The initial stage of the plane unsteady flow caused by the impulsive vertical motion of a wedge initially floating on an otherwise flat free surface is investigated with the help of a combination of numerical and asymptotic methods. The liquid is assumed ideal and incompressible and its flow potential. Compressible effects give a negligible contribution to the flow close to the entering body at the stage considered in the present paper. The vertical velocity of the body is constant after the impulsive start. The flow domain is divided into an outer region, where the first-order solution is given by the pressure-impulse theory, and inner regions close to the intersection points between the free surface and the moving body. The relative displacement of the body plays the role of a small parameter. The inner solution is matched with the outer one. The outer solution is given in quadratures but the inner solution, which is shown to be nonlinear and self-similar, can be found only numerically. With the aim of deriving the inner solution, the inner region is divided into three parts. In the far-field zone the solution is given in terms of its asymptotic behavior while, in the jet region, attached to the wedge, the flow is described by a second-order shallow-water approximation. In the intermediate region a boundary-element method is used, which is suitably coupled with the solutions in both the jet and the far-field regions through an iterative pseudo-time stepping procedure. The procedure is dependent on the deadrise angle of the wedge. If the angle is equal or smaller than π/4, eigensolutions appear in the far-field asymptotics and their amplitudes are recovered together with the solution. The approach is applied to different values of the wedge deadrise angle. The obtained results can be used to improve the prediction of the hydrodynamic loads acting on floating bodies, the velocity of which changes rapidly.