Editing Mass (section)

=== Galilean free fall ===
[[File:Galileo.arp.300pix.jpg|left|upright|thumb|Galileo Galilei (1636)]]
[[File:Falling ball.jpg|thumb|right|upright|Distance traveled by a freely falling ball is proportional to the square of the elapsed time.]]
Sometime prior to 1638, Galileo turned his attention to the phenomenon of objects in free fall, attempting to characterize these motions. Galileo was not the first to investigate Earth's gravitational field, nor was he the first to accurately describe its fundamental characteristics.  However, Galileo's reliance on scientific experimentation to establish physical principles would have a profound effect on future generations of scientists. It is unclear if these were just hypothetical experiments used to illustrate a concept, or if they were real experiments performed by Galileo,<ref>{{cite journal |last=Drake |first=S. |date=1979 |title=Galileo's Discovery of the Law of Free Fall |journal=Scientific American |volume=228 |issue=5 |pages=84–92 |bibcode=1973SciAm.228e..84D |doi=10.1038/scientificamerican0573-84}}</ref> but the results obtained from these experiments were both realistic and compelling.  A biography by Galileo's pupil [[Vincenzo Viviani]] stated that Galileo had dropped [[ball]]s of the same material, but different masses, from the [[Leaning Tower of Pisa]] to demonstrate that their time of descent was independent of their mass.<ref group="note">At the time when Viviani asserts that the experiment took place, Galileo had not yet formulated the final version of his law of free fall. He had, however, formulated an earlier version that predicted that bodies ''of the same material'' falling through the same medium would fall at the same speed. See {{cite book |last=Drake |first=S. |date=1978 |title=Galileo at Work |pages=[https://archive.org/details/galileoatwork00stil/page/19 19–20] |publisher=University of Chicago Press |isbn=978-0-226-16226-3 |url=https://archive.org/details/galileoatwork00stil/page/19 }}</ref> In support of this conclusion, Galileo had advanced the following theoretical argument: He asked if two bodies of different masses and different rates of fall are tied by a string, does the combined system fall faster because it is now more massive, or does the lighter body in its slower fall hold back the heavier body?  The only convincing resolution to this question is that all bodies must fall at the same rate.<ref>{{cite book |last=Galileo |first=G. |date=1632 |title=Dialogue Concerning the Two Chief World Systems|title-link=Dialogue Concerning the Two Chief World Systems }}</ref>

A later experiment was described in Galileo's ''Two New Sciences'' published in 1638.  One of Galileo's fictional characters, Salviati, describes an experiment using a bronze ball and a wooden ramp.  The wooden ramp was "12 cubits long, half a cubit wide and three finger-breadths thick" with a straight, smooth, polished [[Groove (engineering)|groove]].  The groove was lined with "[[parchment]], also smooth and polished as possible".  And into this groove was placed "a hard, smooth and very round bronze ball".  The ramp was inclined at various [[angle]]s to slow the acceleration enough so that the elapsed time could be measured.  The ball was allowed to roll a known distance down the ramp, and the time taken for the ball to move the known distance was measured.  The time was measured using a water clock described as follows:
:a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.<ref>{{cite book |last1=Galileo |first1=G. |date=1638 |title=Discorsi e Dimostrazioni Matematiche, Intorno à Due Nuove Scienze |volume=213 |publisher=[[House of Elzevir|Louis Elsevier]]}}, translated in {{cite book |date=1954 |editor1-last=Crew |editor1-first=H. |editor2-last=de Salvio |editor2-first=A. |title=Mathematical Discourses and Demonstrations, Relating to Two New Sciences |url=http://oll.libertyfund.org/index.php?option=com_staticxt&staticfile=show.php%3Ftitle=753&Itemid=99999999 |publisher=[[Dover Publications]] |isbn=978-1-275-10057-2 |access-date=11 April 2012 |archive-date=1 October 2013 |archive-url=https://web.archive.org/web/20131001015122/http://oll.libertyfund.org/index.php?option=com_staticxt&staticfile=show.php%3Ftitle=753&Itemid=99999999 |url-status=dead }} and also available in {{cite book |editor1-last=Hawking |editor1-first=S. |date=2002 |title=On the Shoulders of Giants |pages=[https://archive.org/details/isbn_9780762413485/page/534 534–535] |publisher=[[Running Press]] |isbn=978-0-7624-1348-5 |url=https://archive.org/details/isbn_9780762413485/page/534 }}</ref>

Galileo found that for an object in free fall, the distance that the object has fallen is always proportional to the square of the elapsed time:
: <math>{\text{Distance}} \propto {\text{Time}^2}</math>

Galileo had shown that objects in free fall under the influence of the Earth's gravitational field have a constant acceleration, and Galileo's contemporary, Johannes Kepler, had shown that the planets follow elliptical paths under the influence of the Sun's gravitational mass.  However, Galileo's free fall motions and Kepler's planetary motions remained distinct during Galileo's lifetime.