Difference between velocity vectors, relative speed

Question

Consider two particles moving in the same direction instantaneously but one of them follows a circular path and one follows a straight path.

If I adopt a rotating frame of reference in which the particle $A$ is stationary, math tells me that the relative speed between $A$ and $B$ is $V_{rel}=V_B-V_A-\frac{V_B h}{R}$, where $h$ is the distance between the particles and $R$ is the radius of the circular path.

I understand that this is right, I can infer this from the formulas of rotating reference systems. But can someone please explain why it doesn't work to simply take $V_A-V_B$. The relative velocity should be the difference between two velocity vectors, I think that's very intuitive. But in this case it isn't? Why? When I see a physical situation like this (e.g. they could be airplanes, one doing a loop), how can I know if I can simply take the difference between the velocity vectors to find the relative velocity or not?

Answer 1

The relative velocity between two points depends on the frame.

In general, $A$ will see $B$ going at a different velocity than $B$ sees $A$.

Imagine that, in your example, $\mathbf{V_B}=\mathbf{V_A}$ (zero relative velocity) in the inertial frame. In the rotating frame, $B$ will lag a bit behind, because it is further than $A$ from the rotation center. This shows that the relative velocity is not zero in that frame.

Another example: Your friend is immobile a few meters away, while you are rotating around yourself. He will see you immobile, but you will see him turn around you. The relative velocities differ.

You should solve your problem in a given reference frame of your choice, and use the relative velocity in that frame.