17.
A light signal is sent from the origin of a system K at t = 0 to the point x = 3 m, y = 5 m, z = 10
m. (a) At what time t is the signal received? (b) Find (x0 , y 0 , z 0 , t0 ) for the receipt of the signal in
a frame K 0 that is moving along the x axis of K at a speed of 0.8c. (c) From your results in (b)
verify that the light traveled with a speed c as measured in the K 0 frame.
1. The location of the receiver as measured in K:
x=3
y=5
z = 10
2. The distance traveled by the light signal as
√ measured in K:
d = 32 + 52 + 102 = 11.5758 m
3. The time t when the signal was received as measured in K:
11.5758 m
−8
t = dc = 2.99792×10
s
8 m/s = 3.86129 × 10
4. The time t0 when the signal was received as measured in K 0 :
γ=q
1
1−
t0 = γ(t −
vx
c2 )
=γ t−
0.8x
c
(0.8c)2
c2
=
√
1
1−0.64
= √136 = 5/3
= 5/3 3.86129 × 10−8 s −
100
0.8×3
2.99792×108
s = 5.10122 × 10−8 s
5. The coordinates where the signal was received as measured in K 0 :
x0 = γ(x − vt) = 5/3 3 m − (0.8 × 2.99792 × 10−8 )(3.86129 × 10−8 ) m = −10.4345 m
y0 = y = 5 m
z 0 = z = 10 m
6. The distance the light traveled as measured
in K 0 :
√
0
d = −10.43452 + 52 + 102 = 15.2931 m
7. The speed of light as measured in K 0 :
c = d0/t0 =
15.2931 m
5.10122×10−8 s
1
= 2.99793 × 108 m/s