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. 3913 and 397
(
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
′
=
∆
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. 3913 and 397
(
b
)
∆
x
=
v
∆
t
=
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
s? (
b
) What is this distance in the laboratory’s frame?
(
a
)
∆
x
=
v
∆
t
(
b
)
∆
x
′
=
∆
x
∆
x
= 0.999
×
3
×
10
8
×
2
×
10
6
m = 599.4 m
∆
x
′
= 13.4 km
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View Full DocumentChapter 39
Relativity
6
∙
Jay has been posted to a remote region of space to monitor traffic. Toward the end of a quiet shift, a spacecraft
goes by, and he measures its length using a laser device, which reports a length of 85 m. He flips open his handy
reference catalogue and identifies the craft as a CCCNX22, which has a proper length of 100 m. When he phones in
his report, what speed should Jay give for this spacecraft?
From Equ. 3914,
γ
=
L
p
/
L
; solve for
V
= 100/85;
m/s
.
c
=
.
=
/
V
=
c
10
58
1
527
0
1
1
8
2
×
−
7
∙
A spaceship travels to a star 95 lightyears away at a speed of 2.2
×
10
8
m/s. How long does it take to get there (
a
)
as measured on earth and (
b
) as measured by a passenger on the spaceship?
(
a
) As measured on earth,
∆
t
=
∆
x
/
V
(
b
) Use Equ. 3913;
∆
t
p
=
∆
t
/
∆
t
= (95 c
.
y)/[(2.2/3) c] = 129.5 y
= 1.47;
∆
t
p
= (129.5/1.47) y = 88 y
8
∙
The mean lifetime of a pion traveling at high speed is measured to be 7.5
×
10
8
s. Its lifetime when measured at
rest is 2.6
×
10
8
s. How fast is the pion traveling?
Use Equ. 3913; solve for
V
= 7.5/2.6;
m/s
.
c
=
.
=
ã
/
V
=
c
10
81
2
938
0
1
1
8
2
×
−
9*
∙
A meterstick moves with speed
V
= 0.8
c
relative to you in the direction parallel to the stick. (
a
) Find the length of
the stick as measured by you. (
b
) How long does it take for the stick to pass you?
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 Spring '11
 Virgil.E.Barnes
 Special Relativity, γ, reference frame, s′

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