Physics Asked by Darkenin on December 3, 2020
It seems to be taken trivially, and it is indeed intuitive, that the relative velocity $v$ of some frame $O$ with respect to $O’$ is the same in magnitude to the relative velocity of $O’$ with respect to $O$. Is there any way to prove this? I’ve seen some tricks with rotating the relative motion axis by 180 degrees, but i’m not sure I get that.
It's a symmetry argument.
There is no absolute difference between $O$ and $O'$.
But if the speed of $O$ as seen from $O'$ is not the same as the speed of $O'$ seen from $O$, $|v'| ne |v|$, then it must be larger or smaller: $|v'|>|v|$ or $|v|>|v'|$. That would give an absolute difference between the two frames, so it can't happen and the speeds, $|v|$ and $|v'|$, must be the same.
When you extend this to consider velocities the minus sign is allowable as that's just about the choice of direction of the $x$ axis
Correct answer by RogerJBarlow on December 3, 2020
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