We propose a new explicit energy-momentum conserving scheme for the time integration of Hamiltonian systems. The scheme, which is formally second-order accurate, is based on two key ideas: the integration during the time-steps of forces between free-flight particles and the use of momentum jumps at the discrete time nodes leading to a two-step formulation for the acceleration. Moreover, the scheme can be used in the framework of slow-fast strategies. The scheme is validated against classical benchmarks and on a nonlinear wave propagation problem.