  (Short abstract). In Galilean physics, the universality of free fall implies
an inertial frame, which in turns implies that the mass m of the falling body
is omitted. Otherwise, an additional acceleration proportional to m/M would
rise either for an observer at the centre of mass of the system, or for an
observer at a fixed distance from the centre of mass of M. These elementary,
but overlooked, considerations fully respect the equivalence principle and the
identity of an inertial or a gravitational pull for an observer in the Einstein
cabin. They value as fore-runners of the self-force and gauge dependency in
general relativity. The approximate nature of Galilei's law of free fall is
explored herein. When stepping into general relativity, we report how the
geodesic free fall into a black hole was the subject of an intense debate again
centred on coordinate choice. Later, we describe how the infalling mass and the
emitted gravitational radiation affect the free fall motion of a body. The
general relativistic self-force might be dealt with to perfectly fit into a
geodesic conception of motion. Then, embracing quantum mechanics, real black
holes are not classical static objects any longer. Free fall has to handle the
Hawking radiation, and leads us to new perspectives on the varying mass of the
evaporating black hole and on the varying energy of the falling mass. Along the
paper, we also estimate our findings for ordinary masses being dropped from a
Galilean or Einsteinian Pisa-like tower with respect to the current state of
the art drawn from precise measurements in ground and space laboratories, and
to the constraints posed by quantum measurements. The appendix describes how
education physics and high impact factor journals discuss the free fall.
Finally, case studies conducted on undergraduate students and teachers are
reviewed.
