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If I perform a Galilean boost $$x' = x - vt \\ t'=t$$ between two frames $S$ and $S'$, observers in each frame would disagree on the velocity of a particle because $$ \frac{dx'}{dt'} = \frac{dx}{dt} - v. $$

Well Galilean transformations preserve the spatial intervals $\Delta x$ and time intervals $\Delta t$, so surely they should preserve velocity

$$ u = \frac{\Delta x}{\Delta t}? $$

There is obviously something going wrong here with my reasoning. I know the spatial interval $\Delta x$ is defined at constant time $t$, but if I was in the Galilean boosted frame $S'$ observing an object moving between two points $x_1$ and $x_2$ in $S$, I would observe that the interval $\Delta x$ between these two points would be a constant over time anyway so I could still conclude that the particle travelled a distance of $\Delta x$ in time $\Delta t$. What is going on?

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You are mixing two things. $dx$ and $\Delta x$. When you say "Gallilean Transformation preserves $\Delta x$ it means when measured by any observer", the relative distance between any two fixed points is same. But that is not the $dx$ in the definition of velocity. Its the change of position of a particle when we measure the distance between that particle and the origin of co-ordinates for that observer. Since the "origin" of the two co-ordinate system are not same(they have a relative velocity with respect to each other) that's why your measured velocity will be different.

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The confusion is in what $\Delta x$ really is.

Let's say two observers, one in S and the other in S', see a meter stick. They will both agree it is one meter long. So if $\Delta x$ is a set length then yes, this is invariant under the transformation.

Now let's say an observer in S is holding this meter stick. Over any time interval, this observer does not view the meter stick as moving. However, an observer in S' will definitely see the meter stick as moving. So in a given time $\Delta t$, the distance covered by the meter stick ($\Delta x$) will depend on which frame you are in.

Your discussion about velocity should focus on the latter scenario. This is in fact just relative velocity. Different frames will measure different velocities of an object depending on their velocity relative to the object (this is somewhat of a redundant or circular statement). Their measurements are related by their velocities relative to each other.

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