Intro: The Cage Match
Do heavier objects fall faster?
Once upon a time, this question was presented as a cage match between Aristotle and Galileo (Galileo winning). As Carlo Rovelli puts it:
…[Aristotle’s physics] is commonly said to state that heavier objects fall faster when every high-school kid should know they fall at the same speed. (Do they??)
and Thony Christie at The Renaissance Mathematicus says:
As is generally well known, having defined fall as natural motion, Aristotle now goes on to elucidate his laws of fall, which, of course, everybody knows were wrong being first brilliantly corrected by Galileo in the seventeenth century. Firstly, Aristotle’s laws of fall are not as wrong as people think, and secondly, they were, as we shall see in later episodes, challenged and corrected much earlier than Galileo.
These quotations already indicate cracks in the old consensus, with better marks for Aristotle. Indeed, Rovelli goes on to say:
I argue here that contrary to common claims Aristotle’s physics is counterintuitive, based on observation, and correct (in its domain of validity) in the same sense in which Newtonian physics is correct (in its domain).
Wikipedia (citing Rovelli) remarks, in a caption:
In Physics he states that objects fall at a speed proportional to their weight and inversely proportional to the density of the fluid they are immersed in. This is a correct approximation for objects in Earth’s gravitational field moving in air or water.
And to quote Thony again:
Aristotle’s laws of fall are actually based on simple everyday empirical observation. If I drop a lead ball from an oak tree it evidently falls to the ground faster than an acorn that I dislodge whilst dropping the ball. In real life not all objects fall at the same speed. It is only in a vacuum that this is the case.
Rovelli also goes to bat for Aristotle as an observer. He writes, “Hard to claim [his law of fall] is not based on good observation.”
Aristotle’s “law of fall” is not purely qualitative. As the Wikipedia caption says, it can be summarized
v ∝ W/ρ
with v being speed, W the weight of the object, and ρ the density of the medium. Aristotle expressed his law in terms of the time to fall a given distance. So, as T ∝ 1/v:
T ∝ ρ/W.
Rovelli devotes a whole section to defending this law. Thony also signs on (“a good first approximation for objects on the Earth falling through air or water”).
I’m sorry, but this is a modern myth. Easy experiments show it’s utterly false that objects fall through air with a speed proportional to their mass.
To give Aristotle his due, the assertion “all objects falling in air fall at exactly the same rate under all circumstances” is also quite false. To cite a famous example, a hammer vs. a feather inspired this NASA video. But Galileo never said that. His Dialogues Concerning Two New Sciences includes an extensive discussion on the effects of air resistance. (Buoyancy in air also gets a nod: see what he says about leather bladders.) For a huge swath of everyday experience, Galileo (and his predecessors) win the cage match.
Thony’s post covers a lot of ground, as does Rovelli’s paper which in addition gets pretty technical. This post is limited to the topic of objects falling through air. I address three questions:
- Does Aristotle’s law of fall survive even a crude confrontation with experience?
- Do we believe Aristotle based his law on careful observations?
- Do Rovelli’s technical arguments pass muster?
As you might have guessed, I will be answering No to all three. The rest of this post is divided into sections:
- Observations
- A historical argument
- Proportionality
- Two kinds of air friction
- Rovelli’s arguments
- Summing up
- Other matters
Let’s start with Thony’s acorn example. I didn’t have an acorn handy, but a crumpled up piece of paper has mass about 4g, which is the average mass of an acorn (according to the ever-trustworthy internet). As the paper ball has a much larger volume, that should exaggerate the effects of air friction.
Not wanting to drop a lead ball onto the floor, I used an onyx paperweight instead (468g). According to Aristotle, the falling times over a given distance should be inversely proportional to the masses, that is 117:1. OK, let’s view the results:
Here’s another drop in slow motion:
The distance is roughly 1.5m, and the time maybe ½ sec. Let’s say that’s the time for the paperweight. So the ball of paper should have taken over a minute to fall!
In the slow motion video, I didn’t release the two objects at quite the same time. (Not that easy to do by hand!) But can you seriously claim that the paperweight is catching up, let alone passing it?
(Incidentally, one professor has used this difficulty in dropping the two objects simultaneously, to argue that Galileo really did perform the tower experiment. This remains a minority viewpoint among historians.)
Oh, but you say, that’s only a meter and a half drop! What about greater heights?
When I used to teach high-school physics, I regularly performed a lab dropping a bunch of objects from a upper window of the school to the grass outside, ∼7m. Students took measurements with timers providing a precision of 0.01 secs. The heaviest object was a bowling ball, the lightest a fake golf ball (something like a ping-pong ball). In between was a tennis ball with a fuzzy surface, and a real golf ball. I think you’d expect the tennis ball or the fake golf ball to experience the most air friction. The results, averaged over many students:
- Bowling ball: 6200g, 1.17s
- Tennis ball: 59g, 1.27s
- Golf ball: 45g, 1.15s
- Fake golf ball: 5g, 1.64s
The usual s=4.9t2 formula gives a time of 1.2s to fall 7m, so these are rather good. (I’d expected reaction time to make more of a difference, but perhaps some students had trigger-happy fingers.) In any case, Aristotle’s law predicts 1240:1 for the ratio of bowling ball to fake golf ball times. So the fake golf ball should have taken over 20 minutes to reach the ground! Or else the bowling ball should have reached it in about a 1000th of a second.
The time difference between the fake golf ball and the bowling ball was around ½ sec, easily perceptible. (According to this, 0.1s is about the “threshhold where actions are perceived as ‘instant’”.) The substitutions from Thony’s example (acorn⇒fake golf ball, lead ball⇒bowling ball, tree⇒window) seem reasonable, so that claim is confirmed. But to say, “Aristotle’s laws of fall are actually based on simple everyday empirical observation”—I think that’s stretching things a bit.
No one denies that Aristotle used his eyes. But the historical debate is between an empirical and a priori origin for his law. Thony’s description seems to preclude an a priori origin. I don’t buy an empirical base for a theory that gets such easy examples so very wrong.
How about the much weaker claim that “heavier objects fall faster”? (After all, this is technically true in air.) Well, we have some instances supporting that argument, but many examples go the other way. With the bowling ball/golf ball comparison, the time difference would not be perceptible, despite the 138:1 ratio. For really everyday experience, say 2m heights, even the acorn/lead ball time difference would not be noticeable.
And the notion that the “heavier falls faster” meme got its start from such acorn/lead ball races—that falls apart when you think about it. How often do you see, in everyday life, a lead ball (or a stone) and an acorn dropped simultanously from a tree? But suppose you did. What if the acorn hit the ground a fraction of a second before the lead ball? Would you exclaim, “Hey, lighter objects fall faster than heavier ones!” Or would you conclude that the acorn must have had a slight headstart? The meme derives not from common experience, but from psychological factors. In other words, a priori, not empirical.
Galileo gave an eloquent rebuttal to this Aristotelian defense:
But, Simplicio, I trust you will not follow the example of many others who divert the discussion from its main intent and fasten upon some statement of mine which lacks a hair’s-breadth of the truth and, under this hair, hide the fault of another which is as big as a ship’s cable. Aristotle says that “an iron ball of one hundred pounds falling from a height of one hundred cubits reaches the ground before a one-pound ball has fallen a single cubit.” I say that they arrive at the same time. You find, on making the experiment, that the larger outstrips the smaller by two finger-breadths, that is, when the larger has reached the ground, the other is short of it by two finger-breadths; now you would not hide between these two fingers the ninety-nine cubits of Aristotle, nor would you mention my small error and at the same time pass over in silence his very large one.”
—Dialogues Concerning Two New Sciences, Day 1
You may be wondering, “If Aristotle’s law of fall was so obviously wrong, why did it take almost 2000 years for people to realize it was bad?”
It didn’t. As far as I know, the earliest recorded refutation comes from John Philoponus, who lived c.490–c.570CE. As quoted in Clagett, Greek Science in Antiquity:
But this [view of Aristotle] is completely erroneous, and our view may be corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights of which one is many times as heavy as than the other, you will see that the ratio of the times required for the motion does not depend on the weights, but that the difference in time is a very small one. And so, if the difference in the weights is not considerable, that is, if one is, let us say, double the other, there will be no difference, or else an imperceptible difference, in time, though the difference in weight is by no means negligible, with one body weighing twice as much as the other.
(Wikipedia includes part of this quote.)
Let’s sub Philoponus in for Galileo in the cage match. Rovelli writes:
Aristotle’s physics is the correct approximation of Newtonian physics in a particular domain, which happens to be the domain where we, humanity, conduct our business.
and
The terrestrial physics of Aristotle matches perfectly the Newtonian one in the appropriate regime. It is definitely not true that objects with different weights fall at the same speed, in any reasonable terrestrial regime.
and in support of Aristotle the observer:
Aristotle’s detailed theory however, as well as —seems reasonable— Aristotle’s detailed observations leading to it …
If these all hold true, then how did Philoponus come to his realization? Why did his observations directly contradict Aristotle’s supposed “detailed observations”? You can’t claim that Aristotle was a good observer without saying Philoponus was a bad one.
Philoponus wrote roughly 900 years after Aristotle, still pretty long. Why no objections earlier? A few thoughts:
- Was Aristotle’s law of fall universally accepted throughout that period? Probably not. The historical record is incomplete but suggestive. Clagett mentions Strato the Physicist (aka Strato of Lampsacus, c.335–c.269 BCE), “whose views often differed radically from those of Aristotle … Strato had an apparently deserved reputation as an experimenter. … Strato composed a work, now lost, called On Motion.” The astronomer Hipparchus (c.190–c.120 BCE) wrote a treatise, now lost, called On Bodies Carried Down by Weight. Philoponus may have gotten many of his ideas from Hipparchus, according to Clagett.
- How much attention was devoted to this topic? Aristotle left writings on nearly everything; plenty of fodder for later scholars without going down this path.
- The traditional explanation: medieval thinkers followed Aristotle uncritically, repeating his positions without ever testing them. We now know this is far from true. But that doesn’t mean it’s 100% false.
I’ve seen this defence of Aristotle’s law of fall (paraphrasing): “Don’t take him literally! He wasn’t big on math; when he says ‘inversely proportional’ he’s just using it metaphorically—the same way people say ‘exponentially greater’ when they have only the vaguest notion of y=ex.” I call this the poor math skills defence.
Okay, let’s look at what Aristotle wrote. First, in On the Heavens (Bk 1 Part 6):
A given weight moves a given distance in a given time; a weight which is as great and more moves the same distance in a less time, the times being in inverse proportion to the weights. For instance, if one weight is twice another, it will take half as long over a given movement.
That last sentence sure looks like what a teacher says to illustrate the concept. Aristotle mentions the 2:1 ratio each time he refers to proportionality. Even if you claim (absurdly) that this is the only ratio he really understood, in this very case the difference in times will generally be imperceptible. Moreover, if you look at the surrounding text, you’ll find the discussion conversant with more sophisticated aspects of proportionality theory, like commensurability and anthyphairesis.
In the Physics (Bk IV Part 8), Aristotle discusses the T∝ρ part of his law:
We see the same weight or body moving faster than another for two reasons, either because there is a difference in what it moves through, as between water, air, and earth, or because, other things being equal, the moving body differs from the other owing to excess of weight or of lightness. … For let B be water and D air; then by so much as air is thinner and more incorporeal than water, A will move through D faster than through B. Let the speed have the same ratio to the speed, then, that air has to water. Then if air is twice as thin, the body will traverse B in twice the time that it does D…
Again, no problems with the idea of proportionality.
In the two passages, he gives (confusingly worded) arguments that ρ=0 and W=∞ are impossible. Basic idea: the proportionality T ∝ ρ/W would imply T=0. Aristotle is in full command of the limiting behavior of his law.
Say what you will about Aristotle, he was highly educated and intelligent. He hung out at Plato’s Academy for a couple of decades, a place associated with some first-rate mathematicians (Theaetetus, Eudoxus, and Archytas), and with the famous (apocryphal) geometry entrance requirement. Ratio and proportion were well-established in Greek math before Aristotle was born. The noted historian Sir Thomas Heath had this to say in his A Manual of Greek Mathematics:
Aristotle was clearly not a professional mathematician … but he was fond of mathematical illustrations … The text-book in use in the Academy would no doubt be that of Theudius, and this may have Aristotle’s source. … Aristotle was apparently acquainted with Eudoxus’ theory of proportion; he frequently uses the terminology of proportions, and he defines similar figures as Euclid does.
There’s only one reason for pleading mathematical incompetence on Aristotle’s behalf: to excuse the abject failure of his law of fall.
Before plunging into Rovelli’s arguments, we need to clarify some stuff about fluid friction. (The website of Michael Fowler is an excellent resource.)
Fluid friction (or drag) comes in two varieties: inertial and viscous. Consider an object falling through the air or water. It has to push the fluid out of the way: this causes inertial drag. The fluid will also cling to the surface of the object in a very thin layer, and forces between the molecules of the fluid will serve to hinder the object’s motion. That’s the cause of viscous drag.
Viscous drag is governed by Stokes’ law:
Fviscous ∝ aηv
where a is a characteristic size for the object (e.g., the radius for a sphere), η is the viscosity of the fluid, and v is the speed of the body. The constant of proportionality depends on the shape of the object, and its orientation as it falls.
Inertial drag is governed by a law attributed to Rayleigh:
Finertial ∝ a2ρv2
where ρ is the density of the fluid. The constant of proportionality will not generally be the same as for viscous drag, but it will depend on the shape and orientation of the object.
The ratio of inertial to viscous drag is the Reynolds number R ∝ avρ/η. The definition of the Reynolds number is
R = 2avρ/η.
The Reynolds number, as the ratio of two forces, is dimensionless. So a high Reynolds number means that inertial drag predominates, while a low Reynolds number says that viscous drag is the most important.
In 1976, E. Purcell gave a lecture on “Life at low Reynolds number” (Amer. J. of Physics, v.45 no.1 Jan. 1977) that included this passage:
Now consider things that move through a liquid. The Reynolds number for a man swimming in a liquid might be 104, if we put in reasonable dimensions, for a goldfish or a tiny guppy it might get down to 102. For the animals that we’re going to be talking about, as we’ll see in a moment it’s about 10−4 or 10−5. For these animals inertia is totally irrelevant. We know that F=ma, but they could scarcely care less. … we are going to be taking about objects which are the order of a micron in size. … If I have to push that animal to move it, and suddenly I stop pushing, how far will it coast before it slows down? The answer is, about 0.1 angstrom. And it takes it about 0.6 microsec to slow down. I think this makes it clear what low Reynolds number means. Inertia plays no role whatsoever. If you are at very low Reynolds number, what you are doing at the moment is entirely determined by the forces that are exerted on you at that moment, and by nothing in the past.
to which he later added the footnote:
In that world, Aristotle’s mechanics is correct!
Purcell does not, however, credit Aristotle with anticipating bacteria.
The ratio ρ/η varies by orders of magnitude depending on the medium and other factors like temperature. For air and water at around 20°C, the water ratio is about 20 times the air ratio. (Water is 1000 times more dense, but only about 50 times more viscous.) Purcell’s goldfish, gasping in air, would experience a Reynolds number as low as 5. So viscosity still makes a minor contribution.
For viscosity to be important in air, we need av to be small. Millikan’s oil droplets, motes of dust, or Purcell’s micron-sized objects all qualify. For roughly spherical objects a couple of centimeters in size (like an acorn), the speed must be very small. If you work it out, you’ll find that for air at the usual temperatures and pressures, R∼1 when v is around a millimeter per second. (For comparison, snails crawl at an average 13 mm/sec.) Except for motes of dust and such, we can neglect viscous friction for almost all of an object’s fall.
Aristotle’s physics stands accused of gross inaccuracy, with its author accused of having arrived at his laws a priori, rather than from careful observation. Rovelli rejects both charges.
The heart of Rovelli’s argument is an analogy:
Aristotle:Newton=Newton:Einstein
I show in this note that the technical relation between Aristotle’s physics and Newton’s physics is of the same nature as the relation between Newton’s physics and Einstein’s.
At a speed of 13,000 km/sec, special relativistic corrections remain less than 1/10th of 1%. The corrections from general relativity are even smaller for terrestial and solar-system phenomena; that’s why expermental evidence for the theory remained slight for the first 40 years, and really began to accumulate only in the 1970s.
How does the Aristotle:Newton side of the analogy compare? In Butterfield’s classic The Origins of Modern Science, he described the crucial change as “picking up the opposite end of the stick”. He means that instead of starting with the idea that heavier falls faster, begin with the notion that all weights fall at the same speed. Then account for the exceptions with friction and buoyancy. (Likewise for the two contestants for motion on level ground: “Things stop when you stop pushing”, vs. “Things glide on forever.” But I will stick to falling objects.)
As I said before, no one denies that Aristotle used his eyes. But Rovelli’s proposal of “detailed observations” should mean more than this. It should definitely include dropping, say, an acorn and a rock a few feet (or a crumpled ball of paper and an onyx paperweight), and noticing that they hit the ground at practically the same time.
Of course, that’s just one example. We’ve a wide variety of cases to consider: motes of dust dancing in the sunlight, leaves fluttering down, maple whirligigs spinning to the ground, balloons floating along… Neither Aristotle nor Galileo could have hoped to capture this diversity in a single formula. I think it’s clear that the relevant “domain of discourse” consists of objects that fall straight down, not in zig-zag paths. Thus, relatively compact and at least moderately heavy. The T ∝ ρ/W law is a very bad approximation—hardly worthy of the name—for distances of a few meters. What is Rovelli’s defence?
Rovelli, like a good attorney, indulges in some clever distractions. Here’s one example. After writing Aristotle’s law as v ∼ W/ρ, he notes that for Aristotle’s proof of the impossibility of a vacuum, we need only this: v→∞ as ρ→0. Rovelli takes this as justification to change the formula to
Why a power law, rather than some other function with the same limiting behavior? The reason becomes clear in the next section. Rovelli finds a regime in which
With a flourish, he addresses the jury: “What Aristotle does not have is only the square root, namely n=½, which would have been hard for him to capture given the primitive mathematical tools he was using. His factual statements are all correct.”
How well does this hold up? First off, a square root dependence (or quadratic dependence, viewed from the other side) was no stranger to Greek mathematics of the period.
This is a carefully calibrated version of the “poor math skills” defence. On the one hand, we are not to believe that Aristotle meant what he wrote. On the other hand, had Aristotle only known about square roots, he’d have gotten it right! (Guess he skipped class that day.) The argument also has the flavor of cherry-picking: we can derive a formula using Newtonian physics that matches something Aristotle might have said (but didn’t)—so give him credit for anticipation! To quote Butterfield again:
…the whole fabric of our history of science is lifeless and its whole shape is distorted if we seize upon this particular man in the fifteenth century who had an idea that strikes us as modern, now upon another man of the sixteenth century who had a hunch or an anticipation of some later theory…
But the modified formula still doesn’t work. The bowling ball/fake golf ball mass ratio was 1240:1; taking the square root, we get 35:1. The predicted time for the fake golf ball becomes about 40 seconds instead of 20 minutes. An improvement, but nowhere near the observed time (under 2 seconds). The crumpled paper would take over 5 seconds to hit the floor, flatly contradicted by the video.
Rovelli fixes this with a trick I call regime change; lawyers might refer to it as forum shopping.
He begins by noting the Newtonian forces on a falling object: gravity, buoyancy, fluid friction, and any external forces. For “natural motion” the external forces are 0. He makes the obvious approximation of mg for gravity near the earth’s surface. He includes (quite properly) only inertial friction, implicitly assuming a high Reynolds number. For the part of the discussion that concerns us, he excludes buoyancy.
He now sets up and solves the resulting differential equation for v as a function of time. The result:
![v(t)=\sqrt{\frac{W}{C\rho}}\tanh\left[\sqrt{WC\rho}t\right]](https://s0.wp.com/latex.php?latex=v%28t%29%3D%5Csqrt%7B%5Cfrac%7BW%7D%7BC%5Crho%7D%7D%5Ctanh%5Cleft%5B%5Csqrt%7BWC%5Crho%7Dt%5Cright%5D&bg=ffffff&fg=333333&s=0&c=20201002)
He points out the two phases of fall, the so-called transient phase that lasts about 
Certainly we have to consider the steady state regime. The classic coffee filter experiment of intro physics labs takes place there. Rovelli offers the example of “a stone left at high altitude by a bird”—maybe Alexander had an eagle that Aristotle trained to drop pairs of stones?
But Rovelli goes much further, demoting the transient phase to irrelevance. Some quotes, with my bolding:
Aristotle’s physics is the correct approximation of Newtonian physics in a particular domain, which happens to be the domain where we, humanity, conduct our business.
… everything around us is immersed in a fluid. Aristotle’s physics is a highly nontrivial correct description of these phenomena, without mistakes, and consistent with Newtonian physics, in the same manner in which Newtonian physics is consistent with Einstein physics in its domain of validity…
The terrestrial physics of Aristotle matches perfectly the Newtonian one in the appropriate regime. It is definitely not true that objects with different weights fall at the same speed, in any reasonable terrestrial regime.
Virtually everything of Aristotle’s theory of motion is still valid.
And finally a dig at Galileo:
(Two heavy balls with the same shape and different weight do fall at different speeds from an aeroplane, confirming Aristotle’s theory, not Galileo’s.)
As noted earlier, Galileo nowhere said “precisely equal speeds in all cases”, discussing air resistance and buoyancy and finally concluding with the explicit statement, “we are justified in believing it highly probable that in a vacuum all bodies would fall with the same speed” (my emphasis). Calling the abbreviated form “Galileo’s theory” after grafting a square-root onto Aristotle’s law is a bit much.
Of course Rovelli is well aware of transient phase. Here’s how he justifies disregarding it:
The transient phase during which a body reaches the constant falling velocity is generally too short for a careful observation. For a piece of metal falling in water its duration is often below our ability to resolve it. For an heavy object (like a stone) falling for a few meters, the time taken to fall is comparable with the transient phase time, therefore the stone does not have the time to reach the steady phase. But such a phenomenon implies fast velocities which again are hard to resolve with direct observations (unless one is as smart as Galileo to guess, correctly, that an incline would slow the fall without affecting its qualitative features.)
and further on:
This phase is either too short (in water) or too rapid (for very heavy objects in air) for any careful observation. This phase, on the other hand, is relevant for the short fall of heavy objects, which is the regime on which Galileo (fruitfully) concentrated, circumventing the difficulty of observation by the ingenious trick of the incline. For this regime, it was already pointed out as early as by Philoponus in the VIth century, that the speed of fall is not proportional to the weight…
Notice the phrases “heavy object (like a stone)”, “very heavy objects in air”. I wouldn’t call my 4 grams of crumpled paper, or my 5 gram fake golf ball, “very heavy”. Nor even the golf ball or tennis ball, each about 50 grams.
We’ve seen Rovelli deploy the “poor math skills” defence. To this he now adds the “weak observational skills” defence. It won’t wash. As we’ve seen, Aristotle phrases his law in terms of times. It’s quite easy to observe that the times are nearly the same for objects with very different weights—certainly nowhere close to the mass ratio. And his call out to Philoponus reminds us of the historical argument I presented in section 2.
Just how common is the steady state phase in “the domain where we, humanity, conduct our business”? Using one of the convenient free fall calculators on the internet, I came up with some scenarios.
I modeled an acorn by letting mass = 5g, radius = 1cm, and a roughly spherical shape (using a drag coefficient of 0.5). This yields a terminal speed of 25 m/s (according to the calculator). Then I let it fall from various heights:
“Acorn” (mass=5g, radius=1cm)
| Height (m) | Time (s) | Final speed (m/s) |
|---|---|---|
| 10 | 1.5 | 13.0 |
| 20 | 2.1 | 17.1 |
| 30 | 2.7 | 19.5 |
| 40 | 3.2 | 21.1 |
| 50 | 3.6 | 22.2 |
| ∞ | ∞ | 25.0 |
The internet tells me that some oaks can grow as tall as 44m, though the average is 15m–21m. Few acorns will make it to the steady state phase.
How about my crumpled paper ball? This should reach the steady state phase faster. Indeed it does.
Paper ball (mass=4g, radius=3cm)
| Height (m) | Time (s) | Final speed (m/s) |
|---|---|---|
| 2 | 0.7 | 5.3 |
| 4 | 1.0 | 6.5 |
| 6 | 1.3 | 7.0 |
| 8 | 1.6 | 7.3 |
| 10 | 1.9 | 7.4 |
| ∞ | ∞ | 7.5 |
Many, many everyday objects will take much longer. For example, a lime.
Lime (mass=100g, radius=3cm)
| Height (m) | Time (s) | Final speed (m/s) |
|---|---|---|
| 20 | 2.1 | 18.5 |
| 40 | 3.0 | 24.5 |
| 60 | 3.8 | 28.2 |
| 80 | 4.4 | 30.7 |
| 100 | 5.1 | 32.4 |
| ∞ | ∞ | 37.4 |
You could quibble with some of my choices. Should I have used another figure for the drag coefficient? Some sources suggest a value less than 0.5 for roughly spherical objects, which would increase the transient phase. What cutoff percentage defines the transient/steady state boundary? But it’s clear enough that the transient phase has a much better claim on the realm of humanity’s everyday business. In reality, excluding either of these phases from ‘everyday life’ is artificial and unjustified.
Objects don’t have to reach the steady state phase for a difference in falling times to become perceptible. But then, most everyday drops are from shoulder height or less, not even 2m for most people. And remember: we are evaluating which is a better approximation, a better starting point: times are equal versus time is inversely proportional to weight. (Or the square-root of weight, if you buy Rovelli’s reasoning.)
Here is Rovelli’s argument for the defence, as I see it. (In my role as the prosecutor, I will be snarky.)
- Aristotle based his law on detailed observations.
- But he didn’t notice that dropping a heavy/light pair of objects usually results in them hitting the ground at almost the same time.
- The only appropriate examples to consider, when talking about everyday experience, are dropping fairly light objects at least several meters.
- (Make that 100 meters for something like a lime.)
- That’s because things happen too fast for the first couple of meters.
- Aristotle didn’t care much about math, so we shouldn’t take his passages literally.
- But he anticipated the law
, leaving out the square-root because he hadn’t heard of them.
You may ask, if everyday experience speaks so loudly for the Philoponus-Galileo position, why did Aristotle ever propose such a bad law? And why do so many people find it plausible? I’m not a cognitive psychologist, but I do have some thoughts. These come partly from reading the work of Hestenes’ group on the Force Concept Inventory (FCI), and from my time teaching high-school physics (and talking with my students).
Our intutive beliefs about how the physical world works do not derive simply and directly from our sense impressions. Psychological factors play a major role. A heavy object has more “heft” than a light one. Most people feel that if you push something harder, it will move faster. Why should gravity be any different? The FCI has documented many misconceptions associated with the “active force” notion.
Perhaps linguistic factors matter too. Let’s play the game of “which one doesn’t belong?”
bigger; stronger; heavier; equal; higher; faster
I confess that even now, knowing all I know, I still feel the pull of “heavier falls faster in proportion to weight”. It seems natural. Rovelli lists several counterintuitive aspects of Aristotle’s physics. His law of fall is not one of them.
Rovelli concludes his essay:
Why don’t you just try [it]: take a coin and piece of paper and let them fall. Do they fall at the same speed? … It is curious to read everywhere “Why didn’t Aristotle do the actual experiment?”. I would retort: “Those writing this, why don’t they do the actual experiment?”. They would find Aristotle right.
I am probably over-sensitive in feeling this slanders Philoponus and Galileo and generations of physics teachers. Anyway, I did this experiment. (I took the liberty of crumpling the paper.) The result looked just like the videos of section 1.
My focus is Aristotle’s law of fall for objects in air, as I said at the top. Both Thony and Rovelli discuss much else: objects in water, the four element theory, Aristotle’s astronomy, his four causes, and his role in the history of science. In some of these areas Aristotle cuts a better figure.
Just because Aristotle blew it with falling objects doesn’t mean he hurt the development of physics. That’s one common position, but good arguments can be offered in the other direction. Even his errors may have acted as a prod for the emergence of better theories. Rovelli makes this point, along with many other intriguing ideas. It’s especially valuable to read the views of a research physicist on the nature of scientific progress.
Steven Weinberg in his book To Explain the World: The Discovery of Modern Science has a much more adverse evaluation of Aristotle’s influence. Even so, I was struck by many points of agreement in their two takes.
Thony’s post is, as usual, a masterful survey, giving a historian’s viewpoint (and I believe, the consensus among modern historians). Unlike Rovelli’s full-throated championing of Aristotle’s physics, Thony aims mainly to correct the old-fashioned “Aristotle gets an F” perspective. I suspect my points of disagreement come entirely from his relying on the Wikipedia quote.
So: Aristotle, good or bad for physics? To determine the answer experimentally, we should invent a time machine, go back and have Alexander kill Aristotle with a javelin, and then check what’s happening in Italy in the 1500s…
Diagonal Argument 