I will present a new solution method for the 5d Einstein-brane equations and solve for both the background and cosmological perturbations of the Randall-Sundrum model. The cosmological solutions describe two separating or colliding branes, and the key to our method is an expansion in the rapidity of this collision. The 5d solution reveals interesting effects beyond the scope of 4d effective theory; in particular the mixing of 4d perturbation modes as the branes separate, from the close-brane regime described by Kaluza-Klein theory to the well-separated limit in which gravity is confined to the positive-tension brane.