PHY2061
R. D. Field
Solutions
Relativity
Page 1 of 6
Relativity Solutions
Problem 1:
A rod whose length at rest is
10 meters
is
moving to the right along the positive x-axis of
the
O-frame
with velocity
V = 0.8c
.
An
observer riding on the rod (
i.e.
in the
O'-frame
)
sets off two explosions simultaneously – so he
says.
If the rod is parallel to the x-axis how far
apart are the two explosions (
in meters
) in the
O-frame
?
Is this length the same as the length
of the rod as measured in the
O-frame
?
If your answer is no, then what is
the length of the rod (
in meters
) as measured in the
O-frame
?
Answers:
16.67, no, 6
Solution:
The two explosions occur simultaneously in the
O'-frame
which
implies that
∆
t' = 0
and thus
m
m
L
t
c
x
x
67
.
16
)
10
)(
667
.
1
(
)
(
0
=
=
=
′
∆
+
′
∆
=
∆
γ
β
γ
where I used
∆
x' = L
0
. This is
not
the same as the length of the rod as
measured in the
O-frame
.
The length of the rod as measured in the
O-
frame
corresponds to marking both ends of the rod simultaneously in the
O-
frame
(
i.e.
∆
t = 0
) as follows:
L
t
c
x
L
x
γ
β
γ
=
∆
−
∆
=
=
′
∆
)
(
0
and
m
m
L
L
6
667
.
1
10
0
=
=
=
γ
.
Problem 2:
In the previous problem, what is the time interval (
in seconds
) between the
two explosions as measured in the
O-frame
?
Which explosion occurred
first in the
O-frame
, the one at the left or right end of the rod?
Answers:
4.44x10
-8
, left one first
Solution:
The two explosions occur simultaneously in the
O'-frame
which
implies that
∆
t' = 0
and thus
m
m
L
x
t
c
t
c
3
.
13
)
10
)(
33
.
1
(
)
(
0
=
=
=
′
∆
+
′
∆
=
∆
γβ
β
γ
where I used
∆
x' = L
0
. This,
y
x
z
y’
z’
x’
V=0.8c
O'
O
Explosions
simultaneously in
the O'-frame
L
0

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