Relativity, reference frames and Lorentz transformations

[Petar Mali]

Homework Statement

The system S' moves in relation to the system S with velocity \upsilon along the -x- axis. At the time when the beginnings of the coordinate system are in the same point, clocks in both system shows t=t'=0. Which coordinates will have a reference point during the motion in every of these systems, which has the property that in some next moment clocks in systems S, S' shows the same time t=t'. Determine the law of motion of motion of this point.

Homework Equations

Lorentz transformation

x'=\frac{x-\upsilon t}{\sqrt{1-\frac{{\upsilon}^2}{c^2}}}

y'=y

z'=z

t'=\frac{t-\frac{\upsilon}{c^2}x}{\sqrt{1-\frac{{\upsilon}^2}{c^2}}}

The Attempt at a Solution

I tried like this

t'=\frac{t-\frac{\upsilon}{c^2}x}{\sqrt{1-\frac{{\upsilon}^2}{c^2}}}

t=\frac{t'+\frac{\upsilon}{c^2}x'}{\sqrt{1-\frac{{\upsilon}^2}{c^2}}}

t=t'

t-\frac{\upsilon}{c^2}x=t'+\frac{\upsilon}{c^2}x'

t-t'=\frac{\upsilon}{c^2}(x+x')

0=\frac{\upsilon}{c^2}(x+x')

and get

x=\frac{\upsilon t}{\sqrt{1-\frac{\upsilon^2}{c^2}}+1}

Is this correct?

 

[Petar Mali]

Any answer?