1
Phy 3101 Modern physics, Spring 2008
Homework set #3. Due at the start of class on Monday, January 21, 2008.
For numerical answers give two significant digits. The first should be correct, the second can
be approximate.
1. Here are three Taylor expansions
1
= 1 + x + O(x2 ),
1−x
√
1
1 − x = 1 − x + O(x2 ),
2
√
1
1
= 1 + x + O(x2 ).
2
1−x
Use your calculator to find the absolute value of the error in each of the following
approximations:
(a) 1/(1 − x) ≈ 1 + x for values of x
(b)
√
1 − x ≈ 1 − 12 x for values of x
√
(c) 1/ 1 − x ≈ 1 + 12 x for values of x
10−2 , 10−3 and
10−2 , 10−3 and
10−2 , 10−3 and
10−4 .
10−4 .
10−4 .
2. 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
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 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
2
dilation slows the clocks on the satellites down relative to a clock at rest at the center
of the Earth. 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? Hint: recall Newtonian gravity and the fact that centripetal acceleration
for circular motion is v 2 /R.
(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.
3. A muon is created at the top of a mountain in a cosmic ray collision. Assume that
the muon lives, in its frame of reference, for precisely 2 × 10−6 s before it decays into
an electron and two neutrinos. The mountain is L0 =
q 800 m high, and the muon is
traveling straight down with a speed v = 0.8c so that 1 − v 2 /c2 = 0.6 = 3/5.
(a) How long, as measured in the muon’s frame of reference, does the muon live
before it decays?
(b) What distance down the mountain, as measured in the muon’s frame of reference,
does the muon travel before it decays? Perhaps this should be phrased: As
measured in the muon’s frame of reference, how far past the muon does the
mountain travel before the decay?
(c) How long do we (standing at the bottom of the mountain) observe the muon to
live before it decays?
(d) Does the muon live at least until it reaches the bottom of the mountain?