1
Phy 3031 Modern physics, Spring 2007
Homework set #2. Due at the start of class on Wednesday, January 17, 2007.
1. Tarzan can swim through still water with a speed c 0 . To impress Jane, he swims directly across (at a right angle to the shore) the Zambesi river and then returns to his starting place. Where he is swimming, the Zambesi is a distance d wide, but flows with a speed of v .
(a) How long does it take Tarzan to cross the river and then return?
(b) Jane appears bored by all of this, so now Tarzan decides to swim a distance d directly upstream and then swim back downstream to the original spot on the shore. How long does it take for Tarzan to make this round trip?
(c) What is the difference in these times in the limit that v ¿ c 0 ? This means: subtract the two answers, and write your answer in terms of the small quantity v/c 0 . Then take a Taylor expansion of the result, and keep only the dominant term in powers of v/c 0
2. The current value of Hubble’s “constant” is about H = 72 (km/s) / Mpc , and the current age of the universe is about 13.7 billion years. What was Hubble’s “constant” 10 6 years after the big bang? Give your answer in units of (km/s)/Mpc . I am sure that you recall from the first lecture that 1 pc ≈ 3 lt-yr, and that 1 Mpc = 10 6 pc. You can verify these by googling “1 pc in light years”, for example.
3. Respond to the following question: If Einstein were to hold a mirror in front of himself while running faster than the speed of light, then what would he see in the mirror?
4. Hint: This problem is similar to problem 41, in Chapter 1 of the textbook, which is included in the “Student Solutions Manual.” The Global Positioning System is a network of 24 satellites which orbit the Earth each with a period of about 12 hours.
The actual orbits are in many different orientations, but at any given time no matter where you happen to be standing on Earth at least six or seven of the satellites would be visible if your eyesight were good enough. In general terms, each satellite regularly broadcasts a message which identifies itself and very accurately gives the current time t
1 and its own location at the moment that the message is broadcast. Your own GPS receiver (They are available for under $100.) detects this message and accurately notes the time t
2 of reception. The difference in these times t
2
− t
1 is a result of the radio signal traveling at the finite speed of light c . And the brain inside your GPS receiver quickly calculates that distance to be d = ( t
2
− t
1
) /c . Generally in three dimensions if you know your distance to three different particular places (three satellites), then you can uniquely determine your position. The GPS knows trigonometry and does the hard work for you. The redundant information from the rest of the satellites in view is used to keep your clock at the correct time and to increase accuracy. In the end the GPS system can give your latitude, longitude and altitude to an accuracy of
2 meters—if you are in the U.S. military. The military designed this system, and they add a random error into the timing information to limit the accuracy to 10 or 20 meters

2 for just an ordinary person or some terrorist—the military know how to remove this extra random error to get the 2 meter accuracy.
Very accurate clocks are required to make this scheme work. Light travels about a foot in one nanosecond, 1 ns = 1 × 10 − 9 s. So, for 2 meter (6 foot) accuracy the clocks on the satellites must know the correct time to an accuracy of 6 ns. Relativistic time dilation slows the clocks on the satellites down. If the GPS system did not take this into account then eventually the error in the clocks would accumulate to be more than
6 ns and the system would fail.
(a) The orbital period of a GPS satellite is 12 hours. What is the radius of the orbit?
(b) What is the speed of a GPS satellite?
(c) If relativistic time dilation were not taken into account, then how long would it take for the clocks to be in error by 6 ns? It should now be clear that the success of the GPS confirms one of the more surprising consequences of special relativity.