Explicit Asymptotic Velocity of the Boundary between Particles and Antiparticles

On the real line initially there are infinite number of particles on the positive half line, each having one of K-negative velocities v1(+),…,vK(+). Similarly, there are infinite number of antiparticles on the negative half line, each having one of L-positive velocities v1(-),…,vL(-). Each particle moves with constant speed, initially prescribed to it. When particle and antiparticle collide, they both disappear. It is the only interaction in the system. We find explicitly the large time asymptotics of β(t)—the coordinate of the last collision before t between particle and antiparticle.