I am designing a 2d physics engine that uses Verlet integration for moving points (velocities mentioned below can be derived), constraints to represent moving line segments, and continuous collision detection to resolve collisions between moving points and static lines, and collisions between moving/static points and moving lines.

I already know how to calculate the Time of Impact for both types of collision events, and how to resolve moving point static line collisions. However, I can't figure out how to resolve moving/static point moving line collisions.

Here are the initial conditions in a point and moving line collision event. We have a line segment joined by two points, A and B. At this instant, point P is touching/colliding with line AB. These points have unit mass and some might have an initial velocity, unless point P is static. The line is massless and has no explicit rotational component, since points A and B could freely move around, extending or contracting the line as a result (which will be fixed later by the constraint solver). Collision is inelastic.

What are the final velocities of the points after collision?