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Main Question or Discussion Point
OK, I'm sure I must be doing something wrong here, because I've run into a seeming contradiction. Please refer to the image I've attached below: an ellipse with the center and a focus marked. If there is a massive body at point P, then an object can orbit in the ellipse shown. If I know the velocity of the object at point B, and want to compute the velocity at point A, I can use conservation of angular momentum,
[tex]mv_{A}r_{PA} = mv_{B}r_{PB}[/tex]
[tex]v_{A} = v_{B} \dfrac{r_{PB}}{r_{PA}}[/tex]
Here, [tex]r_{PA}[/tex] and [tex]r_{PB}[/tex] are the distances from the focus to points A and B respectively.
However, if I switch coordinate systems and measure angular momentum with respect to the center of the ellipse, then I can write,
[tex]mv_{A}a = mv_{B}a[/tex]
[tex]v_{A} = v_{B}[/tex]
Where a is the semimajor axis of the ellipse.
OK, clearly I can't do this, since switching coordinate systems gives me a different result. Can someone explain why the latter is an invalid means to solve this problem? Thanks.
[tex]mv_{A}r_{PA} = mv_{B}r_{PB}[/tex]
[tex]v_{A} = v_{B} \dfrac{r_{PB}}{r_{PA}}[/tex]
Here, [tex]r_{PA}[/tex] and [tex]r_{PB}[/tex] are the distances from the focus to points A and B respectively.
However, if I switch coordinate systems and measure angular momentum with respect to the center of the ellipse, then I can write,
[tex]mv_{A}a = mv_{B}a[/tex]
[tex]v_{A} = v_{B}[/tex]
Where a is the semimajor axis of the ellipse.
OK, clearly I can't do this, since switching coordinate systems gives me a different result. Can someone explain why the latter is an invalid means to solve this problem? Thanks.
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