Derive length contraction formula given a two system experiment.

1. The problem statement, all variables and given/known data

Show that the experiment depicted in Figure 2.11 and
discussed in the text leads directly to the derivation of
length contraction.

Figure 2.11:

2. Relevant equations

d=v*t

Requested result: L=L0[itex]\sqrt{1-\frac{v^2}{c^2}}[/itex]

3. The attempt at a solution

In K the distance the light pulse travels is 2t1v+L and the total time for the pulse to return to its origin is that distance over c.

In K’ the distance the light pulse travels is 2L0 and the total time for the pulse to return to its origin is that distance over c.

I can also say that t1=[itex]\frac{vt_1+L}{c}[/itex] then solve for t1 to substitute that into total time in K frame. At this point I’m not sure where to go or even if what I’ve done is useful. I am tempted to set the total times equal to one another and solve for L but I when I’ve tried this I can get it to look like it is supposed to although there are familiar pieces.

Any suggestions? Thank you!

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