
442 Classical Mechanics
velocities can be deduced easily. By denition, the components of velocity in S and S′ are given by,
respectively,
v
x
t
x
==
−
−
d
d
and
v
y
t
y
==
−
−
d
d
′
=
′
=
′
−
′
′
−
′
v
x
t
x
d
d
and
′
=
′
=
′
−
′
′
−
′
v
y
t
y
d
d
and so on.
Applying the Lorentz transformations to x
1
and x
2
and then taking the difference, we get
d
dd
,x
xu
c
=
′
+
′
−
=
1
2
β
β
.
Similarly,
d
′
=
−
−
x
xut
1
2
β
.
Do the same for the time intervals dt and dt′:
d
t
t udx c
=
′
+
′
−
2
2
1 β
.
(a)
(b)
D
C
A
B
A´
l
l
u
D´
l
l
B
lβ
1 − βl
2
θ
FIGURE 13.5 Visual apparent shape of a rapidly moving object. (a) A cub moving perpendicular to an
observer’s sight, (b) the observer view the cube r