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

Situation of relative motion

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?

  • $\begingroup$ What is $V_B$ exactly? $\endgroup$ – David Z May 21 '13 at 19:33
  • $\begingroup$ @DavidZaslavsky $V_B$ and $V_A$ are velocities. Imagine for example two airplanes. One flying in a loop, how would that pilot perceive his relative velocity to the pilot in the other airplane? (This is the question which I worked on when I got stuck with this conceptual problem) $\endgroup$ – user714 May 21 '13 at 19:35
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    $\begingroup$ Yes, it's clear that they are velocities, but how exactly is $V_B$ defined? Velocity of what (presumably object B) with respect to what? $\endgroup$ – David Z May 21 '13 at 19:37
  • $\begingroup$ @DavidZaslavsky Sorry, I didn't understand the question. Velocities of objects, for example airplanes, with respect to an inertial frame. $\endgroup$ – user714 May 21 '13 at 19:49

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.


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