1. From the time dilation equation
∆
t
=
γ
∆
t
0
(where
∆
t
0
is the proper time interval,
γ
β
=
−
1
1
2
/
, and
β
=
v
/
c
), we obtain
β
=
−
F
H
G
I
K
J
1
0
2
∆
∆
t
t
.
The proper time interval is measured by a clock at rest relative to the muon. Specifically,
∆
t
0
= 2.2000
µ
s. We are also told that Earth observers (measuring the decays of moving
muons) find
∆
t
= 16.000
µ
s. Therefore,
2
2.2000
s
1
0.99050.
16.000
s
µ
β
µ
⎛
⎞
=
−
=
⎜
⎟
⎝
⎠

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2. (a) We find
β
from
γ
β
=
−
1
1
2
/
:
(
)
2
2
1
1
1
1
0.14037076.
1.0100000
β
γ
=
−
=
−
=
(b) Similarly,
(
)
2
1
10.000000
0.99498744.
β
−
=
−
=
(c) In this case,
(
)
2
1
100.00000
0.99995000.
β
−
=
−
=
(d) The result is
(
)
2
1
1000.0000
0.99999950.
β
−
=
−
=