# Is any elastic collision disallowed if the only force of interaction is gravity?

A gravity assist, or gravitational slingshot, is often described as being an elastic collision between a spacecraft and a planet. Basically you have two particles, each with a mass and an initial velocity; they approach, interact gravitationally, and move apart again with final velocities satisfying conservation of momentum and energy. My question is whether there are sets of initial and final velocities that are valid solutions for, say, billiard balls bouncing off each other, but can't happen with only gravity.

Said another way, as suggested in a comment, I am asking whether, in gravitational scattering, any scattering angle is possible. With repulsive forces, in the center of mass frame, each particle can be deflected by any angle (including bouncing straight back the way they came). With on gravity (an attractive $1/r^2$ force) only, is that still the case? If masses and radii of the interacting particles are given, which angles are not possible?

• To my understanding a slingshot is a three body problem. Feb 15, 2018 at 8:20
• -1 Not clear what you are asking. Your title seems to contradict your 1st & 2nd sentences. Your final sentence (and your title) requires explanation. Are you asking if it is possible for the collision between 2 billiard balls to be elastic? ... Also see Related questions in the column on the right, which might answer your question. Feb 16, 2018 at 14:08
• @sammygerbil it seems you are being intentionally obtuse. There is no contradiction, and I think the question is clear. I am asking whether, in gravitational scattering, any scattering angle is possible. With repulsive forces, in the center of mass frame, each particle can be deflected by any angle. With attractive $1/r^2$ force only, is that still the case? Feb 16, 2018 at 14:34
• The question (v1) might be clear in your mind, but it is not clear to someone who cannot see into your mind. There is no mention of scattering angles in your question. According to your 1st statement a slingshot is gravitational, elastic and 'allowed' (since you seem to accept that it happens). This contradicts your title, which suggests that such a collision is not allowed (cannot happen). ... I recommend that you edit your last comment into your question. Feb 16, 2018 at 14:52
• The Rutherford scattering formula works for any $1/r^{2}$ force. Feb 19, 2018 at 16:12