This paper has a didactic aim. Indeed, starting from the well-known principle of free fall discovered by Galileo Galilei, we want to make some reflections that we consider useful for secondary school students. Being the acceleration invariant with respect to Galilei's transformations, it is useful to underline that the free fall in a gravitational field is independent of the mass of the body that falls only if the attractive mass is an inertial frame. In general, the corrections to the motion of a test particle falling in a gravitational field due to its mass can be easily calculated avoiding misinterpretations of Galileo's principle. Moreover, it may be useful to observe that, in the case of periodic motions, these corrections increase with time and would lead, over the centuries, to not negligible effects.