The minimum-time problem frequently arises in the design of control for actuators, and is usually solved assuming to know the correct model of the system. In industrially important cases, however, important parts of the dynamics, like friction forces or disturbances by exosystems, are hardly known or even unknown. Against this background, this paper presents an iterative approach to achieve the minimum-time control for a nonlinear, single input second-order system with constrained input and partly unknown dynamics, effectively removing the requirement of perfect knowledge of the system and its parameters to achieve the minimum-time solution in application. First it is shown that, under reasonable assumptions about the unknown part of the dynamics, the optimal control exists for the presented class of systems and that it is a bang-bang control, with at most one switch. Then this property is exploited in the proposed algorithm, that finds the single optimal switching time by an iterative method, without involving any kind of identification of the unknown system parts.