Free Fallin'

In Galileo Galilei’s time (the 16th century), and in his place (Italy), in addition to “generally despis[ing]” mathematics, most people believed in Aristotle’s view of motion [1, pg 78].

Aristotle had the idea that if you drop something, it will fall. Coincidentally, I agree. He follows this idea by saying that the falling object moves at a constant speed, and the heavier the object is, the faster that constant speed will be.

Galileo, it seems, loved contradicting people. According to his pupil and biographer, he was “rewarded with their hatred, as they could not bear that a young student... rejected and discarded in new ways and with such evidence the doctrines they had swallowed...” [1, pg 77].

There’s a lesson here. You can be right. Totally and unequivocally correct. But if you choose to be obnoxious about how very smart you are, people will not enjoy your company. People might not enjoy it anyways, but as my old roommate would say, “you win some, you lose most”.

And so we come to the classic story of Galileo throwing things out of a tower in Pisa. As with practically every juicy and dramatic story in history, there is a debate about whether this was legend or truth, but since his pupil Viviani wrote it, I’ll continue to perpetuate the apocryphal story. As it goes, Galileo gathered lecturers, philosophers, and other students to watch as he repeatedly dropped objects of different size but similar element (ie, rocks compared to rocks) off of a high tower [1, pg 84].

Something tells me that Galileo wanted to revel in the humiliation of some wig wearing intellectuals. So I think you’ll agree with me in classifying this as a literal throw-down. If microphones had been around at the time, I have no doubt that this dramatic, disputatious youth would’ve conducted a several-story mic-drop.

Figure 1: Aristotle’s view of free fall compared to Galileo’s view. Aristotle believed objects fall with a constant velocity which increases as the object increases in mass. Galileo believed that all objects experience the same acceleration regardless of mass.

This experiment was conducted again in 1971, though it likely resulted in far fewer Shakespearean insults. During the Apollo 15 Lunar mission, Commander David Scott stood on the surface of the moon, held out a feather and a hammer, and dropped them [2]. They hit the ground at the same time. I would have liked to sit in on the meeting where someone fought against all the bureaucracy and squabbling scientists to get a single feather on the lunar packing list. The feather advocate and I would’ve high-fived at the end of the meeting, and likely become good friends.

Free Fall on Earth

If Galileo was so very right, that the velocity of falling objects doesn’t depend on the mass of the object, then how do objects free fall?

Galileo constructed the following equation for a free-falling object which starts at rest.

where ∆x is the change in position (an initial height, such as the top of the tower in Pisa, to a final height, such as the ground), g is the gravitational acceleration, and t is time. A similar equation was constructed to obtain the final velocity of an object that falls from rest,

where v_f is the velocity the object reaches upon the instant it crashes at the ground. For all of Galileo’s sneering, Aristotle was no fool; we’ve all witnessed how the feather takes its sweet time falling to the floor, while the Steinway piano plummets onto some cartoon character’s head. It really seems like heavier objects fall faster. The thing which Aristotle could not pull off, and Commander David Scott could, was dropping objects where there is no air. Out on the surface of the moon, there is but the faintest whiff of an atmosphere. Feathers are designed to catch and curve air, keeping birds aloft; but without ‘the wind beneath [their] wings’ birds might as well try to fly using hammers.

References

[1] Stefano Gattei and Vincenzio Viviani. On the life of Galileo: Viviani’s historical account and other early biographies. Princeton University Press, 2019.

[2] David R Williams. Apollo 15 hammer-feather drop, Mar 2023. URL: https://nssdc.gsfc.

nasa.gov/planetary/lunar/apollo_15_feather_drop.html.