This study investigates the computational efficacy of HHT-alpha and TR-BDF2 schemes in addressing dynamic frictionless unilateral contact challenges between an elastic structure and a rigid obstacle. The application of combinations of Nitsche's method with these schemes is explored for managing unilateral contact conditions. An examination of the convergence behavior involving the parameter α in the HHT-alpha method is conducted. Additionally, the mass redistribution method is tested and compared against the standard mass matrix. The numerical outcomes, based on 1D and 3D benchmarks, demonstrate the effectiveness of the employed combinations of schemes and methods.