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Let's say an aircraft is traveling with speed $|\mathbf v|$ with respect to an inertial frame. The aircraft is able to execute turns by producing an acceleration $\mathbf a$ that is normal to $\mathbf v$ at all times.

For simplicity we will assume the aircraft is flying at level altitude above a flat surface (the $x-y$ plane) and that all motion is in the plane of constant altitude ($z$ constant).

If we denote the aircraft's heading (which we will define as the angle $\mathbf v$ makes with the x-axis) as $\theta$, is it correct to say that the heading rate magnitude $|\dot\theta|$ is: $$ |\dot\theta| = \frac{|\mathbf a|}{|\mathbf v|} $$ This definition of heading assumes that the wind axis and body axis are co-incidental, i.e. zero angle of attack and zero sideslip angle. This an approximation, but not a terrible one for a simple analysis.

Justification:

If the aircraft is executing a turn by applying normal acceleration $\mathbf a$, the turn radius $R$ is given by: $$ R = \frac{|\mathbf v|^2}{|\mathbf a|}. $$ The magnitude of the attitude rate $\dot\theta$ is the same as the angular velocity of the aircraft about the center of curvature of the flight path, therefore: $$ R |\dot\theta| = |\mathbf v|, $$ which, after substituting, yields $|\dot\theta| = |\mathbf a|/|\mathbf v|$.

Is this train of reasoning correct? If not, what assumptions are incorrect?

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  • $\begingroup$ Everything seems right, I don’t think there are any problems with your reasoning, and the result looks correct. Is there any reason why you would think it’s not right? $\endgroup$ – Hugo V Oct 10 '18 at 0:10
  • $\begingroup$ I think there is some terminology confusion en.m.wikipedia.org/wiki/Heading_(navigation) heading is where the plane points. $\endgroup$ – Emil Oct 10 '18 at 1:57
  • $\begingroup$ @Emil-- this is the intended use. For purposes of this question, we will assume the velocity vector is co-incident with the aircraft nose direction. I'll edit the question to clarify. $\endgroup$ – Robert L. Oct 10 '18 at 13:11

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