You are very entertaining, I must give it to you. I hope you'll understand why I have difficulty taking anything you say seriously.
Using your own expressions, calculate \(y^"_B-y^"_A\). In \(S''\): \(\begin{align} A'' &= \left(x_A''(t''), \ y_A''(t'') \right)\\ &= \left(-Vt'', \ -u\gamma'(t'' + \frac{Vx_A''}{c^2}) \right ) \end{align}\) \(\begin{align} B'' &= \left(x_B''(t''), \ y_B''(t'') \right) \\ &= \left(\frac{1}{\gamma'} - Vt'', \ -u\gamma'(t'' + \frac{Vx_B''}{c^2}) \right) \end{align}\)
Ok, I'll bite. \(y_B''(t'')-y_A''(t'')=-u\gamma'(t'' + \frac{Vx_B''}{c^2})+u\gamma'(t'' + \frac{Vx_A''}{c^2})=\frac{Vu\gamma'}{c^2}(x_A''-x_B'')\) Agree so far? I'd keep going and plug in the expressions for x, but I'm afraid that if I try to do more than one step at a time, you'll just shut down and go back to mocking me for vaguely wrong algebra.
You are missing part of the expression. I hope you would understand why I can't take your posts seriously.
Oh, sorry! When I posted that, I realized I'd made a typo and immediately edited it. It looks like you got around to responding in the ~10 seconds before I'd made the correction. It should be right now.
When you're done goofing around give me the times in the train frame. Give me the acceleration of the rod in the train frame. Give me the duration of travel of the rod. I'm gonna run circles around you, Tach, and I have to say, I'm gonna enjoy it. Please Register or Log in to view the hidden image!
Nah, it was a lot longer than that, I made three posts giving you hints as to the error. Now, can you explain to Pete his (and your) error?
I'll let your buddy explain it to you, I don't think that you'll accept the correction from me. The fact that you STILL can't (or won't) see the error speaks volumes about how you think.
NO I CAN'T YOU ELITIST TWIT. Your "hints" have been that something is missing from my answer, and that my mistake is laughable. If you call those hints, I pity anyone who actually tries to learn something from you. At the beginning, demanding that we find the error ourselves might have been pedagogical. It's long since crossed the line into just sadistic, even if there is an error to be found, which I'm convinced there's not. Specify a mistake or naff off.