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Introductory Physics Homework Help
Solution to the Two-Body Problem: Cross-Product and Dot-Product Integration
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[QUOTE="TimeRip496, post: 6003809, member: 536130"] [h2]Homework Statement [/h2] Two-body problem given as $$\ddot{\textbf{r}}+\frac{GM}{r^2}\frac{\textbf{r}}{r}=0$$ $$\textbf{h}=\textbf{r}\times\dot{\textbf{r}}$$ where the moment of the momentum vector mh [h2]Homework Equations[/h2] The vector solution to the above equation may be obtained by first taking the cross-product with the constant h and integrating once with respect to time. This yields $$\dot{\textbf{r}}\times\textbf{h}=GM(\frac{\textbf{r}}{r}+\textbf{e})$$ The final solution to the equation is then obtained by taking the dot product of above equation with r, is $$r=\frac{h^2/(GM)}{1+ecos\theta}$$ [h2]The Attempt at a Solution[/h2] I have no idea what the author is doing. How does cross product with the momentum and then dot product with r solve the equation? I try following his step but I get a different integration result $$\dot{\textbf{r}}\times\textbf{h}=GM(\frac{\textbf{r}}{r^3}t\times\textbf{h}+\textbf{e})$$ I have no idea how his integration w.r.t. time for the GM/r[SUP]2[/SUP] reduces it to GM/r and how to cross product GM/r[SUP]2[/SUP] with h since they are unknown variables? [/QUOTE]
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Introductory Physics Homework Help
Solution to the Two-Body Problem: Cross-Product and Dot-Product Integration
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