The problem of the vertical entry of a rigid horizontal cylinder into an incompressible inviscid fluid initially at rest is addressed. The contributions of previous researchers are presented and discussed in the light of various assumptions introduced and the validity of the results obtained. Based on this review, realistic simplifying assumptions are introduced and the problem formulated. The method of matched asymptotic expansions is used to solve the resulting boundary-value problem. A new formula for the impact force is obtained, which differs from the classical von Kármán’s formula by a corrective term. The results obtained are compared with those of experimental observations and numerical calculations. The method may be extended to different geometries and nonvertical velocities to provide an estimate of the impact forces on the partially emerged pontoons of damaged semi-submersibles.