The classical problem of buckling of a thin rod subjected to axial compressive force is studied. Two cases are studied in detail: (i) case of short loading and (ii) case of long-lasting loading. Dynamic buckling of a thin rod subjected to a continuously acting longitudinal load at the initial stage of the movement is studied. If the applied static load significantly exceeds the critical Euler force, one of the higher buckling modes has the maximum rate of amplitude growth at the initial stage of the motion. This result is obtained in the framework of a linear statement of the problem, and an explanation of the paradoxical result by Lavrentyev and Ishlinsky is provided. A possibility of the appearance of buckling due to a suddenly applied longitudinal load which is smaller than the Euler critical force is found out. This buckling can occur only for the rod length from a certain range and is caused by the parametric resonance. In the linear approximation, the amplitude increases unboundedly while a small resistance leads to a significant increase in the amplitude. Introduction of nonlinear terms into consideration results in beats with energy exchange between longitudinal and transverse vibrations. Axial impact on the rod by an impactor is considered as a way of reproduction of the jump force in the experiment. The contact force is determined analytically and by means of finite element analysis. The results of these two approaches are confirmed by test results on the example of the impact time.