In this paper, we consider the asymptotic behavior of solutions to the p -system with time-dependent damping on the half-line R + = 0 , + ∞ , v t − u x = 0 , u t + p v x = − α / 1 + t λ u with the Dirichlet boundary condition u x = 0 = 0 , in particular, including the constant and nonconstant coefficient damping. The initial data v 0 , u 0 x have the constant state v + , u + at x = + ∞ . We prove that the solutions time-asymptotically converge to v + , 0 as t tends to infinity. Compared with previous results about the p -system with constant coefficient damping, we obtain a general result when the initial perturbation belongs to H 3 R + × H 2 R + . Our proof is based on the time-weighted energy method.