THIS IS A PRACTICE ASSESSMENT. Show formulas, substitutions, answers, and units!
Option A.1 – The beginnings of relativity (Core)
1. What is a frame of reference?
2. Your frame of reference is S. At the instant S’ (the moving van) is coincident with S you start your stopwatch. The cone is exactly 225 m from you. If the van is traveling at 45.5 m s-1 find the distance the van is from the cone at t = 0.00 s and t = 2.75 s.
3. What is the velocity of the cone with respect to you in S?
4. What is the velocity of the cone with respect to S’?
5. What is the acceleration of the cone in S? What is its acceleration in S’?
6. What is a Galilean transformation?
7. Write the Galilean transformation equations.
8. Use the Galilean transformations and the picture in 2. above to answer the following question. Suppose the cone is traveling at 20 ms-1 to the right (it is on wheels!) and the van is traveling at 30 ms-1 to the right (both relative to you). Find v, u, and u’.
9. Two trains traveling at a speed of 0.5c approach each other on the same track. At the same instant, both trains turn on their headlights. How fast do the two beams of light approach one another?
10. Describe what is meant by an inertial frame of reference.
11. What does it mean to say that the laws of physics are invariant across two frames of reference?
12. Do you think acceleration (and thus force) is invariant across two inertial reference frames? Explain.
13. Do you think acceleration (and thus force) is invariant across an inertial and a non-inertial reference frame? Explain.
Option A.2 – Lorentz transformations (Core)
14. Explain how Maxwell’s electromagnetic theory “upset” the philosophical thinking of physicists in the late 19th century. Give an example of how the magnetic force laws are different in different inertial reference frames.
15. Explain how the null results of the Michelson-Morley experiment (MME) contradicted two wave behaviors physicists thought all waves had.
16. Explain what the luminiferous ether was and how the MME demonstrated that it was not necessary.
17. Explain how the MME led to the Lorentz-FitzGerald contraction theory.
18. According to the Lorentz-FitzGerald contraction, what would be the length of a rocket ship which was traveling at 0.75c in the direction of its length, if its rest length were 85 m?
19. What are Einstein’s two postulates of special relativity? Which one was probably based on the results of Maxwell’s electromagnetic theory?
20. Two trains traveling at a speed of 0.5c approach each other on the same track. At the same instant, both trains turn on their headlights. How fast do the two beams of light approach one another? Use Einstein’s postulates, not the Galilean transformations.
21. Describe the concept of a light clock. Include a sketch.
22. Derive the time dilation formula using the concept of the light clock. Include a precise sketch (or sketches).
23. During the time dilation derivation, you have to make a choice. The choice is that either time is absolute, or the speed of light is absolute. What do each of these mean? Which one did you choose? Why did you choose it?
24. Define proper time interval.
25. Suppose S0 has relative speed of v = 0.866c with respect to S. (a) Find the value of g. (b) If Dobson measures the time to cook a 1.5-minute egg in S0, how long does Nosbod measure the same event in S? Who, therefore, is ageing more quickly?
26. In a laboratory in which it is at rest a particular particle has a lifetime of 10-8 seconds. If it is now traveling at 0.998c through the same laboratory reference frame, how long will it last as observed by someone in the laboratory?
27. A scientist constructs two boxes with lights on top that flash 5.0 times each minute. One box is left in the laboratory and the other is stuck to the side of a manned rocket that is sent into space. Some years later the rocket returns, flying past the laboratory at 0.90 c.
(a) According to the scientist
(i) What is the time between the flashes of his own box in the laboratory?
(ii) What is the time between the flashes of the box on the rocket?
(b) According to the astronaut
(i) What is the time between the flashes of his own box on the rocket?
(ii) What is the time between the flashes of the box in the laboratory?
(c) Why don’t they agree?
28. Sketch a graph showing v/c on the horizontal axis, and the Lorentz factor g on the vertical axis. Be sure to include crucial values and asymptotes on your sketch.
29. By referring to the Lorentz factor, explain that happens to g as v approaches c.
30. By referring to the Lorentz factor, explain that happens to g if v exceeds c.
31. Suppose the Lorentz factor for a spaceship observed from Earth is g = 4.75. What is the speed of the ship relative to the earth observer?
32. Explain the Twin Paradox of Special Relativity, and how it is resolved in General Relativity.
33. Define proper length.
34. Describe the phenomenon of length contraction.
35. Show that Einstein’s prediction of length contraction L = L0 / g is equivalent to the Lorentz-FitzGerald contraction. Why is Einstein’s derivation preferable to that of Lorentz and FitzGerald, even though the formulas are identical?
36. Dobson is in the spaceship of 31. The spaceship’s length, as measured by Dobson (who is on board the ship), is 125 m. What is the length of the spaceship in the Earth reference frame of Nosbod? What is the length of Dobson’s meter stick in the Earth reference frame? What is the length of the meter stick in Dobson’s reference frame? Who measured proper length?
A muon m- is created during a collision of a cosmic ray with a molecule 3.125 km above a detector in a laboratory in the upper atmosphere. Muons decay in 2.197 ms according to the reaction m- ® nm + e- + ne.
37. Draw a Feynman diagram of this reaction.
38. If this muon is traveling toward the ground at 0.9725c relative to the laboratory, what is its lifetime in the laboratories frame of reference? How far can the muon thus travel before decay?
39. In the muon’s frame of reference: how fast is the laboratory approaching it? How far can the muon travel before decay? What is the distance the lab is from the muon?
40. John is in the center of a train car that is moving to the north. Barbara is standing on the side of the tracks watching John’s car roll by. At the precise instant John is opposite Barbara lightning strikes at both the north and south ends of the car. Barbara sees the bolts strike simultaneously. Explain why John does not. Which bolt does he see first? Is it possible for two events to be simultaneous in one reference frame and not another?
41. Write the Lorentz transformation equations for x’ and t’.
Clocks in two IRFs S and S’ are synchronized so that x = x’ = 0.00 m at t = t’ = 0.00 s. Frame S’ has a speed of v = 0.75c relative to S. Two events occur in frame S:
Event 1: x1 = 10.0 m, t1 = 0.35 ms. Event 2: x2 = 40.0 m, t2 = 0.55 ms.
42. Find the distance between the events in IRF S.
43. Find the time interval between the events in IRF S.
44. Find the distance between the events in IRF S’.
45. Find the time interval between the events in IRF S’.
46. Using the Lorentz transformation equations, show that if two events are simultaneous for one observer but happen at different points in space, then the events are not simultaneous for an observer in a different reference frame.
47. The relative velocity between S’ and S is 0.866c. A satellite is launched from S’ with a speed of 0.750c relative to S’. Find the speed of the satellite relative to S.
48. The relative velocity between S’ and S is 0.75c. A photon is launched from S’ with a speed of c relative to S’. Find the speed of the photon relative to S.
49. The relative velocity between S’ and S is 0.75c. A satellite is launched from S’ with a speed of 0.50c relative to S. Find the speed of the satellite relative to S’.
IRF S’ has a speed of v = 0.750c relative to IRF S. An event occurs in frame S having spacetime coordinates x = 10.0 m, t = 0.300 ms.
50. Find the values of x’ and t’.
51. Show that the spacetime interval is invariant for this scenario.
52. An observer from Earth sees a meteorite traveling at 0.50c on a head-on collision with a spaceship traveling at 0.60c. What is the velocity of the meteorite as measured by the spaceship?
Option A.3 – Spacetime diagrams (Core)
53. Compare and contrast a space, and a spacetime diagram for a planet in orbit about a star. Include sketches.
54. Why did Minkowski believe that spacetime diagrams were necessary?
55. Why is there a ct axis instead of a t axis in a spacetime diagram?
56. Compare and contrast Newtonian spacetime distance S with Minkowski spacetime distance s. Which one is Euclidean? Which one is correct?
57. Sketch a spacetime diagram showing a stationary object at x = 2 units. Sketch in a light line. Now add the worldline of an object traveling at half the speed of light in the positive x-direction and having a starting position of x = 2 units.
Consider the three points plotted in Minkowski spacetime:
58. Find the spacetime distance between each set of points.
59. Which pair of points is spacelike separated? Null separated, timelike separated?
60. What is the shortest distance between A and B?
61. Could a particle travel from A to C? Explain. Could a particle travel from C to A? Explain.
62. Could a particle travel from C to B? Explain. Could a particle travel from B to A? Explain.
63. Sketch in a spacetime diagram showing S, which is stationary, and S’, which is traveling to the right at c / 4. Assume at ct = ct’ = 0 units that x = x’ = 0 units. Give two reasons that the x’ axis is drawn at the same angle above the x axis as the ct’ axis is drawn to the right of the ct axis. If spacetime were Euclidean, how would the x’ axis be drawn? Do the Lorentz transformations work in Euclidean spacetime?
64. Find the relative velocity in terms of c between S and S’. Which IRF is stationary? Which is moving? Which way is it moving?
65. Draw a similar diagram for the same scenario, only from S’s perspective.
Two flashes occur in S at the times and positions shown.
66. Which flash occurs first? How do you know?
67. Two observers are located in S at x = 0 units at t = 0 units. Observer 1 is stationary. Observer 2 is traveling at c/4 to the right. Which observer sees the flashes simultaneously?
68. In what order will the other observer see the flashes? Make a good estimate from the graph how many times units this observer will measure between flashes.
69. Make a spacetime diagram illustrating the Twin Paradox. Show that the twin that has two IRFs travels a shorter spacetime distance than the twin on Earth. (This shows that the paradox is resolved by the asymmetry of the two twins’ worldlines.)
70. Write the Lorentz transformations in two forms: First form, the speed of light is c. Second form, the speed of light is 1. What is the maximum value for v?
71. Write the spacetime distance formula in two forms: First form, the speed of light is c. Second form, the speed of light is 1. What is the maximum value for v?
72. Write the formula for q in two forms: First form, the speed of light is c. Second form, the speed of light is 1. What is the maximum value for v?
Option A.4 – Relativistic mechanics (AHL)
The following questions are about a nuclear power plant.
73. State the formula representing the equivalence of mass and energy.
74. Define rest mass. Write the formula for relativistic mass. Why can the units MeVc-2 be used for mass, as well as kilograms?
75. A nuclear power plant converts 75 kg of matter into energy each year. How many joules is this? How many watts? Explain why this wattage is not the actual wattage of the electricity produced by the power plant.
The following questions are about bodies in motion.
76. A proton is accelerated until its mass has increased to that of a 12C atom. What is its speed?
77. Distinguish between the energy of a body at rest and its total energy when it is moving. What single form of energy does a body at rest have (ignore potential energy). What two forms of energy does a body in motion have?
78. Explain why no object can ever attain the speed of light in a vacuum.
79. Consider the kinetic energy EK of a particle having a charge e and mass m accelerated through a potential difference V. Explain why the formula EK = eV works under relativistic conditions, but the formula (1/2)mv2 = eV does not.
A proton is accelerated from rest through a p.d. of 1.0 MV.
80. Find the kinetic energy of the proton.
81. Find the total energy of the proton.
82. Find the speed of the proton.
83. Find the momentum of the proton.
The following questions are about accelerating electrons.
84. Calculate the total energy of an electron moving with a velocity of 0.90 c.