CHAPTER
39
Relativity
1*
∙
You are standing on a corner and a friend is driving past in an automobile. Both of you note the times when the
car passes two different intersections and determine from your watch readings the time that elapses between the two
events. Which of you has determined the proper time interval?
By definition, the proper time is measured by the clock in the rest frame of the car, i.e., by the clock in the car.
2
∙
The proper mean lifetime of pions is 2.6
×
10
-8
s. If a beam of pions has a speed of 0.85
c
, (
a
) what would their
mean lifetime be as measured in the laboratory? (
b
) How far would they travel, on average, before they decay? (
c
)
What would your answer be to part (
b
) if you neglect time dilation?
(
a
)
Use Equs. 39-13 and 39-7
(
b
)
∆
x
=
v
∆
t
(
c
)
Neglecting time dilation,
∆
t
= 2.6
×
10
-8
s
γ
=
s
.
;
Ä Ät
=
.
=
.
/
10
94
4
90
1
85
0
1
1
8
2
−
×
−
∆
x
= 0.85
×
3
×
10
8
×
4.94
×
10
-8
m = 12.6 m
∆
x
= 0.85
×
3
×
10
8
×
2.6
×
10
-8
m = 6.63 m
3
∙
(
a
) In the reference frame of the pion in Problem 2, how far does the laboratory travel in a typical lifetime of
2.6
×
10
-8
s? (
b
) What is this distance in the laboratory’s frame?
(
a
)
∆
x
=
v
∆
t
π
(
b
)
∆
x
′
=
γ
∆
x
∆
x
= 6.63 m
∆
x
′
= 12.6 m
4
∙
The proper mean lifetime of a muon is 2
µ
s. Muons in a beam are traveling at 0.999
c
. (
a
) What is their mean
lifetime as measured in the laboratory? (
b
) How far do they travel, on average, before they decay?
(
a
)
Use Equs. 39-13 and 39-7
(
b
)
∆
x
=
v
∆
t
γ
=
s
ì
.
;
Ä Ät
=
.
=
/
7
44
37
22
)
999
.
0
(
1
1
2
−
∆
x
= 0.999
×
3
×
10
8
×
44.7
×
10
-6
m = 13.4 km
5*
∙
(
a
) In the reference frame of the muon in Problem 4, how far does the laboratory travel in a typical lifetime of
2
µ
s? (
b
) What is this distance in the laboratory’s frame?
(
a
)
∆
x
=
v
∆
t
µ
(
b
)
∆
x
′
=
γ
∆
x
µ
∆
x
µ
= 0.999
×
3
×
10
8
×
2
×
10
-6
m = 599.4 m
∆
x
′
= 13.4 km

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