{"id":3099,"date":"2017-04-04T03:34:29","date_gmt":"2017-04-04T01:34:29","guid":{"rendered":"http:\/\/le-bars.net\/yoann\/?p=3099"},"modified":"2018-01-23T06:01:44","modified_gmt":"2018-01-23T04:01:44","slug":"falling-bodies-galileo-sets-up-aristotle-to-the-fall","status":"publish","type":"post","link":"https:\/\/le-bars.net\/yoann\/en\/2017\/04\/04\/falling-bodies-galileo-sets-up-aristotle-to-the-fall\/","title":{"rendered":"Falling bodies: Galileo sets up Aristotle to the fall"},"content":{"rendered":"<figure style=\"width: 2500px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Justus_Sustermans_-_Portrait_of_Galileo_Galilei,_1636.jpg?uselang=en\" title=\"The image on Wikimedia Common\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/d\/d4\/Justus_Sustermans_-_Portrait_of_Galileo_Galilei%2C_1636.jpg\" width=\"2500\" height=\"3176\" alt=\"Galileo GALILEI\" class=\"size-large\" \/><\/a><figcaption class=\"wp-caption-text\">Portrait of Galileo\u00a0Galilei by <a href=\"https:\/\/en.wikipedia.org\/wiki\/Giusto_Sustermans\" title=\"Giusto Sustermans\" target=\"_blank\">Giusto\u00a0Sustermans<\/a> in 1636 \u2013 image from public domain via <a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Justus_Sustermans_-_Portrait_of_Galileo_Galilei,_1636.jpg?uselang=en\" title=\"The image on Wikimedia Common\" target=\"_blank\">Wikimedia\u00a0Common<\/a>.<\/figcaption><\/figure>\n<p>As <a href=\"http:\/\/le-bars.net\/yoann\/en\/2015\/08\/20\/how-i-extended-galileos-work\/\" title=\"How I extended Galileo\u2019s work!\">seen twice<\/a> with <a href=\"http:\/\/le-bars.net\/yoann\/en\/2015\/09\/15\/everything-is-relative-my-dear-bruno\/\" title=\"Everything is relative, my dear Bruno!\">a view from here<\/a>, works from <a href=\"https:\/\/en.wikipedia.org\/wiki\/Nicolaus_Copernicus\" title=\"Nicolaus\u00a0Copernicus\" target=\"_blank\">Nicolaus\u00a0Copernicus<\/a> (1473 \u2013 1543), <a href=\"https:\/\/en.wikipedia.org\/wiki\/Giordano_Bruno\" title=\"Giordano\u00a0Bruno\" target=\"_blank\">Giordano\u00a0Bruno<\/a> (1548 \u2013 1600), <a href=\"https:\/\/en.wikipedia.org\/wiki\/Johannes_Kepler\" title=\"Johannes\u00a0Kepler\" target=\"_blank\">Johannes\u00a0Kepler<\/a> (1571 \u2013 1630) and <a href=\"https:\/\/en.wikipedia.org\/wiki\/Galileo_Galilei\" title=\"Galileo\u00a0Galilei\" target=\"_blank\">Galileo\u00a0Galilei<\/a> (1564 \u2013 1642), which have been preceded by those of <a href=\"https:\/\/en.wikipedia.org\/wiki\/Nicole_Oresme\" title=\"Nicole\u00a0Oresme\" target=\"_blank\">Nicole\u00a0Oresme<\/a> (circa 1320-1322 \u2013 1382), completely challenged the Aristotelian model. In this questioning, Galileo did not settle with astronomy. He also tackled several other fundamental subjects, one of which has a great influence on my areas of research: the fall of bodies.<\/p>\n<p><!--more--><\/p>\n<h2>What Aristotle thought<\/h2>\n<p>The physical system of <a href=\"https:\/\/en.wikipedia.org\/wiki\/Aristotle\" title=\"Aristotle\" target=\"_blank\">Aristotle<\/a><span class=\"footnote_referrer\"><a role=\"button\" tabindex=\"0\" onclick=\"footnote_moveToReference_3099_1('footnote_plugin_reference_3099_1_1');\" onkeypress=\"footnote_moveToReference_3099_1('footnote_plugin_reference_3099_1_1');\" ><sup id=\"footnote_plugin_tooltip_3099_1_1\" class=\"footnote_plugin_tooltip_text\">1<\/sup><\/a><span id=\"footnote_plugin_tooltip_text_3099_1_1\" class=\"footnote_tooltip\">\u1f08\u03c1\u03b9\u03c3\u03c4\u03bf\u03c4\u03ad\u03bb\u03b7\u03c2. <em>\u03a6\u03c5\u03c3\u03b9\u03ba\u1f74 \u1f00\u03ba\u03c1\u03cc\u03b1\u03c3\u03b9\u03c2<\/em>. An English translation: Robin\u00a0Waterfield, 1999. <em>Physics<\/em>, ed. David\u00a0Bostock, Oxford\u00a0University\u00a0Press.<\/span><\/span><script type=\"text\/javascript\"> jQuery('#footnote_plugin_tooltip_3099_1_1').tooltip({ tip: '#footnote_plugin_tooltip_text_3099_1_1', tipClass: 'footnote_tooltip', effect: 'fade', predelay: 0, fadeInSpeed: 200, delay: 400, fadeOutSpeed: 200, position: 'top right', relative: true, offset: [10, 10], });<\/script> (384 \u2013 322 BC) already addressed the fall of bodies. It is generally claimed that in this system the heavier a body is the faster it falls on Earth. Actually, it is a little more sophisticated.<\/p>\n<p>According Aristotle, the speed of a falling object depends on its ability to split the medium \u2013 to rephrase it modernly, to split the atmosphere. Thus, the speed of fall depends not only on the weight of the object, but also on its shape and orientation. Moreover, the medium can carry out a return action, opening the way to the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Archimedes\" title=\"Archimedes\" target=\"_blank\">Archimedes<\/a>\u2019 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Archimedes%27_principle\" title=\"Archimedes\u2019 principle\" target=\"_blank\">principle<\/a> (287 \u2013 212 BC).<\/p>\n<p>Nowadays, it is usually highlighted that the Aristotelian system is integrally invalidated, something undeniable. However, long after Galileo\u2019s work, it still was in conformity with observations. Yet, Galileo had the intuition that something was wrong.<\/p>\n<h2>Falling objects from the tower of Pisa<\/h2>\n<p>It is commonly alleged that Galileo dropped two objects with different weight from the top of the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Leaning_Tower_of_Pisa\" title=\"Leaning tower of Pisa\" target=\"_blank\">tower of Pisa<\/a> to show that they would reach the bottom at the same time. Actually, this is simply impossible as I hope the following will convince you. Indeed, the tower of Pisa length 55.8 meters. If we let a tennis ball and a <a href=\"https:\/\/en.wikipedia.org\/wiki\/P%C3%A9tanque\" title=\"P\u00e9tanque\" target=\"_blank\">p\u00e9tanque boule<\/a> (as I am French, I use French references) fall from the top of this tower, the effect of air friction will lead them to fall at different speed. The theoretical study (I detail it in an <a href=\"#appendix\" title=\"Appendix\">appendix to this article<\/a>, but if calculations repel you, feel free to skip it) indicates the p\u00e9tanque boule will reach the ground after falling for about 3.4\u202fs. At this moment, the tennis ball will be about 10.8\u202fm above the ground, a height human eyes can perfectly perceive. The total duration of the tennis ball fall is about 3.9\u202fs, that is to say about 0.5\u202fs more than the p\u00e9tanque boule.<\/p>\n<p>Thus, contradictory to what is too often said, Galileo did not prove the invalidity of the Aristotelian system by making two different balls fall from the top of the tower of Pisa. Had he done so, everyone would have noticed what the Aristotelian system foresaw: the heaviest would have hit the ground first.<\/p>\n<p>Then, what contradiction did Galileo bring to Aristotle?<\/p>\n<h2>Galileo and thought experiments<\/h2>\n<p>Shortly before 1600, Galileo wrote about falling objects without initial impulse<span class=\"footnote_referrer\"><a role=\"button\" tabindex=\"0\" onclick=\"footnote_moveToReference_3099_1('footnote_plugin_reference_3099_1_2');\" onkeypress=\"footnote_moveToReference_3099_1('footnote_plugin_reference_3099_1_2');\" ><sup id=\"footnote_plugin_tooltip_3099_1_2\" class=\"footnote_plugin_tooltip_text\">2<\/sup><\/a><span id=\"footnote_plugin_tooltip_text_3099_1_2\" class=\"footnote_tooltip\">Galileo\u00a0Galilei, 1590. <em>De Motu<\/em><\/span><\/span><script type=\"text\/javascript\"> jQuery('#footnote_plugin_tooltip_3099_1_2').tooltip({ tip: '#footnote_plugin_tooltip_text_3099_1_2', tipClass: 'footnote_tooltip', effect: 'fade', predelay: 0, fadeInSpeed: 200, delay: 400, fadeOutSpeed: 200, position: 'top right', relative: true, offset: [10, 10], });<\/script>. Using mathematical reasoning, he showed that if the action of the medium (that is to say of the atmosphere) is negligible, then the object falling will have a speed proportional to the duration of the fall. In other words, he gave a first mathematical analysis of free fall without interaction with the atmosphere, that is to say concretely in a vacuum. He thus confirmed some of Nicole\u00a0Oresme\u2019s speculations<span class=\"footnote_referrer\"><a role=\"button\" tabindex=\"0\" onclick=\"footnote_moveToReference_3099_1('footnote_plugin_reference_3099_1_3');\" onkeypress=\"footnote_moveToReference_3099_1('footnote_plugin_reference_3099_1_3');\" ><sup id=\"footnote_plugin_tooltip_3099_1_3\" class=\"footnote_plugin_tooltip_text\">3<\/sup><\/a><span id=\"footnote_plugin_tooltip_text_3099_1_3\" class=\"footnote_tooltip\">Nicole\u00a0Oresme, between 1400 et 1420. <em>Trait\u00e9 de la sph\u00e8re<\/em>.<\/span><\/span><script type=\"text\/javascript\"> jQuery('#footnote_plugin_tooltip_3099_1_3').tooltip({ tip: '#footnote_plugin_tooltip_text_3099_1_3', tipClass: 'footnote_tooltip', effect: 'fade', predelay: 0, fadeInSpeed: 200, delay: 400, fadeOutSpeed: 200, position: 'top right', relative: true, offset: [10, 10], });<\/script>.<\/p>\n<p>The most noticeable novelty is that Galileo studied the fall of bodies in a vacuum. However, in his days it was not possible to realize such a vacuum. Hence, Galileo studied an abstract situation that could not be observed. In order to tackle this phenomenon, he therefore made a purely intellectual exercise in which he tried to understand an idealized situation. Doing so, he rejected any phenomenon which could have prevented him to properly understand the subject he was interested in. This way of reasoning is call a thought experiment, a methodology Galileo introduced<span class=\"footnote_referrer\"><a role=\"button\" tabindex=\"0\" onclick=\"footnote_moveToReference_3099_1('footnote_plugin_reference_3099_1_4');\" onkeypress=\"footnote_moveToReference_3099_1('footnote_plugin_reference_3099_1_4');\" ><sup id=\"footnote_plugin_tooltip_3099_1_4\" class=\"footnote_plugin_tooltip_text\">4<\/sup><\/a><span id=\"footnote_plugin_tooltip_text_3099_1_4\" class=\"footnote_tooltip\">Ernst\u00a0Mach, 1883. <em>Die Mechanik in ihrer Entwicklung. Historisch-kritisch dargestellt<\/em>. An English translation by Thomas\u00a0J.\u00a0McCormack: Ernst\u00a0Mach, 1960. <em>The Science of Mechanics<\/em>, 6th edition, LaSalle, Illinois: Open Court, pp. 32 \u2013 41, 159 \u2013 162<\/span><\/span><script type=\"text\/javascript\"> jQuery('#footnote_plugin_tooltip_3099_1_4').tooltip({ tip: '#footnote_plugin_tooltip_text_3099_1_4', tipClass: 'footnote_tooltip', effect: 'fade', predelay: 0, fadeInSpeed: 200, delay: 400, fadeOutSpeed: 200, position: 'top right', relative: true, offset: [10, 10], });<\/script> and which became regularly used afterwards.<\/p>\n<p>However, Galileo gave his most complete analysis of falling bodies in his last published work<span class=\"footnote_referrer\"><a role=\"button\" tabindex=\"0\" onclick=\"footnote_moveToReference_3099_1('footnote_plugin_reference_3099_1_5');\" onkeypress=\"footnote_moveToReference_3099_1('footnote_plugin_reference_3099_1_5');\" ><sup id=\"footnote_plugin_tooltip_3099_1_5\" class=\"footnote_plugin_tooltip_text\">5<\/sup><\/a><span id=\"footnote_plugin_tooltip_text_3099_1_5\" class=\"footnote_tooltip\">Galileo\u00a0Galilei, 1638. <em>Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze<\/em>. An English translation by Stillman\u00a0Drake: Galileo\u00a0Galilei, 1974. <em>Two new sciences<\/em>, University\u00a0of\u00a0Wisconsin\u00a0Press. <a href=\"http:\/\/galileoandeinstein.physics.virginia.edu\/tns_draft\/index.html\" title=\"Two new sciences\" target=\"_blank\">Available on-line<\/a>.<\/span><\/span><script type=\"text\/javascript\"> jQuery('#footnote_plugin_tooltip_3099_1_5').tooltip({ tip: '#footnote_plugin_tooltip_text_3099_1_5', tipClass: 'footnote_tooltip', effect: 'fade', predelay: 0, fadeInSpeed: 200, delay: 400, fadeOutSpeed: 200, position: 'top right', relative: true, offset: [10, 10], });<\/script>.<\/p>\n<h2>Aristotle falling down<\/h2>\n<p>Galileo considered the case of two stones, but I propose to adapt the experience by considering a statue of Aristotle: as this experience takes place in thought, we can easily adopt the material we want. If we drop this statue without any initial velocity from the top of the Tower of Pisa, it will reach the ground with a velocity I propose to call Roger \u2013 let us note it <em>v<sub>s<\/sub><\/em>.<\/p>\n<p>By considering two stones, Galileo had the idea of studying the fall of not only one body, but two. To be able to do the same, let us saw the head of the statue of Aristotle to detach it from the body.<\/p>\n<figure style=\"width: 1700px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Aristotle_Altemps_Inv8575.jpg?uselang=en\" title=\"Bust of Aristotle\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/a\/ae\/Aristotle_Altemps_Inv8575.jpg\" width=\"1700\" height=\"2275\" alt=\"Bust of Aristotle\" class=\"size-large\" \/><\/a><figcaption class=\"wp-caption-text\">Aristotle finds this thought experiment not really respectful \u2013 picture of the public domain of a marble bust of Aristotle, a Roman copy of a Greek original bronze of <a href=\"https:\/\/en.wikipedia.org\/wiki\/Lysippos\" title=\"Lysippos\" target=\"_blank\">Lysippos<\/a> (circa 330 BC) from the old <a href=\"https:\/\/en.wikipedia.org\/wiki\/Ludovisi_(family)\" title=\"The Ludovisi family\" target=\"_blank\">Ludovisi<\/a> collection, via <a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Aristotle_Altemps_Inv8575.jpg?uselang=en\" title=\"The picture in Wikimedia\u00a0Commons\" target=\"_blank\">Wikimedia\u00a0Commons<\/a>.<\/figcaption><\/figure>\n<p>Now, let us drop the head from the top of the Tower of Pisa: it reaches the ground with the speed <em>v<sub>h<\/sub><\/em>. Under the same conditions, the body will reach the ground with the velocity <em>v<sub>b<\/sub><\/em>. According to Aristotle\u2019s physical theory, the speeds at ground impact of the two pieces of the statue, each being less heavy than the entire statue, should be less than the speed at the impact of the complete statue. Therefore, we have the following mathematical relations: <em>v<sub>h<\/sub><\/em>\u00a0<\u00a0<em>v<sub>s<\/sub><\/em> and <em>v<sub>b<\/sub><\/em>\u00a0<\u00a0<em>v<sub>s<\/sub><\/em>. Moreover, as the head weight less than the rest of the body, according to Aristotle we should have <em>v<sub>h<\/sub><\/em>\u00a0<\u00a0<em>v<sub>b<\/sub><\/em>. Finally, insofar as the two parts of the statue constitute all the material of the statue, this same theory gives <em>v<sub>s<\/sub><\/em>\u00a0=\u00a0<em>v<sub>t<\/sub><\/em>\u00a0+\u00a0<em>v<sub>c<\/sub><\/em>.<\/p>\n<p>Lastly, let us attach the two parts of the statue with a sufficiently strong rope, which weight is sufficiently small to be negligible in comparison with the weights of each element of the statue: once again, as the experiment takes place in our mind, such a rope exists. Let us drop without initial speed the whole set, composed of the head and the body of the statue connected by the rope. Such a combination is now called a system in physics, as the use of words varies with time. It will reach the ground with the speed <em>v<sub>w<\/sub><\/em>. As a consequence, still in agreement with the Aristotelian theory, we should have <em>v<sub>w<\/sub><\/em>\u00a0=\u00a0<em>v<sub>h<\/sub><\/em>\u00a0+\u00a0<em>v<sub>b<\/sub><\/em>\u00a0=\u00a0<em>v<sub>s<\/sub><\/em>. In other words, the falling speed of this set should be the same as the falling speed of the complete statue. Hence, we have <em>v<sub>w<\/sub>\u202f>\u202fv<sub>b<\/sub><\/em>. However, once again following the same physical theory, insofar as <em>v<sub>h<\/sub><\/em>\u00a0<\u00a0<em>v<sub>b<\/sub><\/em>, the string should tighten and the head, being the element with the lowest speed, should slow down the fall of the body. Therefore, the falling speed of the set should be less than the one of the body alone, which leads to the relation <em>v<sub>w<\/sub>\u00a0<\u00a0v<sub>b<\/sub><\/em>.<\/p>\n<p>With this thought experiment, Galileo demonstrated that the Aristotelian system simultaneously foresees <em>v<sub>w<\/sub><\/em>\u00a0>\u00a0<em>v<sub>b<\/sub><\/em> and <em>v<sub>w<\/sub><\/em>\u00a0<\u00a0<em>v<sub>b<\/sub><\/em>. In other words, the falling speed of the set should be greater to the falling speed of the body of the statue alone and lower to that speed at the same time. Therefore, it contains an internal contradiction.<\/p>\n<h2>Towards a new physics<\/h2>\n<p>To solve this contradiction, Galileo concludes that, when there is no action due to the medium (the atmosphere), all bodies fall at the same rate, whatever their forms, weights and compositions. More than four centuries later, using a hammer and a feather on the Moon, Apollo\u00a015 commander Dave\u00a0Scott showed that the speed of fall in a vacuum is indeed independent of the weight:<\/p>\n<div style=\"width: 640px;\" class=\"wp-video\"><video class=\"wp-video-shortcode\" id=\"video-3099-1\" width=\"640\" height=\"360\" preload=\"metadata\" controls=\"controls\"><source type=\"video\/ogg\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e8\/Apollo_15_feather_and_hammer_drop.ogv?_=1\" \/><a href=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e8\/Apollo_15_feather_and_hammer_drop.ogv\">https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e8\/Apollo_15_feather_and_hammer_drop.ogv<\/a><\/video><\/div>\n<p>For now on, let\u2019s use the terms of classical physics, of which Galileo is a precursor and which will be formalized by <a href=\"https:\/\/en.wikipedia.org\/wiki\/Isaac_Newton\" title=\"Isaac\u00a0Newton\" target=\"_blank\">Isaac\u00a0Newton<\/a> (1642 or 1643 \u2013 1727). Of this physics and of Newton, I will soon discuss as viewed from here. In the meantime, we still can notice that the Aristotelian model confuses the force which is the cause of the fall of bodies, namely the gravitation, and the forces which oppose this fall, namely the friction of the air and possibly the buoyant force from Archimedes\u2019 principle (generally negligible in the case of the fall of a body in the atmosphere of the Earth).<\/p>\n<p>Several essential things have been established in this case. First, Galileo used mathematics to express the physical problem he was addressing. Even if Nicolaus\u00a0Copernicus had already begun this evolution, it was with Galileo that the use of mathematics tended to become systematic.<\/p>\n<p>Moreover, to contradict the Aristotelian system, Galileo used arguments internal to this system. Indeed, <a href=\"http:\/\/le-bars.net\/yoann\/en\/2015\/09\/15\/everything-is-relative-my-dear-bruno\/\" title=\"Everything is relative, my dear Bruno!\">as I have already referred to<\/a>, other systems had been proposed before. In particular, Nicolaus\u00a0Copernicus\u2019 system was widely used at the time of Galileo for the calculation of the positions of the stars because of its precision, although it generally was not envisaged as being able to translate reality (writings of Nicolaus\u00a0Copernicus will not be put to the Index until 1616<span class=\"footnote_referrer\"><a role=\"button\" tabindex=\"0\" onclick=\"footnote_moveToReference_3099_1('footnote_plugin_reference_3099_1_6');\" onkeypress=\"footnote_moveToReference_3099_1('footnote_plugin_reference_3099_1_6');\" ><sup id=\"footnote_plugin_tooltip_3099_1_6\" class=\"footnote_plugin_tooltip_text\">6<\/sup><\/a><span id=\"footnote_plugin_tooltip_text_3099_1_6\" class=\"footnote_tooltip\">Pierre-No\u00ebl\u00a0Mayaud, 1997. <em>La Condamnation des livres coperniciens et sa r\u00e9vocation \u00e0 la lumi\u00e8re de documents in\u00e9dits des Congr\u00e9gations de l'Index et de l'Inquisition<\/em>, Universit\u00e9 pontificale gr\u00e9gorienne, 1997. <a href=\"https:\/\/books.google.fr\/books?id=yE2G1xjWkG8C&printsec=frontcover&hl=fr#v=onepage&q&f=false\" title=\"La Condamnation des livres coperniciens et sa r\u00e9vocation \u00e0 la lumi\u00e8re de documents in\u00e9dits des Congr\u00e9gations de l'Index et de l'Inquisition\" target=\"_blank\">Partially available on-line.<\/a><\/span><\/span><script type=\"text\/javascript\"> jQuery('#footnote_plugin_tooltip_3099_1_6').tooltip({ tip: '#footnote_plugin_tooltip_text_3099_1_6', tipClass: 'footnote_tooltip', effect: 'fade', predelay: 0, fadeInSpeed: 200, delay: 400, fadeOutSpeed: 200, position: 'top right', relative: true, offset: [10, 10], });<\/script>). However, these different systems did not induce a lasting questioning of the dominant system. In contrast, the demonstration of its internal inconsistency was an important cause of its forsaking.<\/p>\n<p>To highlight this inconsistency, Galileo had the idea to move from one to two bodies. The study of two bodies in mutual interaction is a case since often studied in mechanics. It allowed several theoretical advances.<\/p>\n<p>Lastly, Galileo used great scientific caution. Indeed, he has repeatedly gathered elements that each questioned the Aristotelian system \u2013 the case of the fall of the bodies discussed in this article, his astronomical observations <a href=\"http:\/\/le-bars.net\/yoann\/en\/2015\/09\/15\/everything-is-relative-my-dear-bruno\/\" title=\"Everything is relative, my dear Bruno!\">I have already presented<\/a>, among others. However, in particular to ensure that he was not misleading, he waited until he had accumulated enough different elements to corroborate the fact that this system had to be abandoned before actually taking a position against it. This prudence, notably with the idea of not concluding too fast, is now at the heart of scientific approaches.<\/p>\n<p>These elements have thus led to the foundation of a new physics, which is now called classical physics and which I will continue to develop in these pages.<\/p>\n<h2 id=\"appendix\">Appendix: calculation of the falling time from a tower<\/h2>\n<p>The calculations presented here are based on classical mechanics, of which Galileo is an initiator and which will be formalised by Isaac\u00a0Newton<span class=\"footnote_referrer\"><a role=\"button\" tabindex=\"0\" onclick=\"footnote_moveToReference_3099_1('footnote_plugin_reference_3099_1_7');\" onkeypress=\"footnote_moveToReference_3099_1('footnote_plugin_reference_3099_1_7');\" ><sup id=\"footnote_plugin_tooltip_3099_1_7\" class=\"footnote_plugin_tooltip_text\">7<\/sup><\/a><span id=\"footnote_plugin_tooltip_text_3099_1_7\" class=\"footnote_tooltip\">Isaac\u00a0Newton, 1687. <em>Philosophiae naturalis principia mathematica<\/em>, Joseph\u00a0Streater, London. <a href=\"http:\/\/www.gutenberg.org\/ebooks\/28233?msg=welcome_stranger\" title=\"Philosophiae naturalis principia mathematica\" target=\"_blank\">Available on-line.<\/a> An English version by Florian\u00a0Cajori: Isaac\u00a0Newton, 1946. <em>Principia in Modern English, Sir Isaac\u00a0Newton\u2019s Mathematical Principles<\/em>, University of California Press. <a href=\"http:\/\/www.17centurymaths.com\/contents\/newtoncontents.html\" title=\"Isaac\u00a0Newton: Principia\" target=\"_blank\">Available on-line.<\/a><\/span><\/span><script type=\"text\/javascript\"> jQuery('#footnote_plugin_tooltip_3099_1_7').tooltip({ tip: '#footnote_plugin_tooltip_text_3099_1_7', tipClass: 'footnote_tooltip', effect: 'fade', predelay: 0, fadeInSpeed: 200, delay: 400, fadeOutSpeed: 200, position: 'top right', relative: true, offset: [10, 10], });<\/script>. Upcoming articles of popularization on this website will detail it.<\/p>\n<p>We will try to determine the moment when two balls, namely a p\u00e9tanque boule and a tennis ball, will hit the ground after being dropped simultaneously without initial speed from the top of the tower of Pisa. As this fall takes place without an initial velocity, it is rectilinear in the geocentric frame of reference (that is to say with respect to the centre of the Earth). Therefore, we shall only consider the velocity along the vertical. We will take the bottom of the tower as our origin and our axis will be directed upwards.<\/p>\n<p>Let <em>t<\/em> be the considered instant (in s), <em>v<\/em> the speed of fall of the object (in m\u202fs<sup>\u22121<\/sup>), m its mass (in kg), <em>g<\/em> the acceleration due to terrestrial gravity (at sea-level <em>g<\/em>\u00a0=\u00a09.81\u202fm\u202fs<sup>\u22122<\/sup>) and <em>K<\/em> a constant I will specify. In the case of a free-falling ball in the atmosphere without initial velocity, we have the following relation:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+m+%5Cfrac%7B%5Cmathrm%7Bd%7D+v%7D%7B%5Cmathrm%7Bd%7D+t%7D+%3D+m+g+-+K+v%5E2.%5Cqquad+%281%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;displaystyle m &#92;frac{&#92;mathrm{d} v}{&#92;mathrm{d} t} = m g - K v^2.&#92;qquad (1)\" class=\"latex\" \/><\/p>\n<p>For <em>S<\/em> the area of the surface of the ball perpendicular to the direction of motion (in m<sup>2<\/sup>), <em>C<sub>x<\/sub><\/em> the aerodynamic coefficient of the moving body (dimensionless) and <em>\u03c1<\/em> the density of the air \u2013 at sea-level and at 15\u202f\u00b0C, <em>\u03c1<\/em>\u00a0=\u00a01.225\u202fkg\u202fm<sup>\u22123<\/sup>, \u2013 we have:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+K+%3D+%5Cfrac%7BC_x+%5Crho+S%7D%7B2%7D.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;displaystyle K = &#92;frac{C_x &#92;rho S}{2}.\" class=\"latex\" \/><\/p>\n<p>For a sphere, the experiment gives <em>C<sub>x<\/sub><\/em>\u00a0=\u00a00.44. In addition, the mass of a p\u00e9tanque boule is 700\u202fg (that is 0.7\u202fkg) and its diameter (<em>d<\/em>) 7.5\u202fcm (that is 0.075\u202fm), while the mass of a tennis ball is 55\u202fg and its diameter 6.7\u202fcm. For <em>S<\/em>, in the case of a sphere and at the speeds considered, we will take the area of the disc of the same diameter as the sphere:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+S+%3D+%5Cpi+%5Cfrac%7Bd%5E2%7D%7B4%7D.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;displaystyle S = &#92;pi &#92;frac{d^2}{4}.\" class=\"latex\" \/><\/p>\n<p>All this gives for the p\u00e9tanque boule <em>K<\/em> \u2248\u00a01.19\u202f\u22c5\u202f10<sup>\u22123<\/sup>\u202fkg\u202fm<sup>\u22121<\/sup> and for the tennis ball <em>K<\/em>\u00a0\u2248\u00a09.50\u202f\u22c5\u202f10<sup>\u22124<\/sup>\u202fkg\u202fm<sup>\u22121<\/sup>.<\/p>\n<p>For those who already know differential equations (that is, equations involving derivatives of functions, all notions that I would present in a forthcoming article), notice that equation (1) is an equation with separable variables. It implies that the ball will not be able to exceed a maximum velocity (obtained when the derivative of the velocity, that is to say the acceleration, is zero), called the limit velocity and which I propose you to note <em>v<sub>l<\/sub><\/em>, determined by:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+v_l+%3D+%5Csqrt%7B%5Cfrac%7Bm+g%7D%7BK%7D%7D.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;displaystyle v_l = &#92;sqrt{&#92;frac{m g}{K}}.\" class=\"latex\" \/><\/p>\n<p>According to our previous calculations, we have for the p\u00e9tanque boule <em>v<sub>l<\/sub><\/em>\u00a0\u2248\u00a075.94\u202fm\u202fs<sup>\u22121<\/sup> and for the tennis ball <em>v<sub>l<\/sub><\/em>\u00a0\u2248\u00a023.83\u202fm\u202fs<sup>\u22121<\/sup>. As the limit velocity of the p\u00e9tanque boule is much higher than the one of the tennis ball, it is clear that the former will reach the ground more quickly than the later.<\/p>\n<p>If you know differential equations with separable variables, then you can solve equation (1). Otherwise, well, I give you the solution:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+v%5Cleft%28t%5Cright%29+%3D+v_l+%5Ctanh%5Cleft%28%5Cfrac%7Bg+t%7D%7Bv_l%7D%5Cright%29.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;displaystyle v&#92;left(t&#92;right) = v_l &#92;tanh&#92;left(&#92;frac{g t}{v_l}&#92;right).\" class=\"latex\" \/><\/p>\n<p>Let <em>y<\/em> be the height of the centre of the ball with respect to the ground and <em>y<sub>0<\/sub><\/em> its initial height, that is <em>y<sub>0<\/sub><\/em>\u00a0=\u00a055.8\u202fm in our case. We can then determine the height where the ball is located at time t:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+y%5Cleft%28t%5Cright%29+%3D+y_0+-+%5Cfrac%7Bv_l%5E2%7D%7Bg%7D+%5Cln%5Ccosh%5Cleft%28%5Cfrac%7Bg+t%7D%7Bv_l%7D%5Cright%29.+%5Cqquad+%282%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;displaystyle y&#92;left(t&#92;right) = y_0 - &#92;frac{v_l^2}{g} &#92;ln&#92;cosh&#92;left(&#92;frac{g t}{v_l}&#92;right). &#92;qquad (2)\" class=\"latex\" \/><\/p>\n<p>To determine when the ball touches the ground, equation (2) is solved when <em>y<\/em>\u00a0=\u00a00, that is:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+t+%3D+%5Cfrac%7Bv_l%7D%7Bg%7D+%5Cmathrm%7Bacosh%7D%5Cexp%5Cleft%28%5Cfrac%7Bg+y_0%7D%7Bv_l%5E2%7D%5Cright%29.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;displaystyle t = &#92;frac{v_l}{g} &#92;mathrm{acosh}&#92;exp&#92;left(&#92;frac{g y_0}{v_l^2}&#92;right).\" class=\"latex\" \/><\/p>\n<p>In the case of the p\u00e9tanque boule, we obtain a falling time of approximately 3.43\u202fs, and about 3.93\u202fs in the case of the tennis ball. Then, with equation\u00a0(2) we can determine at what height the tennis ball is when the p\u00e9tanque boule hit the ground, that is at 3.43\u202fs: about 10.85\u202fm.<\/p>\n<div class=\"speaker-mute footnotes_reference_container\"> <div class=\"footnote_container_prepare\"><p><span role=\"button\" tabindex=\"0\" class=\"footnote_reference_container_label pointer\" onclick=\"footnote_expand_collapse_reference_container_3099_1();\">Notes<\/span><span role=\"button\" tabindex=\"0\" class=\"footnote_reference_container_collapse_button\" style=\"display: none;\" onclick=\"footnote_expand_collapse_reference_container_3099_1();\">[<a id=\"footnote_reference_container_collapse_button_3099_1\">+<\/a>]<\/span><\/p><\/div> <div id=\"footnote_references_container_3099_1\" style=\"\"><table class=\"footnotes_table footnote-reference-container\"><caption class=\"accessibility\">Notes<\/caption> <tbody> \r\n\r\n<tr class=\"footnotes_plugin_reference_row\"> <th scope=\"row\" class=\"footnote_plugin_index_combi pointer\"  onclick=\"footnote_moveToAnchor_3099_1('footnote_plugin_tooltip_3099_1_1');\"><a id=\"footnote_plugin_reference_3099_1_1\" class=\"footnote_backlink\"><span class=\"footnote_index_arrow\">&#8593;<\/span>1<\/a><\/th> <td class=\"footnote_plugin_text\">\u1f08\u03c1\u03b9\u03c3\u03c4\u03bf\u03c4\u03ad\u03bb\u03b7\u03c2. <em>\u03a6\u03c5\u03c3\u03b9\u03ba\u1f74 \u1f00\u03ba\u03c1\u03cc\u03b1\u03c3\u03b9\u03c2<\/em>. An English translation: Robin\u00a0Waterfield, 1999. <em>Physics<\/em>, ed. David\u00a0Bostock, Oxford\u00a0University\u00a0Press.<\/td><\/tr>\r\n\r\n<tr class=\"footnotes_plugin_reference_row\"> <th scope=\"row\" class=\"footnote_plugin_index_combi pointer\"  onclick=\"footnote_moveToAnchor_3099_1('footnote_plugin_tooltip_3099_1_2');\"><a id=\"footnote_plugin_reference_3099_1_2\" class=\"footnote_backlink\"><span class=\"footnote_index_arrow\">&#8593;<\/span>2<\/a><\/th> <td class=\"footnote_plugin_text\">Galileo\u00a0Galilei, 1590. <em>De Motu<\/em><\/td><\/tr>\r\n\r\n<tr class=\"footnotes_plugin_reference_row\"> <th scope=\"row\" class=\"footnote_plugin_index_combi pointer\"  onclick=\"footnote_moveToAnchor_3099_1('footnote_plugin_tooltip_3099_1_3');\"><a id=\"footnote_plugin_reference_3099_1_3\" class=\"footnote_backlink\"><span class=\"footnote_index_arrow\">&#8593;<\/span>3<\/a><\/th> <td class=\"footnote_plugin_text\">Nicole\u00a0Oresme, between 1400 et 1420. <em>Trait\u00e9 de la sph\u00e8re<\/em>.<\/td><\/tr>\r\n\r\n<tr class=\"footnotes_plugin_reference_row\"> <th scope=\"row\" class=\"footnote_plugin_index_combi pointer\"  onclick=\"footnote_moveToAnchor_3099_1('footnote_plugin_tooltip_3099_1_4');\"><a id=\"footnote_plugin_reference_3099_1_4\" class=\"footnote_backlink\"><span class=\"footnote_index_arrow\">&#8593;<\/span>4<\/a><\/th> <td class=\"footnote_plugin_text\">Ernst\u00a0Mach, 1883. <em>Die Mechanik in ihrer Entwicklung. Historisch-kritisch dargestellt<\/em>. An English translation by Thomas\u00a0J.\u00a0McCormack: Ernst\u00a0Mach, 1960. <em>The Science of Mechanics<\/em>, 6th edition, LaSalle, Illinois: Open Court, pp. 32 \u2013 41, 159 \u2013 162<\/td><\/tr>\r\n\r\n<tr class=\"footnotes_plugin_reference_row\"> <th scope=\"row\" class=\"footnote_plugin_index_combi pointer\"  onclick=\"footnote_moveToAnchor_3099_1('footnote_plugin_tooltip_3099_1_5');\"><a id=\"footnote_plugin_reference_3099_1_5\" class=\"footnote_backlink\"><span class=\"footnote_index_arrow\">&#8593;<\/span>5<\/a><\/th> <td class=\"footnote_plugin_text\">Galileo\u00a0Galilei, 1638. <em>Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze<\/em>. An English translation by Stillman\u00a0Drake: Galileo\u00a0Galilei, 1974. <em>Two new sciences<\/em>, University\u00a0of\u00a0Wisconsin\u00a0Press. <a href=\"http:\/\/galileoandeinstein.physics.virginia.edu\/tns_draft\/index.html\" title=\"Two new sciences\" target=\"_blank\">Available on-line<\/a>.<\/td><\/tr>\r\n\r\n<tr class=\"footnotes_plugin_reference_row\"> <th scope=\"row\" class=\"footnote_plugin_index_combi pointer\"  onclick=\"footnote_moveToAnchor_3099_1('footnote_plugin_tooltip_3099_1_6');\"><a id=\"footnote_plugin_reference_3099_1_6\" class=\"footnote_backlink\"><span class=\"footnote_index_arrow\">&#8593;<\/span>6<\/a><\/th> <td class=\"footnote_plugin_text\">Pierre-No\u00ebl\u00a0Mayaud, 1997. <em>La Condamnation des livres coperniciens et sa r\u00e9vocation \u00e0 la lumi\u00e8re de documents in\u00e9dits des Congr\u00e9gations de l'Index et de l'Inquisition<\/em>, Universit\u00e9 pontificale gr\u00e9gorienne, 1997. <a href=\"https:\/\/books.google.fr\/books?id=yE2G1xjWkG8C&printsec=frontcover&hl=fr#v=onepage&q&f=false\" title=\"La Condamnation des livres coperniciens et sa r\u00e9vocation \u00e0 la lumi\u00e8re de documents in\u00e9dits des Congr\u00e9gations de l'Index et de l'Inquisition\" target=\"_blank\">Partially available on-line.<\/a><\/td><\/tr>\r\n\r\n<tr class=\"footnotes_plugin_reference_row\"> <th scope=\"row\" class=\"footnote_plugin_index_combi pointer\"  onclick=\"footnote_moveToAnchor_3099_1('footnote_plugin_tooltip_3099_1_7');\"><a id=\"footnote_plugin_reference_3099_1_7\" class=\"footnote_backlink\"><span class=\"footnote_index_arrow\">&#8593;<\/span>7<\/a><\/th> <td class=\"footnote_plugin_text\">Isaac\u00a0Newton, 1687. <em>Philosophiae naturalis principia mathematica<\/em>, Joseph\u00a0Streater, London. <a href=\"http:\/\/www.gutenberg.org\/ebooks\/28233?msg=welcome_stranger\" title=\"Philosophiae naturalis principia mathematica\" target=\"_blank\">Available on-line.<\/a> An English version by Florian\u00a0Cajori: Isaac\u00a0Newton, 1946. <em>Principia in Modern English, Sir Isaac\u00a0Newton\u2019s Mathematical Principles<\/em>, University of California Press. <a href=\"http:\/\/www.17centurymaths.com\/contents\/newtoncontents.html\" title=\"Isaac\u00a0Newton: Principia\" target=\"_blank\">Available on-line.<\/a><\/td><\/tr>\r\n\r\n <\/tbody> <\/table> <\/div><\/div><script type=\"text\/javascript\"> function footnote_expand_reference_container_3099_1() { jQuery('#footnote_references_container_3099_1').show(); jQuery('#footnote_reference_container_collapse_button_3099_1').text('\u2212'); } function footnote_collapse_reference_container_3099_1() { jQuery('#footnote_references_container_3099_1').hide(); jQuery('#footnote_reference_container_collapse_button_3099_1').text('+'); } function footnote_expand_collapse_reference_container_3099_1() { if (jQuery('#footnote_references_container_3099_1').is(':hidden')) { footnote_expand_reference_container_3099_1(); } else { footnote_collapse_reference_container_3099_1(); } } function footnote_moveToReference_3099_1(p_str_TargetID) { footnote_expand_reference_container_3099_1(); var l_obj_Target = jQuery('#' + p_str_TargetID); if (l_obj_Target.length) { jQuery( 'html, body' ).delay( 0 ); jQuery('html, body').animate({ scrollTop: l_obj_Target.offset().top - window.innerHeight * 0.2 }, 380); } } function footnote_moveToAnchor_3099_1(p_str_TargetID) { footnote_expand_reference_container_3099_1(); var l_obj_Target = jQuery('#' + p_str_TargetID); if (l_obj_Target.length) { jQuery( 'html, body' ).delay( 0 ); jQuery('html, body').animate({ scrollTop: l_obj_Target.offset().top - window.innerHeight * 0.2 }, 380); } }<\/script>","protected":false},"excerpt":{"rendered":"<p>As seen twice with a view from here, works from Nicolaus\u00a0Copernicus (1473 \u2013 1543), Giordano\u00a0Bruno (1548 \u2013 1600), Johannes\u00a0Kepler (1571 \u2013 1630) and Galileo\u00a0Galilei (1564 \u2013 1642), which have been preceded by those of Nicole\u00a0Oresme (circa 1320-1322 \u2013 1382), completely challenged the Aristotelian model. In this questioning, Galileo did not settle with astronomy. He also &hellip; <a href=\"https:\/\/le-bars.net\/yoann\/en\/2017\/04\/04\/falling-bodies-galileo-sets-up-aristotle-to-the-fall\/\" class=\"more-link\">Continuer la lecture de <span class=\"screen-reader-text\">Falling bodies: Galileo sets up Aristotle to the fall<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[50],"tags":[98,72,107,69],"class_list":["post-3099","post","type-post","status-publish","format-standard","hentry","category-science-en","tag-history-of-sciences","tag-physics","tag-science-en","tag-scientific-popularisation"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Falling bodies: Galileo sets up Aristotle to the fall &#8211; Vu d\u2019ici<\/title>\n<meta name=\"description\" content=\"Falling bodies is among Galileo&#039;s major contributions. 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