# Falling bodies: Galileo sets up Aristotle to the fall

As seen twice with a view from here, works from Nicolaus Copernicus (1473 – 1543), Giordano Bruno (1548 – 1600), Johannes Kepler (1571 – 1630) and Galileo Galilei (1564 – 1642), which have been preceded by those of Nicole Oresme (circa 1320-1322 – 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.

## What Aristotle thought

The physical system of Aristotle1Ἀριστοτέλης. Φυσικὴ ἀκρόασις. An English translation: Robin Waterfield, 1999. Physics, ed. David Bostock, Oxford University Press. (384 – 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.

According Aristotle, the speed of a falling object depends on its ability to split the medium – 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 Archimedesprinciple (287 – 212 BC).

Nowadays, it is usually highlighted that the Aristotelian system is integrally invalidated, something undeniable. However, long after Galileo’s work, it still was in conformity with observations. Yet, Galileo had the intuition that something was wrong.

## Falling objects from the tower of Pisa

It is commonly alleged that Galileo dropped two objects with different weight from the top of the tower of Pisa 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 pétanque boule (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 appendix to this article, but if calculations repel you, feel free to skip it) indicates the pétanque boule will reach the ground after falling for about 3.4 s. At this moment, the tennis ball will be about 10.8 m above the ground, a height human eyes can perfectly perceive. The total duration of the tennis ball fall is about 3.9 s, that is to say about 0.5 s more than the pétanque boule.

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.

Then, what contradiction did Galileo bring to Aristotle?

## Galileo and thought experiments

Shortly before 1600, Galileo wrote about falling objects without initial impulse2Galileo Galilei, 1590. De Motu. 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 Oresme’s speculations3Nicole Oresme, between 1400 et 1420. Traité de la sphère..

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 introduced4Ernst Mach, 1883. Die Mechanik in ihrer Entwicklung. Historisch-kritisch dargestellt. An English translation by Thomas J. McCormack: Ernst Mach, 1960. The Science of Mechanics, 6th edition, LaSalle, Illinois: Open Court, pp. 32 – 41, 159 – 162 and which became regularly used afterwards.

However, Galileo gave his most complete analysis of falling bodies in his last published work5Galileo Galilei, 1638. Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze. An English translation by Stillman Drake: Galileo Galilei, 1974. Two new sciences, University of Wisconsin Press. Available on-line..

## Aristotle falling down

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 – let us note it vs.

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.

Now, let us drop the head from the top of the Tower of Pisa: it reaches the ground with the speed vh. Under the same conditions, the body will reach the ground with the velocity vb. According to Aristotle’s 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: vh < vs and vb < vs. Moreover, as the head weight less than the rest of the body, according to Aristotle we should have vh < vb. Finally, insofar as the two parts of the statue constitute all the material of the statue, this same theory gives vs = vt + vc.

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 vw. As a consequence, still in agreement with the Aristotelian theory, we should have vw = vh + vb = vs. In other words, the falling speed of this set should be the same as the falling speed of the complete statue. Hence, we have vw > vb. However, once again following the same physical theory, insofar as vh < vb, 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 vw < vb.

With this thought experiment, Galileo demonstrated that the Aristotelian system simultaneously foresees vw > vb and vw < vb. 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.

## Towards a new physics

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 15 commander Dave Scott showed that the speed of fall in a vacuum is indeed independent of the weight:

For now on, let’s use the terms of classical physics, of which Galileo is a precursor and which will be formalized by Isaac Newton (1642 or 1643 – 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’ principle (generally negligible in the case of the fall of a body in the atmosphere of the Earth).

Several essential things have been established in this case. First, Galileo used mathematics to express the physical problem he was addressing. Even if Nicolaus Copernicus had already begun this evolution, it was with Galileo that the use of mathematics tended to become systematic.

Moreover, to contradict the Aristotelian system, Galileo used arguments internal to this system. Indeed, as I have already referred to, other systems had been proposed before. In particular, Nicolaus Copernicus’ 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 Copernicus will not be put to the Index until 16166Pierre-Noël Mayaud, 1997. La Condamnation des livres coperniciens et sa révocation à la lumière de documents inédits des Congrégations de l’Index et de l’Inquisition, Université pontificale grégorienne, 1997. Partially available on-line.). 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.

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.

Lastly, Galileo used great scientific caution. Indeed, he has repeatedly gathered elements that each questioned the Aristotelian system – the case of the fall of the bodies discussed in this article, his astronomical observations I have already presented, 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.

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.

## Appendix: calculation of the falling time from a tower

The calculations presented here are based on classical mechanics, of which Galileo is an initiator and which will be formalised by Isaac Newton7Isaac Newton, 1687. Philosophiae naturalis principia mathematica, Joseph Streater, London. Available on-line. An English version by Florian Cajori: Isaac Newton, 1946. Principia in Modern English, Sir Isaac Newton’s Mathematical Principles, University of California Press. Available on-line.. Upcoming articles of popularization on this website will detail it.

We will try to determine the moment when two balls, namely a pétanque 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.

Let t be the considered instant (in s), v the speed of fall of the object (in m s−1), m its mass (in kg), g the acceleration due to terrestrial gravity (at sea-level g = 9.81 m s−2) and K a constant I will specify. In the case of a free-falling ball in the atmosphere without initial velocity, we have the following relation:

$\displaystyle m \frac{\mathrm{d} v}{\mathrm{d} t} = m g - K v^2.\qquad (1)$

For S the area of the surface of the ball perpendicular to the direction of motion (in m2), Cx the aerodynamic coefficient of the moving body (dimensionless) and ρ the density of the air – at sea-level and at 15 °C, ρ = 1.225 kg m−3, – we have:

$\displaystyle K = \frac{C_x \rho S}{2}.$

For a sphere, the experiment gives Cx = 0.44. In addition, the mass of a pétanque boule is 700 g (that is 0.7 kg) and its diameter (d) 7.5 cm (that is 0.075 m), while the mass of a tennis ball is 55 g and its diameter 6.7 cm. For S, 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:

$\displaystyle S = \pi \frac{d^2}{4}.$

All this gives for the pétanque boule K ≈ 1.19 ⋅ 10−3 kg m−1 and for the tennis ball K ≈ 9.50 ⋅ 10−4 kg m−1.

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 vl, determined by:

$\displaystyle v_l = \sqrt{\frac{m g}{K}}.$

According to our previous calculations, we have for the pétanque boule vl ≈ 75.94 m s−1 and for the tennis ball vl ≈ 23.83 m s−1. As the limit velocity of the pétanque 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.

If you know differential equations with separable variables, then you can solve equation (1). Otherwise, well, I give you the solution:

$\displaystyle v\left(t\right) = v_l \tanh\left(\frac{g t}{v_l}\right).$

Let y be the height of the centre of the ball with respect to the ground and y0 its initial height, that is y0 = 55.8 m in our case. We can then determine the height where the ball is located at time t:

$\displaystyle y\left(t\right) = y_0 - \frac{v_l^2}{g} \ln\cosh\left(\frac{g t}{v_l}\right). \qquad (2)$

To determine when the ball touches the ground, equation (2) is solved when y = 0, that is:

$\displaystyle t = \frac{v_l}{g} \mathrm{acosh}\exp\left(\frac{g y_0}{v_l^2}\right).$

In the case of the pétanque boule, we obtain a falling time of approximately 3.43 s, and about 3.93 s in the case of the tennis ball. Then, with equation (2) we can determine at what height the tennis ball is when the pétanque boule hit the ground, that is at 3.43 s: about 10.85 m.

Notes

↑1 Ἀριστοτέλης. Φυσικὴ ἀκρόασις. An English translation: Robin Waterfield, 1999. Physics, ed. David Bostock, Oxford University Press. Galileo Galilei, 1590. De Motu Nicole Oresme, between 1400 et 1420. Traité de la sphère. Ernst Mach, 1883. Die Mechanik in ihrer Entwicklung. Historisch-kritisch dargestellt. An English translation by Thomas J. McCormack: Ernst Mach, 1960. The Science of Mechanics, 6th edition, LaSalle, Illinois: Open Court, pp. 32 – 41, 159 – 162 Galileo Galilei, 1638. Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze. An English translation by Stillman Drake: Galileo Galilei, 1974. Two new sciences, University of Wisconsin Press. Available on-line. Pierre-Noël Mayaud, 1997. La Condamnation des livres coperniciens et sa révocation à la lumière de documents inédits des Congrégations de l’Index et de l’Inquisition, Université pontificale grégorienne, 1997. Partially available on-line. Isaac Newton, 1687. Philosophiae naturalis principia mathematica, Joseph Streater, London. Available on-line. An English version by Florian Cajori: Isaac Newton, 1946. Principia in Modern English, Sir Isaac Newton’s Mathematical Principles, University of California Press. Available on-line.