Converging forces

The corpuscle P is stationary at the point x_P=0, while the bodies A
and B approach at the same velocity from the left and right, arriving
simultaneously on the corpuscle P, as can be seen in the animation
https://www.geogebra.org/classic/fyymjr9s

During the collision, the crushing of the spherical corpuscle P always
occurs (regardless of the mass of the two bodies A and B) because the
forces F1 and F2 are convergent.

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

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

Luigi Fortunati

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