1
Phy 3031 Modern physics, Spring 2007
Homework set #3. Due at the start of class on Wednesday, January 24, 2007.
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
(b)
√
1 − x ≈ 1 − 12 x for
√
(c) 1/ 1 − x ≈ 1 + 21 x for
10−2 ,
10−2 ,
10−2 ,
10−3 and
10−3 and
10−3 and
10−4 .
10−4 .
10−4 .
2. 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?