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S Inertial Frames

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The explanations are written in the attached pdf file.

The two web pages I use as references for formula are
However you may not necessarily need them as you likely have all the formula in your textbook(s) or lecture notes.

Here is the plain TEX source

We shall use the notations
beta = {vover c}
for the dimensionless speed, and
gamma = {1oversqrt{1-beta^2}}
for the Lorentz factor.

bf Q.1rm

Lorentz transformation for the time is
t' = gammal( t - {vxover c^2} r).
As we are requested to have $t'_A = t'_B$ for the two events, from equation (1.1) this requirement

gammal( t_A - {vx_Aover c^2} r) = gammal( t_B - {vx_Bover c^2} r).
From equation (1.2) we find the requested difference in times of the events in frame S:
t_A-t_B = {vover c^2} l(x_A-x_Br).

bf Q.2rm

Suppose a particle at $x_1'=0$ is created at $t_1'=0$,
stays at $x_2'=0$ and decays at $t_2' = tau = 1~mu s$ (microsecond).
From the Lorentz transformation back from frame S' to frame S (the same formula as usually ...

Solution Summary

PDF is attached to answer these 7 questions on inertial frames.