Analysis of security governed by dynamics of power systems, which we refer to as dynamic security analysis, is a primary but challenging task because of its hybrid nature, that is, nonlinear continuous-time dynamics integrated with discrete switchings. In this paper, we formulate this analysis problem as checking the reachability of a mathematical model representing dynamic performances of a target power system. We then propose a computational approach to the analysis based on the so-called RRT (Rapidly-exploring Random Tree) algorithm. This algorithm searches for a feasible trajectory connecting an initial state possibly at a lower security level and a target set with a desirable higher security level. One advantage of the proposed approach is that it derives a concrete control strategy to guarantee the desirable security level if the feasible trajectory is found. The performance and effectiveness of the proposed approach are demonstrated by applying it to two running examples on power system studies: single machine-infinite system and two-area system for frequency control problem.