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RE: "...a second observer in reference frame S moving..."

The moving reference frame should be designated as S' to distinguish it from the stationary reference frame.

t = t0/(1 - v^2/c^2)^1/2

where:

t0 = time in the stationary frame of reference = 10^-4 s = 0.0001 s

t = time in moving frame of reference = to be determined

c = speed of light in a vacuum = 3.00E+08 m/s

v = velocity of moving frame of reference = 0.8c = 2.40E+08 m/s

t = 0.000166667 s

Ratio of t/t0 = 1.666666667

t = t₀√(1-(v/c)²).

Put in the numbers:

t = (1 × 10ˉ⁴ s)√(1-(0.8 c/c)²) = 6 × 10ˉ⁵ s.