# Confusion about sign conventions used in the Doppler formula

$$f = \left( \frac{c \pm v_r}{c \pm v_s} \right) f_0$$

c is the propagation speed of waves in the medium.

$$v_r$$ is the speed of the receiver relative to the medium, added to c if the receiver is moving towards the source.

$$v_s$$ is the speed of the source relative to the medium, added to c if the source is moving away from the receiver.

I am having great difficulty trying to derive the following formula from the Doppler formula, for the relative velocity of an object when a source wave is reflected off it as in police radar: $$\Delta v = \frac{\Delta f}{f_0 } \frac{c}{2}$$ . Either this is an approximation or I am misunderstanding something very basic about the Doppler formula. My main confusion is about the signs assigned to the velocities. I find the statement "$$v_r$$: Speed of the receiver relative to the medium, added to c if the receiver is moving towards the source." ambiguous.

For example if the source is on the left of the receiver and is moving to the right at 20 m/s and the receiver is moving to the right at 30 m/s, then technically the source is moving away from the receiver, because the gap between them is getting larger. Should I base the sign on whether or not the gap between them is increasing or not? If anyone has a clean derivation of the relative velocity calculated from the reflected signal frequency that would help clarify my understanding too.

$$f = \left( \frac{c + v_r}{c + v_s} \right) f_0$$, where the positive direction for $$v_r$$ and $$v_s$$ is from receiver towards source.
Alternatively, the general 3 dimensional vector form without a specific coordinate system is $$f = \left( \frac{c+\vec{v}_{r}\cdot \vec{e}_{rs}}{c+\vec{v}_{s}\cdot \vec{e}_{rs}} \right) f_0$$