Two-dimensional string theory is a fundamental toy model of string theory. It is one of the first examples of a holographic duality, and it has shed new light on non-perturbative effects in string theory. In this talk, I will use this toy model to explore time-dependent backgrounds in string theory. The background consists of an accelerating Liouville wall, which approaches a final velocity in the future. This velocity can be timelike or spacelike. We compute scattering amplitudes of two-dimensional string theory in the presence of this wall in the case where the wall is timelike, using the holographic dual matrix quantum mechanics. We also comment on the interpretation of the observables in the case where the wall becomes spacelike.