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?