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Given two bodies and their positions and velocities at a given time $t_0$, as well as their masses, and the knowledge that their relative velocity is high enough that they will not be gravitationally bound to each other as they pass by (so, hyperbolic orbits), I am trying to figure out a way to calculate their positions and velocities as a function of time.

The masses are such that we can't make the Keplerian approximation that one mass is much greater than the other. I know, though, that the gravitational two-body problem can be reduced to an equivalent one-body Keplerian-like problem.

Once the semimajor axis of the orbit is calculated ($a= -G(m_1 + m_2)/(2E)$, where $E$ is the total specific energy) then the magnitude of the relative velocity $v$ can be calculated as a function of the magnitude of the separation $r$ using the vis viva equation:

$$v^2 = G(m_1+m_2)\left(\frac{2}{r} - \frac{1}{a}\right).$$

This, however, only gives the magnitude of the relative velocity as a function of separation between the two objects. I am hoping to find the full position and velocity vectors of the objects as a function of time.

This is solved in many books and on many websites for the bound case, where the orbits are elliptical. I have not yet found anywhere that does it for the hyperbolic case though.

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closed as off-topic by John Rennie, Kyle Kanos, user36790, ACuriousMind, Gert Jan 27 '16 at 0:27

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  • $\begingroup$ Is it given that the bodies are in a mutual hyperbolic orbit? In a hyperbolic orbit, the two bodies wouldn't collide. You'd need an elliptical orbit (technically a circular orbit) for the two masses to collide. $\endgroup$ – barrycarter Jan 26 '16 at 19:37
  • $\begingroup$ @barrycarter, I disagree. An orbit of any shape that has a periapse that puts the two objects closer together than the sum of their radii will collide. $\endgroup$ – NeutronStar Jan 26 '16 at 22:13
  • $\begingroup$ Just to confirm, "unbound gravitationally" means they are affected only by each others' gravity? Or that their gravitational attraction towards each other is effectively zero? This may just be a simple conservation-of-momentum problem? $\endgroup$ – barrycarter Jan 27 '16 at 1:32
  • $\begingroup$ @barrycarter, affected only by each others' gravity. $\endgroup$ – NeutronStar Jan 27 '16 at 16:25
  • $\begingroup$ Quoting en.wikipedia.org/wiki/Free_fall (which also has the equations the OP needs) "The motion of two objects moving radially towards each other with no angular momentum can be considered a special case of an elliptical orbit" (emphasis added). However, the OP changed the question so it no longer involves collisions. $\endgroup$ – barrycarter Feb 3 '16 at 0:59