yellowpigs.net

Sara Smollett Sara Smollett
December 2002

Free Fall, Parabolic Trajectory, and Inertia

Purpose:

To discuss three important laws of motion - the law of free fall, the law of parabolic trajectory, and the law of inertia. Upon considering the evidence and arguments in support of and in opposition to these laws, I believe that as of 1651 there was the most evidence for accepting the law of free fall and the least for the law of inertia.

Introduction:

Before I discuss each of these three laws of local motion, I would like to address the general evidence problem facing these and other such laws of local motion. The evidence problem for these theories can be contrasted with that for theories in astronomy. Astronomers can observe the universe, but cannot intervene to devise and test experiments. In the mechanics and physics of Galileo, however, experimentation is not only possible, but crucial. The ability to conduct experiments is a substantial advantage, but it is also problematic. An experiment only demonstrates that the results will follow from situations which are like those of the experiment; it does not show that the results hold universally. That is, the range over which the hypothesis holds is limited. It is essential to devise experiments that will replicate the conditions under which the hypothesis has been posited to hold. If experiments are not to be misleading, either they must exactly model the real world (difficult) or it must be possible to accurately determine real-world results from the approximations given by the experiments. In particular, experiments must be devised in such a way that the effects of air resistance can be isolated.

Additionally, the study of local motion in the 1600's suffered from an evidence problem due to the difficulty of accurately measuring very small increments of time. Experiments involving distance covered in a short amount of time were likely to yield inconsistent results with a high degree of error. This made it difficult to determine if errors were coming from problems with the observations alone or if the hypothesis being tested was itself flawed. These errors call into question the reliability of many experiments in local motion.

Free Fall:

I will first consider Galileo's law of free fall, as introduced in his Dialogues Concerning Two New Sciences (Discorsi). According to the law of free fall, in the absence of air resistance the speed (distance per time) of falling objects depends almost solely on time (other factors, such as weight, are only secondary). An object which is dropped gains equal increments of speed in equal amounts of time. Galileo makes a statement involving a 1, 3, 5,¼ progression which, being the difference between consecutive squares, implies a second order equation (Galileo, Discorsi, 198-205). In modern notation: d µ t2.

Galileo describes several experiments (involving dropping objects, rolling balls down inclined planes, and pendular motion) that can be shown to support these claims. Galileo largely focuses not directly on free fall, but on change in speed on inclined planes (where motion is slower, hence easier to observe with accuracy). He devised numerous experiments involving single planes inclined at different angles as well as experiments which used multiple inclined planes or a combination of inclined and horizontal planes. Galileo asserts that the vertical motion of a ball rolling down an inclined plane is equivalent to free fall motion (Galileo, 205).

Though Galileo and Torricelli (c. 1644) were correct that acceleration occurs vertically (in a direction perpendicular to the surface of the Earth) so that acceleration is the same for balls rolling down inclined planes of the same height but different slopes, Galileo's conclusion is incorrect because he misses the fact that the motions described by rolling and falling are not equivalent. The error is minimal for inclined planes at low angles (the slight error being primarily due to friction), but a ball set on a steep inclined plane will trace a path that involves a combination of rolling and falling. Although neither the specific error (5/7), nor the reason for the error was known, Mersenne, providing data, remarked that it was difficult to get well-behaved results for steeply angled inclined planes. It does not appear that Galileo (nor anyone else prior to 1651) performed experiments to verify or falsify the claim that free fall and inclined motion are equivalent. Without some evidence of equivalence or explanation of the difference, the evidence from Galileo's inclined plane experiments cannot be directly used to support the law of free fall. This assumption that free fall and motion on an inclined plane are the same is a serious lacuna in Galileo's reasoning.

Instead of relying on inclined planes, support must come from actual free fall experiments. Data amassed through experimentation is published not by Galileo, who may not have performed these experiments, but by Mersenne (c. 1636) and Riccioli (c. 1651), who carried out many of Galileo's experiments as accurately as they could. Mersenne performed repeated experiments showing that a spherical object falls 12 feet in the first second, 48 feet in the second second, and 110 (or 108) feet in the third second. This confirms (does not contradict) the 1, 3, 5, ¼ progression and lends support to Galileo's law of free fall. However, further experiments in St. Peter's Basilica found the time for the (roughly) 300 foot drop to be slightly larger than implied by the progression. Furthermore, Mersenne's experiments discredited Galileo's claim that an object falls 5 Parisian feet in the first second.

Riccioli conducted free fall experiments in which he dropped balls of different weights from different heights. Riccioli found that the heavier ball actually does land a bit sooner, but that the time to fall is not directly proportional to weight. His results confirmed Galileo's claim that weight is only a secondary factor, but showed that weight had more of an effect than Galileo believed it would. Riccioli also provided evidence for the 1, 3, 5, ¼ progression; his values were 15 Roman feet in the first second, 60 feet in the second, 135 in the third, and 240 in the fourth.

The experiments of Mersenne and Riccioli were not without error. When the same experiments were performed repeatedly (as by Mersenne), different results were recorded. Riccioli's estimate for acceleration is closer to the actual 9.8m/s2 than the estimates of either Galileo or Mersenne, yet it has an error of almost 10 %. This error was probably largely due to the difficulty in obtaining accurate measurements of very small increments of time and to the effects of air resistance. A serious problem for the law of free fall is that the law holds in the absence of air resistance, but there was no way to devise such an environment for experimentation. The best Galileo could do was replace free fall experiments with inclined plane experiments, but Mersenne's data at higher angles called the equivalence of these two motions into question; data from inclined plane experiments could not be used to support free fall claims with any confidence.

Despite these problems, the carefully-conducted free fall experiments of Mersenne and especially Riccioli are reliable enough and achieved accuracy enough not to prove, but to lend substantial and much-needed support to Galileo's law of free fall (and to falsify some of the claims of Aristotelian physics). Both performed the kind of experiments which can be used to prove or disprove laws. They concluded that Galileo's law of free fall held approximately. I believe that as of 1651 (and not before 1651) there was strong support for the law of free fall as an idealized law which holds in limited scope, namely in the absence of air resistance.

Parabolic Trajectory:

The law of free fall was not the only law introduced in Discorsi. Here Galileo also states the law of parabolic trajectory. Galileo proposes that as the motion of a projectile is a combination of two motions - one uniform in a horizontal direction and one accelerated in a vertical direction - its path is that of a semi-parabola (Galileo, 269). If one accepts Galileo's descriptions of horizontal and vertical motion and that these two motions can be combined in this way, the path is a parabola as a simple result of the fact that the horizontal motion is proportional to time, while the vertical motion (free fall) is proportional to the square of time, yielding a curve described by some variation of y = x2, or a parabola. However, we have already seen that the law of free fall is to be accepted only as an approximation. The uniformity of horizontal motion relies on the law of inertia, another law of questionable certainty (Drake, 241), which will be considered later in this paper. Additionally, the idea that the two motions can be combined by a simple summation needs to be supported.

Galileo himself sees several possible objections to the law of parabolic trajectory. One problem lies in the assertion that horizontal and vertical motion are independent and do not affect each other. A second problem is that since the path cannot be a parabola to the center of the Earth, some explanation is needed as to why it is a parabola at all. Thirdly, problems arise because the theory does not take the curvature of the Earth into consideration. And finally, as with the law of free fall, resistance destroys the parabolic trajectory (Galileo, 274). Galileo concedes that horizontal motion is not eternally uniform and that the difference between the Earth's surface and center, its curvature, and air resistance (a minimal effect) are all ignored by his theory. The law of parabolic trajectory, then, (particularly because of the challenge about the center of the Earth), is not an idealization, but is at best an approximation.

There is at least the potential for some evidential verification of the parabolic law. Studies of Galileo's (unpublished) notebooks reveal numerous experiments and give a geometric method for determining the specific parabolic trajectory. As in his study of free fall, Galileo makes use of inclined planes. He explains an experiment involving a bronze ball rolling down an inclined plane, which is then deflected in a horizontal direction and then falls a distance to land on another horizontal surface. This experiment, if performed (there's no evidence to believe it actually was), would allow Galileo to observe a combination of horizontal and vertical motion to study the resulting parabolic curve.

The strength of Galileo's mathematical theory did not come primarily from experimental results, but from its explanatory power. The law of parabolic trajectory provides an explanation for why the maximum range for projectile motion is achieved when a cannon is elevated at a 45 degree angle, a fact already known in military science. Using the law of parabolic trajectory, Galileo is able to provide (symmetric) tables of projectile motion which are useful for military applications (Galileo, 296-307).

While the law of parabolic trajectory has undeniable practical value, it is only an approximate law. Additionally, the law of parabolic trajectory is not foundational; it depends on the law of free fall, an explanation of horizontal motion (inertia), and the lack of cross-talk from the combination of these two types of motion.

In comparison with the evidence for the law of free fall, then, the evidence for the law of parabolic trajectory is weak. Whereas Mersenne and Riccioli provided strong experimental evidence in support of the law of free fall, the strongest evidence for the law of parabolic trajectory may be that a calibrated application of the theory yields usable practical results. This in no way proves the law of parabolic trajectory. The law appears to be an approximation, but there is no strong support for believing it to be an idealization. As the law of parabolic trajectory relies on the law of free fall (for vertical motion), any evidential difficulties facing the law of free fall apply to the law of parabolic trajectory as well. I conclude then that, as of 1651, the law of free fall was the stronger (more supported) of the two laws.

It is worth reiterating that the law of parabolic trajectory relies not only on the law of free fall, but also on the law of inertia (for horizontal motion). As such, it is also subject to the evidential difficulties of the law of inertia. I will then turn to the law of inertia on which the law of parabolic trajectory depends.

Inertia:

Laws of inertia were formulated not only by Galileo (for horizontal motion, particularly as it applies to the parabolic trajectory of projectile motion), but also by Gassendi, Descartes, and Huygens. In this paper I will discuss Descartes' law of inertia (or rather his first two laws of nature, as described in his Principles of Philosophy ( Principia), Part II, 37-44). Descartes' first law is that objects naturally remain in the same state (either in motion or at rest); any change between these two states requires an external cause. His second law states that all movement is in straight lines, not curves. (Descartes also formulated a third law which is typically, though incorrectly, considered to be a precursor to laws of collisions and conservation of momentum.) The primary cause of motion, according to Descartes, is God.

The strength of the law of inertia lies not so much in supporting empirical evidence (the only examples given being that an object thrown continues in motion after it has left the hand, that a rock in a sling attempts to move away in a straight line, and a number of inaccurate examples of collision reaction) but in its explanatory power. Descartes' law of inertia is both simple and comprehensive. Once the principle of inertia is accepted, questions about the continuation of motion disappear. It is not the continuation of motion that needs to be explained; rather, it is the starting and stopping of motion that requires explanation.

Descartes' explanation of inertia is weakened on three points. First, Descartes says that the law of inertia follows from the immutability of God, but he offers no such derivation. This, as well as his appeal to God, appears rather unscientific when compared to the experimentation of Mersenne, Riccioli, and even Galileo. Second, Descartes' law, while focusing on change in speed, does not really cover change in direction. Third, Descartes does not provide much support for the assertion that all motion is in straight lines, rather than allowing for natural curvilinear motion.

Descartes' law of motion does not seem to be an empirical principle (in the same way as the laws of Galileo); instead, it is a possible explanation of change and motion. Descartes offers a way to conceptualize (to make sense of) motion. In some ways, the law of inertia is lawlike: it is comprehensive and holds universally. However, there is no strong evidence in support of the law of inertia, nor any devised method for testing (verifying or falsifying) the law. The law of inertia seems to be at least in part a metaphysical, not scientific, principle.

I will now compare the evidence for the law of inertia with that for the laws of free fall and parabolic trajectory. In many ways Descartes seems to be engaged in a different kind of science than Galileo. Descartes was engaged in a top-down approach to science, attempting to find a single theory to unify a number of phenomena. Whereas the support for Galileo's laws comes largely from experiments, the support for Descartes' law comes from explanatory power. It is then hard to compare Descartes' law with those of Galileo. I think, however, that there was clearly less support for the law of inertia in 1651 than for the law of free fall.

Additionally, I believe that there was less support for the law of inertia than for the law of parabolic trajectory. The law of parabolic trajectory could have been (with some work) verified experimentally. There is no indication that Descartes or others at the time had any methods of providing strong evidential support for the law of inertia. In fact, as the law of parabolic trajectory depends on the law of inertia, demonstrated evidence in support of the parabolic law may have been the strongest indirect evidence for the law of inertia.

Conclusion:

It is important to realize that although there was some evidence for the laws of free fall, parabolic trajectory, and inertia in 1651, the available evidence was not conclusive enough to establish any of these as laws at the time. These laws were at best idealizations or approximations of motion in the absence of resistance effects. If any of the laws had been better established, evidence for the other two might have come from their inter-relatedness. Galileo himself seems to recognize that he is doing something ``new, subtle and conclusive'' with his law of parabolic trajectory by combining the other two laws as he does (Galileo, 273). However, without establishing any of the laws individually, their coherence can only be taken as evidence that the remaining laws are good theories worth advancing.

Although all three laws suffered from similar evidence problems, the status of these laws was different. Strong evidence for the law of free fall came from the careful experiments of Mersenne and Riccioli; evidence for the law of parabolic trajectory came from Galileo's experimentation and useful calibrated results; and perhaps the strongest evidence for the law of inertia was its ability to explain away a question. I conclude that as of 1651 there was the least evidence to support the law of inertia and the most evidence for the law of free fall.

## References

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Descartes, René. Principles of Philosophy. Trans. Valentine Rodger Miller and Reese P. Miller. (Orig 1644.)

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Galilei, Galileo. Dialogues Concerning Two New Sciences. Trans. Henry Crew and Alfonso de Salvio. Prometheus Books, 1991. (Orig 1638.)

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Mersenne, Marin. Harmonie Universelle. 1636.

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Riccioli, Giambattista. New Almagest. 1651.

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Torricelli, Evangelista. De Motu Gravium Naturaliter Descendentium Et Projectorum. 1644.

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Drake, Stillman. Galileo Studies. 1970. pp 214-256.

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Consulted excerpts from essays in Galileo: Man of Science. Ed. Ernan McMullin. 1967. pp 21-22, 71-79, 308-313, 316-319.

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Galileo's Experiments. http://www.fathom.com/feature/122510.

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Projectiles. http://es.rice.edu/ES/humsoc/Galileo/Student_Work/Experiment95/paraintr.html.

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Scientific Laws. http://carnap.umd.edu/phil250/lectures/laws.html.

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On 14 Dec 2002, 20:31.