- #1

etotheipi

Wikipedia gives, "The

Suppose the coordinate system being used in the rest frame of ##A## is has its origin slightly displaced from ##A## itself, such that the vector from the origin to A is ##\vec{R}##. Suppose also that this coordinate system is translating and rotating such that ##\vec{R}## is a function of time.

We can write the relative position of B wrt. A as ##\vec{r_{BA}} = \vec{r_{B}} - \vec{r_{A}}##, such that the position of ##B## as measured by the coordinate system in the rest frame of ##A## is ##\vec{r_{BA}} + \vec{R}##. Finally, since ##\vec{R}## is dependent on time, differentiating this position will give the velocity of ##B## measured by the coordinate system used in A's frame as ##\vec{v_{BA}} + \dot{\vec{R}}##.

This appears to contradict with the statement from wikipedia. It seems as though if we set everything up in a certain manner, the velocity of B as measured by a chosen coordinate system not centred on A in A's rest frame is not necessarily ##\vec{v_{BA}} = \vec{v_{B}} - \vec{v_{A}}##, since there is a possibility of an extra term.

Everything works out fine if, when dealing with relative positions/velocities/accelerations, we centre the coordinate systems on all of the bodies in question. However, if the coordinate systems are not centred as such (which is permissible, since we can set up our coordinate system as we please in any given rest frame), these formulae start to no longer work. A more easily visualisable example would be the relative position of B in A's rest frame, which evidently can take on infinitely many values depending on where we choose the origin of A's coordinate system!

I was wondering if anyone could tell me what the problem with all of the above junk is, as I feel I am misinterpreting some key ideas! Thank you.

*relative velocity*##{\displaystyle {\vec {v}}_{B\mid A}}## is the velocity of an object or observer**B**in the rest frame of another object or observer**A**."Suppose the coordinate system being used in the rest frame of ##A## is has its origin slightly displaced from ##A## itself, such that the vector from the origin to A is ##\vec{R}##. Suppose also that this coordinate system is translating and rotating such that ##\vec{R}## is a function of time.

We can write the relative position of B wrt. A as ##\vec{r_{BA}} = \vec{r_{B}} - \vec{r_{A}}##, such that the position of ##B## as measured by the coordinate system in the rest frame of ##A## is ##\vec{r_{BA}} + \vec{R}##. Finally, since ##\vec{R}## is dependent on time, differentiating this position will give the velocity of ##B## measured by the coordinate system used in A's frame as ##\vec{v_{BA}} + \dot{\vec{R}}##.

This appears to contradict with the statement from wikipedia. It seems as though if we set everything up in a certain manner, the velocity of B as measured by a chosen coordinate system not centred on A in A's rest frame is not necessarily ##\vec{v_{BA}} = \vec{v_{B}} - \vec{v_{A}}##, since there is a possibility of an extra term.

Everything works out fine if, when dealing with relative positions/velocities/accelerations, we centre the coordinate systems on all of the bodies in question. However, if the coordinate systems are not centred as such (which is permissible, since we can set up our coordinate system as we please in any given rest frame), these formulae start to no longer work. A more easily visualisable example would be the relative position of B in A's rest frame, which evidently can take on infinitely many values depending on where we choose the origin of A's coordinate system!

I was wondering if anyone could tell me what the problem with all of the above junk is, as I feel I am misinterpreting some key ideas! Thank you.

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