hammer and feather

In this puzzle we consider a high precision re-enactment of the famous Galileo (thought)experiment and the claim is that hammer and feather dropped simultaneously from a height h will not hit the ground at exactly the same time.





In order to see this, we increase the height h and consider the full 3-body problem with feather (F), hammer (H) and moon (M) approximated as spheres (the famous spherical cow approximation).





Next we increase the distance between F and H and we increase the mass of the hammer H significantly. Therefore the moon will move towards H by a certain displacement a and thus the hammer has to travel the distance h - a until it collides with the moon M, while the feather F has to travel the increased distance sqrt( h^2 + a^2 ), which suggests that F will indeed collide with M slightly later than H.





But we have no full proof yet (notice that the feather is attracted by moon+hammer while the hammer is slightly less attracted by moon+feather and this could compensate for the different distances).

So in order to obtain full proof of our claim (without using too much math) we move the hammer H even further away from the feather F and we increase its mass until it exceeds the mass of the moon M significantly (perhaps it is easier to decrease the mass of the moon until it is more like a hammer).





But now we have transformed this thought experiment into a configuration where F and M are dropped on H, but from very different heights h and 2h. In other words, comparing with the original configuration, the assumption that the three bodies will collide at the same time is disproved by reductio ad absurdum.



I would like to make three more remarks:

1) The equivalence principle is an idealization (notice that in the general theory of relativity we consider test bodies to have infinitesimal mass and distances are small compared to the radius of curvature).

2) If we make the spheres small enough they will in general not collide at all (I leave it as an exercise for the reader to run a 3-body simulation and check this claim), except for symmetric initial configurations like the one in the last picture.

3) The contemporary opponents of Galileo could have made this reductio ad absurdum to counter his argument and discredit his physics. It is interesting to contemplate how science would have progressed in this case ...