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27. (a) One thing Einstein’s relativity has in common with the more familiar (Galilean)
relativity is the reciprocity of relative velocity. If Joe sees Fred moving at 20 m/s
eastward away from him (Joe), then Fred should see Joe moving at 20 m/s westward
away from him (Fred). Similarly, if we see Galaxy A moving away from us at 0.35
c
then
an observer in Galaxy A should see our galaxy move away from him at 0.35
c
, or 0.35 in
multiple of
c
.
(b) We take the positive axis to be in the direction of motion of Galaxy A, as seen by us.
Using the notation of Eq. 3729, the problem indicates
v
= +0.35
c
(velocity of Galaxy A
relative to Earth) and
u
= –0.35
c
(velocity of Galaxy B relative to Earth). We solve for
the velocity of B relative to A:
2
'
/
/
( 0.35) 0.35
0.62
1
/
1 ( 0.35)(0.35)
uu
c
v
c
cu
v
c
−
−−
==
=
−
−
,
or  '/ 
0.62.
uc
=
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View Full Document(b) In the armada’s rest frame (called
S
a
), the velocity of the messenger is
v
vv
vv
c
cc
c
c
a
a
'
/
..
(. )
(. )/
=
−
−
=
−
−
=
1
095
080
10
9
50
8
0
0 625
22
Now, the length of the trip is
0
1.0ly
'1
.
6
0
y
.
'
0.625
L
t
vc
==
=
(c) Measured in system
S
, the length of the armada is
L
L
−
=
0
2
1
060
γ
.(
.
)
.
,
ly
ly
so the length of the trip is
0.60ly
4.00 y .
0.95
0.80
ma
L
t
c
c
=
−−
31. (a) In the messenger’s rest system (called
S
m
), the velocity of the armada is
v
vv
c
c
c
m
m
'
/
/
=
−
−
=
−
−
=−
1
8
00
9
5
0 625
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 Fall '09
 STEIGMAN
 Physics

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