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The flight time from the USA to China may be different to the return trip, but if so I'd gues this is due to the jet stream rather than the rotation of the Earth. Anyhow, your intuition about the train is correct. The two passengers A and B see the A to B and B to A speeds of the ball to be the same, while an external observer sees them to differ by twice ...

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how Galilean transformations which are wrong (are approximately correct) give the correct answer for k? The Lorentz prediction and the Galilean prediction must agree in the limit that $v \to 0$ (or in the limit that $c \to \infty$). This is because $v=0$ corresponds to no transformation at all, so they had better both agree there. So if you take ...

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In the $S'$ frame, your variables are $x' = x - t\cdot u \cos\theta$ and $y' = y - t\cdot u \sin\theta$. If you do the change of variable, you get that the motion now is described by $$x' = 0$$ $$y' = -\frac{g}{2}t^2$$ So in your new frame of reference you have vertical free fall from rest. This is not very helpful in finding out when or where does the ...

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The metric doesn't transform at all. The metric stays fixed even in an accelerating coordinate system, with any arbitrary acceleration. This is what it means to say that time is separate from space in Newtonian mechanics. The structure of Newtonian mechanics, however, includes more than the metric, it includes the acceleration law, that the forces are ...

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