Consider two frame S and S’, where S’ is moving with velocity (0.9c, 0, 0) with
respect to S. At time, t = 2.2 s , a muon decays at coordinates ( 100. 0.8, 1.0) meters
in S frame. Obtain time and position coordinates with respect to S’.
Answers
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Answer:
Proper Length
One thing all observers agree upon is relative speed. Even though clocks measure different elapsed times for the same process, they still agree that relative speed, which is distance divided by elapsed time, is the same. This implies that distance, too, depends on the observer’s relative motion. If two observers see different times, then they must also see different distances for relative speed to be the same to each of them.
The muon discussed in Example 1 in Simultaneity And Time Dilation illustrates this concept. To an observer on the Earth, the muon travels at 0.950 c for 7.05 μs from the time it is produced until it decays. Thus it travels a distance
L0 = vΔt = (0.950)(3.00 × 108 m/s)(7.05 × 10−6 s) = 2.01 km
relative to the Earth. In the muon’s frame of reference, its lifetime is only 2.20 μs. It has enough time to travel only
L0 = vΔt0 = (0.950)(3.00 × 108 m/s)(2.20 × 10−6 s) = 0.627 km.
The distance between the same two events (production and decay of a muon) depends on who measures it and how they are moving relative to it.
Explanation:
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