Why Does Griffiths Use Two Directions of Light in His Length Contraction Proof?

[maxverywell]

In Griffiths book (Intro to Electrodynamics, page 489) he uses one simple gedanken experiment with train, lamp and mirror, to prove the length contraction $$\Delta x'=\gamma \Delta x$$. My question is why he uses two directions of light and not just only one?
For example, when we use, for observer in the train, $$\Delta t'=\frac{\Delta x'}{c}$$ instead of $$\Delta t'=2\frac{\Delta x'}{c}$$ and for observer in the ground $$\Delta t=\frac{\Delta x}{c-u}$$ instead of $$\Delta t=\frac{\Delta x}{c-u}+\frac{\Delta x}{c+u}$$, it gives us incorrect result. Why is this happening?

 

[maxverywell]

No one can explain this?

Let's suppose that the light is moving in direction in which the train is moving. So, for observer in the train:

$$\Delta t'=\frac{\Delta x'}{c}$$ (1)

and for observer in the ground:

$$\Delta t=\frac{\Delta x}{c-u}$$ (2)

Applying the time dilation formula $$\Delta t'=\frac{\Delta t}{\gamma}$$ to (1) and (2) we find that:

$$\Delta x=\frac{\gamma(c-u)\Delta x'}{c}$$

 

[Doc Al]

Applying the time dilation formula $$\Delta t'=\frac{\Delta t}{\gamma}$$ to (1) and (2) we find that:

$$\Delta x=\frac{\gamma(c-u)\Delta x'}{c}$$

Realize that the time dilation formula applies to time measurements recorded on a single moving clock. You can't apply it to the one-way travel time, since multiple clocks on the moving train are involved--those clocks are not in synch (according to the track frame). If you use the round trip time, which is measured on a single clock, then you can apply the time dilation formula.