1
Phy 3221
Due: March 30, 2011
Homework set # 10
Problem 1: Tarzan can swim through still water with a speed c 0 . To impress Jane, he
swims directly across the Zambesi river (his path is at a right angle to the shore) 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
Problem 2: A passenger train 120 m long passes a long railroad platform in Winnipeg.
The train is moving with a speed of 0.8 c by the platform. The stationmaster measures the
length of the train by measuring the arrival times between the front of the train and the
rear of the train.
a. How long does the stationmaster measure the train to be?
b. What is the difference in the arrival time between the front of the train and the rear?
c. Alice is a passenger on the train. How much time does she measure between when the
stationmaster is at the front of the train and when he is at the the rear of the train?
Problem 3:
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 t1 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 t2 of reception. The difference in these times t2 − t1 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 = (t2 − t1 )/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
2
to limit the accuracy to 10 or 20 meters 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.