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?

  • $\begingroup$ To my understanding a slingshot is a three body problem. $\endgroup$ Feb 15, 2018 at 8:20
  • $\begingroup$ -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. $\endgroup$ Feb 16, 2018 at 14:08
  • $\begingroup$ @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? $\endgroup$
    – Ben51
    Feb 16, 2018 at 14:34
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    $\begingroup$ 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. $\endgroup$ Feb 16, 2018 at 14:52
  • $\begingroup$ The Rutherford scattering formula works for any $1/r^{2}$ force. $\endgroup$
    – Jon Custer
    Feb 19, 2018 at 16:12

1 Answer 1


Yes, not all elastic collisions can be attained through gravitational scattering. (Conversely, not all elastic collisions possible in gravitational scattering can be attained in elastic collisions of billiard balls.)

(Since we are talking about space craft and planets, lets assume just Newtonian gravity. Relativity would add further complications possibly changing the answer.)

The scattering angle (the angle between the initial and final velocity of the spacecraft) and in a gravitational encounter is essentially set the impact parameter (the closest distance that the spacecraft will approach the planet) since the planet has a finite size there is a minimum impact parameter, and therefore a maximum scattering angle that can be achieved. For example, it is impossible the recreate a headon collision of billiard balls, where both balls scatter back in the opposite direction with gravitational scattering.

Conversely, typical slingshot scenarios where momentum is transferred from a slowly moving planet to a fast moving spacecraft can normally not be achieved using colliding billiard balls.

  • $\begingroup$ Is it only collisions in the vicinity of head on collisions that are impossible, and is there a quantitative relationship for the excluded scattering angles given masses and radii of scattering particles? Assuming point masses (zero size), is any scattering angle then possible? I think any scattering angle is possible with billiard balls, so how can it be possible that there are scenarios possible with gravitational scattering that aren't with billiard balls? $\endgroup$
    – Ben51
    Feb 15, 2018 at 16:14

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