy at the end of each stage): Cockcroft- Walton (750 keV), Linac (400 MeV), Booster (8 GeV), Main ring or injector (150 GeV), and fi nally the Tevatron itself (1 TeV). What is the speed of the proton at the end of each stage? Get solution
67. Calculate the momentum, kinetic energy, and total energy of an electron traveling at a speed of (a) 0.020c, (b) 0.20c, and (c) 0.90c. Get solution
68. The total energy of a body is found to be twice its rest energy. How fast is it moving with respect to the observer? Get solution
69. A system is devised to exert a constant force of 8 N on an 80-kg body of mass initially at rest. The force pushes the mass horizontally on a frictionless table. How far does the body have to be pushed to increase its mass-energy by 25%? Get solution
70. What is the speed of a proton when its kinetic energy is equal to twice its rest energy? Get solution
71. What is the speed of an electron when its kinetic energy is (a) 10% of its rest energy, (b) equal to the rest energy, and (c) 10 times the rest energy? Get solution
72. Derive the following equation: ... Get solution
73. Prove that .... This is a useful relation to fi nd the velocity of a highly energetic particle. Get solution
74. A good rule of thumb is to use relativistic equations whenever the kinetic energies determined classically and relativistically differ by more than 1%. Find the speeds when this occurs for (a) electrons and (b) protons. Get solution
75. How much mass-energy (in joules) is contained in a peanut weighing 0.1 ounce? How much mass-energy do you gain by eating 10 ounces of peanuts? Compare this with the food energy content of peanuts, about 100 kcal per ounce. Get solution
76. Calculate the energy needed to accelerate a spaceship of mass 10,000 kg to a speed of 0.3c for intergalactic space exploration. Compare this with a projected annual energy usage on Earth of 1021 J. Get solution
77. Derive Equation (2.58) for the relativistic kinetic energy and show all the steps, especially the integration by parts. ... Get solution
78. A test automobile of mass 1000 kg moving at high speed crashes into a wall. The average temperature of the car is measured to rise by 0.5°C after the wreck. What is the change in mass of the car? Where does this change in mass come from? (Assume the average specific heat of the automobile is close to that of steel, 0.11 cal . g-1 . °C-1.) Get solution
79. A helium nucleus has a mass of 4.001505 u. What is its binding energy? Get solution
80. A free neutron is an unstable particle and beta decays into a proton with the emission of an electron. How much kinetic energy is available in the decay? Get solution
81. The Large Hadron Collider at Europe’s CERN facility is designed to produce 7.0 TeV (that is, 7.0 × 1012 eV) protons. Calculate the speed, momentum, and total energy of the protons. Get solution
82. What is the kinetic energy of (a) an electron having a momentum of 40 GeV/c? (b) a proton having a momentum of 40 GeV/c ? Get solution
83. A muon has a mass of 106 MeV/c 2. Calculate the speed, momentum, and total energy of a 200-MeV muon. Get solution
84. The reaction 2H + 2H ...n + 3He (where n is a neutron) is one of the reactions useful for producing energy through nuclear fusion. (a) Assume the deuterium nuclei (2H) are at rest and use the atomic mass units of the masses in Appendix 8 to calculate the mass-energy imbalance in this reaction. (Note: You can use atomic masses for this calculation, because the electron masses cancel out.) This amount of energy is given up when this nuclear reaction occurs. (b) What percentage of the initial rest energy is given up? Get solution
85. The reaction 2H + 3H ...n + 4He is one of the reactions useful for producing energy through nuclear fusion. (a) Assume the deuterium (2H) and tritium (3H) nuclei are at rest and use the atomic mass units of the masses in Appendix 8 to calculate the mass-energy imbalance in this reaction. This amount of energy is given up when this nuclear reaction occurs. (b) What percentage of the initial rest energy is given up? Get solution
86. Instead of one positive charge outside a conducting wire, as was discussed in Section 2.14 and shown in Figure 2.34, consider a second conducting wire parallel to the first one. Both wires have positive and negative charges, and the wires are electrically neutral. Assume that in both wires the positive charges travel to the right and negative charges to the left. (a) Consider an inertial frame moving with the negative charges of wire 1. Show that the second wire is attracted to the first wire in this frame. (b) Now consider an inertial frame moving with the positive charges of the second wire. Show that the first wire is attracted to the second. (c) Use this argument to show that electrical and magnetic forces are relative. ... Get solution
87g. An ... particle has rest energy 1672 MeV and mean lifetime 8.2 × 10-11 s. It is created and decays in a particle track detector and leaves a track 24 mm long. What is the total energy of the ...particle? Get solution
88g. Show that the following form of Newton’s second law satisfies the Lorentz transformation. Assume the force is parallel to the velocity. ... Get solution
89g. Use the results listed in Table 2.1 to find (a) the number of signals Frank receives at the rate f ‘ and the time at which Frank detects Mary’s turnaround, and (b) the number of signals Mary receives at the rate f ‘ and her clock reading when she turns around. (c) From Frank’s perspective, find the time for the remainder of the trip (after he detects Mary’s turnaround), the number of signals he receives at the rate f “, the total number of signals he receives, and Mary’s age, based on that total number of signals. (d) From Mary’s perspective, find the time for the remainder of the trip (after her turnaround), the number of signals she receives at the rate f “, the total number of signals she receives, and Frank’s age, based on that total number of signals. ... Get solution
90g. For the twins Frank and Mary described in Section 2.8, consider Mary’s one-way trip at a speed of 0.8c to the star system 8 lightyears from Earth. Compute the spacetime interval s in the fixed frame and s’ in the moving frame, and compare the results. Get solution
91g. Frank and Mary are twins. Mary jumps on a spaceship and goes to the star system Alpha Centauri (4.30 lightyears away) and returns. She travels at a speed of 0.8c with respect to Earth and emits a radio signal every week. Frank also sends out a radio signal to Mary once a week. (a) How many signals does Mary receive from Frank before she turns around? (b) At what time does the frequency of signals Frank receives suddenly change? How many signals has he received at this time? (c) How many signals do Frank and Mary receive for the entire trip? (d) How much time does the trip take according to Frank and to Mary? (e) How much time does each twin say the other twin will measure for the trip? Do the answers agree with those for (d)? Get solution
92g. A police radar gun operates at a frequency of 10.5 GHz. The officer, sitting in a patrol car at rest by the highway, directs the radar beam toward a speeding car traveling 80 mph directly away from the patrol car. What is the frequency shift of the reflected beam, relative to the original radar beam? Get solution
93g. A spaceship moving 0.80c direction away from Earth fi res a missile that the spaceship measures to be moving at 0.80c perpendicular to the ship’s direction of travel. Find the velocity components and speed of the missile as measured by Earth. Get solution
94g. An electron has a total energy that is 250 times its rest energy. Determine its (a) kinetic energy, (b) speed, and (c) momentum. Get solution
95g. A proton moves with a speed of 0.90c. Find the speed of an electron that has (a) the same momentum as the proton, and (b) the same kinetic energy. Get solution
96g. A high-speed K0 meson is traveling at a speed of 0.90c when it decays into a π+and a π- meson. What are the greatest and least speeds that the mesons may have? Get solution
97g. Frank and Mary are twins, and Mary wants to travel to our nearest star system, Alpha Centauri (4.30 lightyears away). Mary leaves on her 30th birthday and intends to return to Earth on her 52nd birthday. (a) Assuming her spaceship returns from Alpha Centauri without stopping, how fast must her spaceship travel? (b) How old will Frank be when she returns? Get solution
98g. The International Space Federation constructs a new spaceship that can travel at a speed of 0.995c. Mary, the astronaut, boards the spaceship to travel to Barnard’s star, which is the second nearest star to our solar system after Alpha Centauri and is 5.98 lightyears away. After reaching Barnard’s star, the spaceship travels slowly around the star system for three years doing research before returning back to Earth. (a) How much time does her journey take? (b) How much older is her twin Frank than Mary when she returns? Get solution
99g. A powerful laser on Earth rotates its laser beam in a circle at a frequency of 0.030 Hz. (a) How fast does the spot that the laser makes on the moon move across the moon’s landscape? (b) With what rotation frequency should the laser rotate if the laser spot moves across the moon’s landscape at speed c ? Get solution
100g. The Lockheed SR-71 Blackbird may be the fastest non-research airplane ever built; it traveled at 2200 miles/ hour (983 m/s) and was in operation from 1966 to 1990. Its length is 32.74 m. (a) By what percentage would it appear to be length contracted while in flight? (b) How much time difference would occur on an atomic clock in the plane compared to a similar clock on Earth during a flight of the Blackbird over its range of 3200 km? Get solution
101g. A spaceship is coming directly toward you while you are in the International Space Station. You are told that the spaceship is shining sodium light (with an intense yellow doublet of wavelengths 588.9950 and 589.5924 nm). You have an apparatus that can resolve two closely spaced wavelengths if the difference is Δ λ Get solution
102g. Quasars are among the most distant objects in the universe and are moving away from us at very high speeds, as discussed in Chapter 16. Astrophysicists use the redshift parameter z to determine the redshift of such rapidly moving objects. The parameter z is determined by observing a wavelength λ’ of a known spectral line of wavelength λsource on Earth; z = Δ λ’/ λsource= (λ’ - λsource)/ λsource Find the speed of two quasars having z values of 1.9 and 4.9. Get solution
103g. One possible decay mode of the neutral kaon is K0 ... π0 + π0. The rest energies of the K0 and π0 are 498 MeV and 135 MeV, respectively. The kaon is initially at rest when it decays. (a) How much energy is released in the decay? (b) What are the momentum and relative directions of the two neutral pions (π0)? Get solution
104g. The sun radiates energy at a rate of 3.9 × 1026 W. (a) At what rate is the sun losing mass? (b) At that rate, how much time would it take to exhaust the sun’s fuel supply? The sun’s mass is 2.0 × 1030 kg, and you may assume that the reaction producing the energy is about 0.7% efficient. Compare your answer with the sun’s expected remaining lifetime, about 5 Gy. Get solution
105g. One way astrophysicists have identified “extrasolar” planets orbiting distant stars is by observing redshifts or blueshifts in the star’s spectrum due to the fact that the star and planet each revolve around their common center of mass. (See Scientifi c American, August 2010, p. 41.) Consider a star the size of our sun (mass _ 1.99 _ 1030 kg), with a planet the size of Jupiter (1.90 × 1027 kg) in a circular orbit of radius 7.79 × 1011 m and a period of 11.9 years. (a) Find the speed of the star revolving around the system’s center of mass. (b) Assume that Earth is in the planet’s orbital plane, so that at one point in its orbit the star is moving directly toward Earth, and at the opposite point it moves directly away from Earth. How much is 550-nm light redshifted and blueshifted at those two extreme points? Get solution
106g. Small differences in the wavelengths in the sun’s spectrum are detected when measurements are taken from different parts of the sun’s disk. Specifi cally, measurements of the 656-nm line in hydrogen taken from opposite sides on the sun’s equator—one side approaching Earth and the other receding—differ from each other by 0.0090 nm. Use this information to fi nd the rotational period of the sun’s equator. Express your answer in days. (The sun’s equatorial radius is 6.96 × 108 m.) Get solution
