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View Full DocumentProblem 2 (20 pts)
A pion at rest in the lab frame
decays into a muon and a neutrino. The pion has a mass
m
1
, the muon
has a mass
m
2
, and the neutrino is massless. Muons, however, are themselves unstable and will decay
after an average time
T
as measured in the muon’s rest frame
. Express your answers in terms of
m
1
,
m
2
,
T
and any fundamental constants.
a) What is the muon’s momentum (magnitude)? [in the lab frame]
b) How far away from the pion will the muon travel, as measured in the lab frame?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
a)
Let
p
μ
be the momentum of the muon, which is what we’re trying to compute. Let
p
ν
be the
momentum of the neutrino. First we’ll apply conservation of momentum
in the lab frame. The initial
momentum of the system is zero, since the pion is at rest. The final momentum is
p
μ
+
p
ν
. Therefore,
p
μ
= –
p
ν
. What’s important here is that the
magnitudes
of these two momenta are equal, which we’ll write
as
p
μ
=
p
ν
.
Next, we apply conservation of energy
in the lab frame. The initial energy is the energy of a pion at
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 Spring '08
 LIN
 Quantum Physics, Special Relativity, lab frame, pµ

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