The explanations are written in the attached pdf file.

The two web pages I use as references for formula are
http://en.wikipedia.org/wiki/Lorentz_transformation
and
http://www.wbabin.net/hamdan/hamdan4.htm
However you may not necessarily need them as you likely have all the formula in your textbook(s) or lecture notes.

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Here is the plain TEX source

We shall use the notations
$$
beta = {vover c}
eqno(0.1)
$$
for the dimensionless speed, and
$$
gamma = {1oversqrt{1-beta^2}}
eqno(0.1)
$$
for the Lorentz factor.

bf Q.1rm

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

means
$$
gammal( t_A - {vx_Aover c^2} r) = gammal( t_B - {vx_Bover c^2} r).
eqno(1.2)
$$
From equation (1.2) we find the requested difference in times of the events in frame S:
$$
t_A-t_B = {vover c^2} l(x_A-x_Br).
eqno(1.3)
$$

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

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