The author proves the laws of large numbers for a second class particle in one-dimensional totally asymmetric \(K\)-exclusion processes, under hydrodynamic Euler scaling. The main assumption is that initially the ambient particle configuration converges to a limiting profile. The macroscopic trajectories of the second class of particles are charateristics and shocks of the conversation law of the particle density. The proof uses a variational representation of a second class particle (variational coupling method), to overcome the problem of lack of information about invariant distributions. But the flux function of the conservation law may be neither differentiable nor strictly concave. To give a complete picture, the author also discusses the construction, uniqueness, and other properties of the weak solution that the particle density obeys.