Elements of Modern Physics
12
even explain the observation (1.32) from frame F as follows. The observer in
frame F′ will argue that the scale used by the observer in frame F is shorter by
a factor of
1/ 2
2
2
1
v
c
−
, and therefore the length observed by F should be
2
0
2
1
v
l
c
−
. He will also argue from Eq. (1.21), that t 2 –t 1 = 0 corresponds to
2
1
2
1
2
0
2
(
)
v
t
t
x
x
c
v l
c
= = −
−
′
′
′
′
= −
(1.33)
Therefore, the leading end was measured at a time
0
2
vl
c
−
earlier than the
trailing end, giving rise to an additional contribution to the measurement of the
length of an amount
2
0
2
v l
c
. Including both of these corrections, the length of the
rod should be
2
2
0
0
0
2
2
1
l v
v
l l
l
c
c
′ =
−
+
=
(1.34)
in agreement with the measurement from frame F′ !
1.8 TRANSFORMATION OF VELOCITIES
In the preceding sections, the implications of the Lorentz transformations for
the measurement of position and time intervals in different frames were
considered. Here, the relation between velocities of a particle measured in
different inertial frames is obtained.
Let =
d
dt
r
u
be the velocity of a particle in frame F and
′
′ = ′
d
dt
r
u
be the
corresponding velocity in frame F′ which moves with velocity v. Lorentz
transformations (1.20) imply that
1/ 2,
2
2
,
1
−
′
′
′
=
=
=
−
dx v dt
dx
dy dy dz dz
v
c
(1.35)
12
even explain the observation (1.32) from frame F as follows. The observer in
frame F′ will argue that the scale used by the observer in frame F is shorter by
a factor of
1/ 2
2
2
1
v
c
−
, and therefore the length observed by F should be
2
0
2
1
v
l
c
−
. He will also argue from Eq. (1.21), that t 2 –t 1 = 0 corresponds to
2
1
2
1
2
0
2
(
)
v
t
t
x
x
c
v l
c
= = −
−
′
′
′
′
= −
(1.33)
Therefore, the leading end was measured at a time
0
2
vl
c
−
earlier than the
trailing end, giving rise to an additional contribution to the measurement of the
length of an amount
2
0
2
v l
c
. Including both of these corrections, the length of the
rod should be
2
2
0
0
0
2
2
1
l v
v
l l
l
c
c
′ =
−
+
=
(1.34)
in agreement with the measurement from frame F′ !
1.8 TRANSFORMATION OF VELOCITIES
In the preceding sections, the implications of the Lorentz transformations for
the measurement of position and time intervals in different frames were
considered. Here, the relation between velocities of a particle measured in
different inertial frames is obtained.
Let =
d
dt
r
u
be the velocity of a particle in frame F and
′
′ = ′
d
dt
r
u
be the
corresponding velocity in frame F′ which moves with velocity v. Lorentz
transformations (1.20) imply that
1/ 2,
2
2
,
1
−
′
′
′
=
=
=
−
dx v dt
dx
dy dy dz dz
v
c
(1.35)