15
This is a case of dilation.
T
=
"
#
T
in this problem with the proper time
"
T
=
T
0
T
=
1
"
v
c
#
$
%
&
’
(
2
)
*
+
+
,

.
.
"
1 2
T
0
/
v
c
=
1
"
T
0
T
#
$
%
&
’
(
2
)
*
+
+
,

.
.
1 2
;
in this case
T
=
2
T
0
,
v
=
1
"
L
0
2
L
0
#
$
%
&
’
(
2
)
*
+
,
+

.
+
/
+
1 2
=
1
"
1
4
0
1
2
3
4
5
#
$
%
&
’
(
1 2
therefore
v
=
0.866
c
.
16
This is a case of length contraction.
L
=
"
L
#
in this problem the proper length
"
L
=
L
0
,
L
=
1
"
v
2
c
2
#
$
%
&
’
(
"
1 2
L
0
)
v
=
c
1
"
L
L
0
*
+
,

.
/
2
#
$
%
%
&
’
(
(
1 2
; in this case
L
=
L
0
2
,
v
=
1
"
L
0
2
L
0
#
$
%
&
’
(
2
)
*
+
,
+

.
+
/
+
1 2
=
1
"
1
4
0
1
2
3
4
5
#
$
%
&
’
(
1 2
therefore
v
=
0.866
c
.
17
The problem is solved by using time dilation. This is also a case of
v
<<
c
so the binomial
expansion is used
"
t
=
#
" $
t
%
1
+
v
2
2
c
2
&
’
(
)
*
+
" $
t
,
"
t
# " $
t
=
v
2
" $
t
2
c
2
;
v
=
2
c
2
"
t
# " $
t
(
)
" $
t
%
&
’
(
)
*
1 2
;
"
t
=
24 h day
(
)
3 600 s h
(
)
=
86 400 s
;
"
t
=
" #
t
$
1
=
86 399 s
;
v
=
2
86 400 s
"
86 399 s
(
)
86 399 s
#
$
%
%
&
’
(
(
1 2
=
0.004 8
c
=
1.44
)
10
6
m s
.
18
L
=
"
L
#
1
"
=
L
#
L
=
1
$
v
2
c
2
%
&
’
(
)
*
1 2
v
=
c
1
$
L
#
L
+
,

.
/
0
2
%
&
’
’
(
)
*
*
1 2
=
c
1
$
75
100
+
,

.
/
0
2
%
&
’
’
(
)
*
*
1 2
=
0.661
c
110
(a)
"
=
#
$
"
where
"
=
v
c
and
"
#
=
1
$
%
2
(
)
$
1 2
=
#
&
1
$
v
2
c
2
’
(
)
*
+
,
$
1 2
=
2.6

10
$
8
s
(
)
1
$
0.95
(
)
2
[
]
$
1 2
=
8.33

10
$
8
s
(b)
d
=
v
"
=
0.95
(
)
3
#
10
8
(
)
8.33
#
10
8
s
(
)
=
24 m
112
(a)
70 beats/min or
" #
t
=
1
70
min
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