Some chaotic properties of a classical particle interacting with a time-modulated barrier are studied. The dynamics of this problem is obtained by use of a two-dimensional nonlinear area-preserving map. The chaotic low energy region is characterized in terms of Lyapunov exponents. The time that the particle stays trapped in the well is such that the distributions of successive reflections, and of the corresponding successive reflection times, obey power laws with the same exponent. Using time series analysis, we show that the chaotic sea exhibits an interesting scaling property over a large range of control parameters. Our results indicate that the particle experiences unlimited energy growth when the barrier behaves randomly.