371
R
ELATIVITY
37.1.
I
DENTIFY
and
S
ET
U
P
:
Consider the distance
A
to
O
′
and
B
to
O
′
as observed by an observer on the ground
(Figure 37.1).
Figure 37.1
E
XECUTE
:
Simultaneous to observer on train means light pulses from
and
A
B
′
′
arrive at
O
′
at the same time.
To observer at
O
light from
A
′
has a longer distance to travel than light from
B
′
so
O
will conclude that the pulse
from
(
)
A A
′
started before the pulse at
(
).
B B
′
To observer at
O
bolt
A
appeared to strike first.
E
VALUATE
:
Section 37.2 shows that if they are simultaneous to the observer on the ground then an observer on
the train measures that the bolt at
B
′
struck first.
37.2.
(a)
2
1
2.29.
1
(0.9)
=
=
−
6
6
(2.29) (2.20
10
s)
5.05
10
s.
t
τ
−
−
=
=
×
=
×
(b)
8
6
3
(0.900) (3.00
10 m s) (5.05
10
s)
1.36
10 m
1.36 km.
d
vt
−
=
=
×
×
=
×
=
37.3.
I
DENTIFY
and
S
ET
U
P
:
The problem asks for
u
such that
0
1
/
.
2
t
t
Δ
Δ =
E
XECUTE
:
0
2
2
1
/
t
t
u
c
Δ
Δ =
−
gives
(
)
2
2
8
8
0
1
1
/
(3.00
10
m/s)
1
2.60
10
m/s
2
u
c
t
t
⎛
⎞
=
− Δ
Δ
=
×
−
=
×
⎜
⎟
⎝
⎠
;
0.867
u
c
=
Jet planes fly at less than ten times the speed of sound, less than about 3000 m/s.
Jet planes fly at much lower
speeds than we calculated for
u
.
37.4.
I
DENTIFY
:
Time dilation occurs because the rocket is moving relative to Mars.
S
ET
U
P
:
The time dilation equation is
0
t
t
γ
Δ = Δ
, where
t
0
is the proper time.
E
XECUTE
:
(a)
The two time measurements are made at the same place on Mars by an observer at rest there, so
the observer on Mars measures the proper time.
(b)
0
2
1
(75.0 s)
435 s
1
(0.985)
t
t
γ
μ
μ
Δ = Δ
=
=
−
E
VALUATE
:
The pulse lasts for a shorter time relative to the rocket than it does relative to the Mars observer.
37.5.
(a) I
DENTIFY
and
S
ET
U
P
:
8
7
0
2.60
10
s;
4.20
10
s.
t
t
−
−
Δ
=
×
Δ =
×
In the lab frame the pion is created and decays
at different points, so this time is not the proper time.
E
XECUTE
:
2
2
0
0
2
2
2
says 1
1
/
t
u
t
t
c
t
u
c
Δ
Δ
⎛
⎞
Δ =
−
=
⎜
⎟
Δ
⎝
⎠
−
2
2
8
0
7
2.60
10
s
1
1
0.998;
0.998
4.20
10
s
u
t
u
c
c
t
−
−
⎛
⎞
Δ
×
⎛
⎞
=
−
=
−
=
=
⎜
⎟
⎜
⎟
Δ
×
⎝
⎠
⎝
⎠
E
VALUATE
:
,
u
c
<
as it must be, but
u
/
c
is close to unity and the time dilation effects are large.
(b) I
DENTIFY
and
S
ET
U
P
:
The speed in the laboratory frame is
0.998 ;
u
c
=
the time measured in this frame is
,
t
Δ
so the distance as measured in this frame is
d
u t
=
Δ
E
XECUTE
:
8
7
(0.998)(2.998
10
m/s)(4.20
10
s)
126 m
d
−
=
×
×
=
E
VALUATE
:
The distance measured in the pion’s frame will be different because the time measured in the pion’s
frame is different (shorter).
37
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372
Chapter 37
37.6.
1.667
=
(a)
8
0
1.20
10 m
0.300 s.
(0.800 )
t
t
c
Δ
×
Δ
=
=
=
(b)
7
(0.300 s) (0.800 )
7.20
10 m.
c
=
×
(c)
0
0.300 s
0.180 s.
t
Δ
=
=
(This is what the
racer
measures
your
clock to read at that instant.) At
your
origin
you read the original
8
8
1.20
10 m
0.5 s.
(0.800) (3
10 m s)
×
=
×
Clearly the observers (you and the racer) will not agree on the
order of events!
37.7.
I
DENTIFY
and
S
ET
U
P
:
A clock moving with respect to an observer appears to run more slowly than a clock at rest
in the observer’s frame. The clock in the spacecraft measurers the proper time
0
.
t
Δ
365 days
8760 hours.
t
Δ =
=
E
XECUTE
:
The clock on the moving spacecraft runs slow and shows the smaller elapsed time.
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 Spring '11
 Shaefer
 Kinetic Energy, Special Relativity, Arrakis, 2E 2E 2E

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