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I was wondering where does the formula for redshift $$z=\frac vc$$ come from? Can it be derived from the definition of redshift $$z={\lambda_{observed} - \lambda_{emitted}\over \lambda_{emitted}}?$$

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The non-relativistic formula can be derived analogously to the derivation of the Doppler effect for sound. Namely, draw waves as consecutive parallel lines moving at velocity $c$ whose distance is $\lambda$ (if the source is not moving) and let the source move away at $v$. You will find that the distance has increased and became the observed wavelength, which is $\lambda$ - $vt$ = $\lambda + v\lambda/c$ = $\lambda(1+\frac{v}{c})$.
Note that this formula is not relativistic and that there is cosmological (as pointed out by @Alchimista in the comment below) as well as gravitational redshift, so the formula above only describes only one of the effects that cause redshift or blueshift.

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