Converging forces

[[Mod. note -- I'm sorry, I mistakenly posted an earlier draft of this
article.  The mistake was mine, not the author's.  I believe this is the
correct version.
-- jt]]

The small body P is at rest at the point x_P=0, while the bodies A and
B approach at the same speed from the left and right, arriving at P at
the same time, as shown in the animation
https://www.geogebra.org/classic/ptkfwqh5

During the collision, the crushing of the tiny body P is always there
(regardless of the mass of the two bodies A and B) because the forces
F1 and F2 are convergent.

Instead, the acceleration of P may or may not be there because, after
the collision, P can start moving to the right or to the left (changing
its speed from zero to +v or -v) or it can remain at its initial place
x_P=0, leaving its zero speed unchanged.

What conditions must be met for P to accelerate to one side or the
other and what conditions for it to remain at rest in its place?

If the small body P is not there and the points A and B collide
directly with each other, do the forces F1 and F2 stop being
convergent?

Luigi Fortunati

[[Mod. note -- We are given that A and B have the same speed.  So, if
(and only if) A and B have the same mass, then the system is left-right
symmetric, so P will remain stationary.
-- jt]]