Why the velocity $v$ is taken as a value and the definition of velocity not applied on a relativistic equations?


The equations of time dilation and length contractions as we know are

\[L=L_0{\left(1-\frac{v^2}{c^2}\right)}\]

and

\[\delta {t} = \frac{\delta {t_0}}{\left(1-\frac{v^2}{c^2}\right)}\]

Here, ${\Delta {t_0}}$ is time interval between two events, and ${L_0}$ is length in rest frame. An observer moving with velocity $v$ measures the time interval $\Delta t$ and the length $L$.


The velocity $v$ that I am using has some value and for some reason I take it as a value and do not apply the definition

\[v=\frac{dx}{dt}=\lim_{\delta {t} \to 0}{\frac{\delta {x}}{\delta {t}}}=\lim_{t \to t_0}{\frac{x-x_0}{t-t_0}}\]

Velocity as I know is “change in displacement over change in time”. Let’s take length contraction equation, as the length contract there must be some variation for the end points of a rod (say) to move across a point, causing change in $v$. So, why I am not taking some relativistic equation of $v$ here in the equation. I feel, somehow there is some connection between length contraction, time dilation and this velocity. There must be something in this $v$ which is not factored in these equations.


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