When the linear momentum of a system of particles is zero, the velocities of the individual particles may not be zero. The kinetic energy of the system of particles may be non-zero. Thus 1 does not imply 2. When the kinetic energy of the system of particles is zero, then the kinetic energy and hence the velocity of each particle is zero. Therefore the linear momentum of a system of particles is zero. Thus 2 imply 1.