1. Chasing Light.
What fraction of the speed of light does each of the following speeds v represent? That is, what is the value of the ratio v/c?
(a) A typical rate of continental drift, 3 cm/y.
(b) A high way speed limit of 100 km/h.
(c) A supersonic plane flying at Mach 2.5 = 3100 km/h.
(d) The Earth in orbit around the Sun at 30 km/s.
(e) What conclusion(s) do you draw about the need for special relativity to describe and analyze most everyday phenomena?
(Note: Some everyday phenomena can be derived from relativity. For example, magnetism can be described as arising from electrostatics plus special relativity applied to the slow-moving charges in wires.) (v/c= 3.16x10-18. ; v/c= 9.26x10-8 ;v/c= 2.87x10-6 ; v/c = 10-4 )
2. Eruption from the Sun
You see a sudden eruption on the surface of the Sun. From solar theory you predict that the eruption emitted a pulse of particles that is moving toward the Earth at one- eighth the speed of light. How long do you have to seek shelter from the radiation that will be emitted when the particle pulse hits the Earth? Take the light-travel time from the Sun to the Earth to be 8 minutes.
3. Where and When?
Two firecrackers explode at the same place in the laboratory and are separated by a time of 12 years. (a) What is the spatial distance between these two events in a rocket in which the events are separated in time by 13 years? (b) What is the relative speed of the rocket and laboratory frames? Express your answer as a fraction of the speed of light. (4.7x1016 meters; a little more than one-third the speed of light)
4. Fast-Moving Muons
The half-life of stationary muons is mea sured to be 1.6 microseconds. Half of any initial number of station ary muons decays in one half-life. Cosmic rays colliding with atoms in the upper atmosphere of the Earth create muons, some of which move downward toward the Earth's surface. The mean lifetime of high-speed muons in one such burst is measured to be 16 microseconds. (a) Find the speed of these muons relative to the Earth. (b) Moving at this speed, how far will the muons move in one half-life? (c) How far would this pulse move in one half-life if there were no relativistic time stretching? (d) In the relativistic case, how far will the pulse move in 10 half-lives? (e) An initial pulse consisting of 108 muons is created at a distance above the Earth's surface given in part (d). How many will remain at the Earth's surface? Assume that the pulse moves vertically downward and none are lost to collisions. (Ninety-nine percent of the Earth's atmosphere lies below 40 km altitude.)
5. Living a Thousand Years in One Year
Living a Thousand Years in One Year. You wish to make a round trip from Earth in a spaceship, traveling at constant speed in a straight line for 6 months on your watch and then returning at the same constant speed. You wish, further, to find Earth to be 1000 years older on your return. (a) What is the value of your constant speed with respect to Earth? (b) How much do you age during the trip? (c) Does it matter whether or not you travel in a straight line? For example, could you travel in a huge circle that loops back to Earth? (v/c = 0.0999995 ; one year)
6. A Box of Light
Estimate the power in kilowatts used to light a city of 8 million inhabitants. If all this light generated during one hour in the evening could be captured and put in a box, how much would the mass of the box increase?
7. Really Simultaneous?
(a) Two events occur at the same time in the laboratory frame and at the laboratory coordinates (x1 = 10 km, y1 = 4 km, z1 = 6 km) and (x2 = 10 km, y2 = 7 km, z2 = -10 km). Will these two events be simultaneous in a rocket frame moving with speed v= 0.8c in the x direction in the laboratory frame? Explain your answer.
(b) Three events occur at the same time in the laboratory frame and at the laboratory coordinates (x0,y1, z1), (x0,y2, z2) and (x0,y1, z3) where x0 has the same value for all three events. Will these three events be simultaneous in a rocket frame moving with speed v in the laboratory x direction? Explain your answer.
(c) Use your results of parts (a) and (b) to make a general statement about simultaneity of events in laboratory and rocket frames.
8. Electron Shrinks Distance
An evacuated tube at rest in the /laboratory has a length 3.00 m as measured in the laboratory. An electron moves at speed v = 0.999 987c in the laboratory along the axis of this evacuated tube. What is the length of the tube measured in the rest frame of the electron?
9. Traveling to the Galactic Center
(a) Can a person, in principle, /travel from Earth to the center of our galaxy, which is 23 000 ly distant, in one lifetime? Explain using either length contraction or time dilation arguments.
(b) What constant speed with respect to the galaxy is required to make the trip in 30 y of the traveler's life time?
10. Separating Galaxies.
Galaxy A is measured to be receding from us on Earth with a speed of 0.3c. Galaxy B, located in precisely the opposite direction, is also receding from us at the same speed. What recessional velocity will an observer on galaxy A measure (a) for our galaxy, and (b) for galaxy B?
11. Transit Time
An unpowered spaceship whose rest length 350 meters has a speed 0.82c with respect to Earth. A micrometeorite, also with speed of 0.82c with respect to Earth, passes spaceship on an antiparallel track that is moving in the opposite direction. How long does it take the micrometeorite to pass spaceship as measured on the ship?