The transient temperature distribution of two semi-infinite media at different temperatures that are suddenly brought into contact is investigated. The effect of thermal radiation in the hot medium is considered. Solutions are obtained by a hybrid technique, using an explicit fourth-order Runge-Kutta method for the time variable and the finite-difference method for the space variable. A variable grid spacing system utilizing hyperbolic sine functions is incorporated to extend the computational boundary as well as to minimize the computation time and the number of nodal points. The technique is shown to be relatively simple and accurate, and illustrative solutions are presented for the transient contact temperature after sudden contact of molten uranium with molten sodium.