Vx (1.49)Similarly, Vy (1.50)Vz (1.51)If V (^) xc,that is, if light is emitted in the moving frame S in its direction of motion
relative to S,an observer in frame Swill measure the speed
Vxc
Thus observers in the car and on the road both find the same value for the speed of
light, as they must.
Example 1.11
Spacecraft Alpha is moving at 0.90cwith respect to the earth. If spacecraft Beta is to pass Alpha
at a relative speed of 0.50cin the same direction, what speed must Beta have with respect to
the earth?
Solution
According to the Galilean transformation, Beta would need a speed relative to the earth of
0.90c0.50c1.40c, which we know is impossible. According to Eq. (1.49), however, with
V (^) x0.50cand 0.90c, the required speed is only
Vx0.97c
which is less than c. It is necessary to go less than 10 percent faster than a spacecraft traveling
at 0.90cin order to pass it at a relative speed of 0.50c.
Simultaneity
The relative character of time as well as space has many implications. Notably, events
that seem to take place simultaneously to one observer may not be simultaneous to
another observer in relative motion, and vice versa.
Let us examine two events—the setting off of a pair of flares, say—that occur at the
same time t 0 to somebody on the earth but at the different locations x 1 and x 2. What
does the pilot of a spacecraft in flight see? To her, the flare at x 1 and t 0 appears at the
time
0.50c0.90c
1 (0.90c
c
)(
2
0.50c)
V (^) x
1
c
V
2
^ x
c(c)
c
c
(^1)
c^2
c
V (^) x
1
V
c^2
(^) x
V (^) z 1 ^2 c^2
1
c
V
2
(^) x
V (^) y 1 ^2 c^2
1
c
V
2
(^) x
V (^) x
1
c
V
2
(^) x
Relativistic velocity
transformation
44 Appendix to Chapter 1
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