Is the radial velocity responsible only for changing distance between objects and the component perpendicular to it only for change in direction? If so why?
Please try to give a different explanation than saying that the radial velocity points in the line of sight can only increase the distance, and radial velocity is not affected by the component perpendicular to it, because I find this difficult to understand as velocity can be decomposed into two vectors that are not perpendicular, by using non-perpendicular coordinate axes.
What is the proof of the relation between tangential and angular velocity along any curve?
The formula for angular velocity (I am referring to proof of this relation) is given by $v \cos(\beta)/R$ (where $v$ is the speed and $R$ distance from the origin or observer). $v \sin(\beta)$ is the radial velocity. Is there a specific name for the $v \cos(\beta)$ component?
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