Environmental Engineering Reference
In-Depth Information
(c) And for the return leg we introduce a third event corresponding to when
the ball returns. In S it has co-ordinates (x
1
,t
1
+
2
L/u). As in part (b) we use
Eq. (6.28b). Notice that x
3
−
x
2
=−
x this time whilst t
3
−
t
2
=
t and so
t
in
=
vx/c
2
),
γ(t
+
γL
u
uv/c
2
).
=
(
1
+
(6.32)
Notice also that
the total
time for the journey is just as we would expect
i.e t
tot
=
from time dilation,
γ(
2
L/u) as it should be since the point of
departure and point of return are one and the same place. This result confirms our
earlier claim that there was nothing special about a light-clock.
6.3 VELOCITY TRANSFORMATIONS
6.3.1 Addition of Velocities
We can use the Lorentz transformations to figure out how the rules for adding
velocities must change in Special Relativity. Consider an object moving with a
velocity
v
in
S
. Let us determine its velocity
v
in
S
. The situation is illustrated
in Figure 6.9. Notice that to avoid confusion the relative speed between the two
frames is now
u
and we have simplified to the case of motion in two dimensions
(in the
x
−
y
plane). It is straightforward to generalize to three-dimensions. Recall
that according to Galilean relativity
v
x
v
x
+
v
y
.
Neither of these
=
u
and
v
y
=
holds true in Special Relativity, as we shall now see.
S
′
y
S
y
′
u
u′
υ′
y
υ′
x
O
x
O
′
x
′
Figure 6.9
Relative velocities.
To determine the
x
-component of the velocity in
S
we make use of the Lorentz
transformation formulae for
x
and for
t
:
γ(
d
x
+
u
d
t
)
(
d
x
/
d
t
)
d
x
d
t
=
+
u
v
x
=
u
d
x
/c
2
)
=
γ(
d
t
+
1
+
u(
d
x
/
d
t
)c
2
v
x
+
u
=
uv
x
/c
2
.
(6.33)
1
+
