## Posts Tagged ‘Two New Sciences’

### Bounce, part 5

January 4, 2010

This post is a digression from the topic of the previous parts (1 2 3 4). We’ll move away from discussing how high a tennis ball should bounce when dropped on top a basketball, and into some metadiscussion of the arguments made in the first four parts. It’s a long post as well, but it’ll be good for you, because half the words are Galileo’s, not mine, and he’s a dude worth reading.

Last time, I cited Galileo as our source for understanding uniformly accelerated motion – the motion of a ball dropped or thrown in the air.

Before introducing his idea of what uniformly accelerated motion is, Galileo gives us an extended prelude. It’s long, but I think it’s worth seeing all at once, rather than piece-by-piece.

For anyone may invent an arbitrary type of motion and discuss its properties; thus, for instance, some have imagined helices and conchoids as described by certain motions which are not met with in nature, and have very commendably established the properties which these curves possess in virtue of their definitions; but we have decided to consider the phenomena of bodies falling with an acceleration such as actually occurs in nature and to make this definition of accelerated motion exhibit the essential features of observed accelerated motions. And this, at last, after repeated efforts we trust we have succeeded in doing. In this belief we are confirmed mainly by the consideration that experimental results are seen to agree with and exactly correspond with those properties which have been, one after another, demonstrated by us. Finally, in the investigation of naturally accelerated motion we were led, by hand as it were, in following the habit and custom of nature herself, in all her various other processes, to employ only those means which are most common, simple and easy.

For I think no one believes that swimming or flying can be accomplished in a manner simpler or easier than that instinctively employed by fishes and birds.

When, therefore, I observe a stone initially at rest falling from an elevated position and continually acquiring new increments of speed, why should I not believe that such increases take place in a manner which is exceedingly simple and rather obvious to everybody?

Galileo is mixing two approaches, and they appear to be intrinsically intertwined in his mind. The first is the ultra-skeptical pure empiricism viewpoint. This line of thought says that the only way to know about a thing is to confirm it by experiment. All scientific theories are to be tested against nature. If the theory and experiment agree, we fail to reject the theory. If the theory and experiment disagree, we reject the theory. Many modern scientists cite this as the true scientific viewpoint. (Note that from this point of view, you never confirm a scientific theory. Many scientists will agree with this – you never prove anything to be true in science. Also, I have called this viewpoint “empiricism”, a term which is sometimes used slightly differently in epistemology, where it refers to the belief that knowledge comes from sensory experience in general, rather than scientific experimentation in particular. Nonetheless, the cores of scientific and epistemological empiricism are similar.)

But, along with his statement that his knowledge of falling bodies comes from experiment, Galileo also has curious references to simplicity, in particular some out-of-place stuff about swimming fish and flying birds. This, to me, is the germ of a new idea – an idea that what we learn about nature ought to make sense to us on a deep level, once we’ve learned it. Greek philosophers (so I hear, not having read them) believed the Universe ought to make sense, and that they could therefore understand it with a priori reasoning. This is not quite what Galileo seems to believe. He holds himself responsible to experiment, unlike Aristotle, but I think that if experiment gave strange or unusual results that Galileo couldn’t understand, he’d be extremely dissatisfied. He feels a deep need to take the mathematical results, back them up with data, but then do even more. He needs them to make sense.

Two New Sciences is written as a dialogue (or, there being three interlocutors, a trialogue?), with Sagredo and Simplicio, two men who haven’t learned the new sciences, questioning Salviati, who has learned them and is explaining them to his friends. Galileo uses this device to explore intuition. He has Sagredo and Simplicio raise all manner of interesting objections to Salviati’s ideas, just so Salviati can find interesting answers to allay their unease. (This format is out of style in modern physics text, with rare exceptions like Spacetime Physics, a book I enjoy much more today than I did when first learning special relativity from it six years ago.)

For example, Sagredo thinks there is a problem with saying that a body dropped from rest has a speed proportional to the time fallen. He objects,

…we must infer that, as the instant of starting is more and more nearly approached, the body moves so slowly that, if it kept on moving at this rate, it would not traverse a mile in an hour, or in a day, or in a year or in a thousand years; indeed, it would not traverse a span in an even greater time; a phenomenon which baffles the imagination, while our senses show us that a heavy falling body suddenly acquires great speed.

He thinks there is a disconnect between the math and experiment, because the math says that when you drop something, it has almost no speed after falling a short distance, but Sagredo thinks that when you drop a heavy thing it starts falling quickly immediately. Maybe you don’t have this difficulty of intuition, but if you do, Salviati replies by appealing to an experiment.

You say the experiment appears to show that immediately after a heavy body starts from rest it acquires a very considerable speed: and I say that the same experiment makes clear the fact that the initial motions of a falling body, no matter how heavy, are very slow and gentle. Place a heavy body upon a yielding material, and leave it there without any pressure except that owing to its own weight; it is clear that if one lifts this body a cubit or two and allows it to fall upon the same material, it will, with this impulse, exert a new and greater pressure than that caused by its mere weight; and this effect is brought about by the [weight of the] falling body together with the velocity acquired during the fall, an effect which will be greater and greater according to the height of the fall, that is according as the velocity of the falling body becomes greater. From the quality and intensity of the blow we are thus enabled to accurately estimate the speed of a falling body. But tell me, gentlemen, is it not true that if a block be allowed to fall upon a stake from a height of four cubits and drives it into the earth, say, four finger-breadths, that coming from a height of two cubits it will drive the stake a much less distance, and from the height of one cubit a still less distance; and finally if the block be lifted only one finger-breadth how much more will it accomplish than if merely laid on top of the stake without percussion? Certainly very little. If it be lifted only the thickness of a leaf, the effect will be altogether imperceptible. And since the effect of the blow depends upon the velocity of this striking body, can any one doubt the motion is very slow and the speed more than small whenever the effect [of the blow] is imperceptible? See now the power of truth; the same experiment which at first glance seemed to show one thing, when more carefully examined, assures us of the contrary. (brackets added by translator)

I get the feeling, while reading this passage, that Galileo cites this experiment simply because it gives him pleasure to do so. But in this case, even the experiment is not enough for him. He continues

But without depending upon the above experiment, which is doubtless very conclusive, it seems to me that it ought not to be difficult to establish such a fact by reasoning alone. Imagine a heavy stone held in the air at rest; the support is removed and the stone set free; then since it is heavier than the air it begins to fall, and not with uniform motion but slowly at the beginning and with a continuously accelerated motion. Now since velocity can be increased and diminished without limit, what reason is there to believe that such a moving body starting with infinite slowness, that is, from rest, immediately acquires a speed of ten degrees rather than one of four, or of two, or of one, or of a half, or of a hundredth; or, indeed, of any of the infinite number of small values [of speed]?

Here we see the second approach to nature. The idea that, once we’ve formulated a theory and tested it, we’re still not done. We need to reason about it, too. We need to go back, take the solution, and make it ours. We need to convince our grandmothers, who don’t know math, that this is the way it ought to be. And both these processes are intertwined. You can use the idea that nature ought to be simple to figure out what the laws are, but if you do, you’re still subject to testing them by experiment. Conversely, you can use experiment to figure out the laws, but if you do, you’re still subject to figuring out why things came out that way.

Galileo is the earliest source I’ve seen with this new, sophisticated attitude. Naturalists wanted to observe, discover, and document what happened around us. Philosophers wanted to talk about it in the abstract and explain its deeper logic. But Galileo wanted to do both. And it’s only when you do both that you’ve accomplished the real goal – understanding.

I’m not saying this attitude sprung up in Galileo’s work with no precedent, but I do think it’s clearly evident here, and since Two New Sciences is a landmark work in terms of the physical ideas it presents, it’s important to examine in terms of the philosophical ones is presents, too.

This Galilean principle still guides us today. Science isn’t about testing hypotheses and controlling experiments and statistical significance. Science is about figuring things out. The methods of modern science evolved over time as the problems scientists dealt with demanded them. (A great deal of statistics was invented specifically to study genetic inheritance, for example). Galileo didn’t have our textbook scientific method, but ultimately he didn’t need it to make great progress.

Today we need things like careful laboratory conditions and error propagation formulas to keep us from screwing up when things get tricky and hard to interpret. But the core of my world outlook, which I am not afraid to claim is also the core of the scientific one, is that you are just trying to figure things out, subject to checking what really happens, and then, once you do that, trying to understand.

Next time, I’ll take a look at one of Galileo’s arguments that didn’t work. That’s the other thing about science that I like. Nobody’s perfect, and you’re expected to screw up at least once in a while.

### Bounce, part 4

January 2, 2010

Previous parts: 1 2 3

Last time we made progress on figuring out how high a tennis ball can bounce in the classic experiment where we drop the tennis ball on top a basketball. We didn’t find the answer, but we said that if the tennis ball picks up a speed $v$ in falling, then immediately after bouncing off the basketball, it could have a maximum upward speed of $3v$.

Today we want to figure out what that means in terms of how high the tennis ball will bounce. It turns out that the tennis ball does not bounce three times as high as it started when it rebounds with three times the speed. In fact it bounces much higher.

After bouncing off the basketball, the tennis ball rises, but slows down under the influence of gravity until it comes to a stop at the top of its trajectory. To understand how high it goes, we must answer the question, “what does the influence of gravity do to the motion of the ball?”

One of the first people to understand this question and its answer was Galileo (although several people came to the correct conclusion before him). We’ll look at a few passages of his famous book, Dialogue Concerning Two New Sciences. (specifically this part)

Galileo begins by stating that he thinks “uniformly accelerated motion”, the motion of a tennis ball thrown into the air, should be very simple.

When, therefore, I observe a stone initially at rest falling from an elevated position and continually acquiring new increments of speed, why should I not believe that such increases take place in a manner which is exceedingly simple and rather obvious to everybody? If now we examine the matter carefully we find no addition or increment more simple than that which repeats itself always in the same manner.

In other words, the way the speed of a falling body changes shouldn’t depend on how high it is, or how long it’s been falling, or how far it’s fallen. It should depend on nothing at all – be always the same.

This may be a lot to swallow, but let’s look at one good reason (not due to Galileo) that we might expect the way gravity acts on a falling object not to change with how high the object is above the Earth’s surface. The radius of the Earth is very large compared to the heights we throw things. We expect that if the effects of Earth’s gravity do change with your distance from the center of the Earth, they ought to do so on a distance scale roughly equal to the radius of the Earth.

That is, if you want a significant difference in the force of gravity, you ought to change your position by something significant compared to the radius of the Earth, since it defines the only natural length scale in this problem. The radius of the Earth is roughly six million meters, so throwing a tennis ball up in the air six meters is completely negligible. We could calculate the effects of gravity using Newton’s gravitational law, but that is unnecessary. Any other reasonable gravity law ought to work out basically the same. Near the surface of the Earth, your height should not affect how gravity acts on you.

This is only one part of what Galileo said. For example, he also believes that how fast an object moves should not affect how gravity acts on it. This belief may have been stimulated by the relativity principle – that all laws of physics should be the same, even when you’re moving. Relativity does not absolutely preclude a force that depends on velocity, though (magnetic forces do this), but velocity-dependent forces are not as simple as velocity-independent forces, and for the time being Galileo is guessing that the way gravity acts ought to be very simple.

We continue with the G-spot’s wise words:

A motion is said to be uniformly accelerated, when starting from rest, it acquires, during equal time-intervals, equal increments of speed.

This is Galileo’s working idea of how things fall. If you drop something, and at the end of one second it goes speed $v$, then at the end of two seconds it will be going $2*v$, and at the end of three seconds $3*v$, etc. A plot of speed over time, if we drop an object from rest, should look like this:

This plot shows the speed of a falling tennis ball. The tennis ball is dropped from rest, and so starts at speed zero. Gaining equal speed in each moment of time, the speed is directly proportional to time.

Now that we have a theory for what the speed of the tennis ball does, we should be able to figure out how high it goes. The tennis ball reaches its highest height when its speed is zero, so we simply need to keep track of its speed until that speed falls to zero. If we know how fast it was going and for how long, we should also know how far it traveled.

I’ll paraphrase Galileo’s arguments here rather than quoting them, since he does not directly answer our exact question. The relevant pages are 171 – 178.

First, let us suppose it takes the tennis ball a time $t$ to fall before bouncing, and it acquires speed $v$ in that time. We know it bounces back up with speed $3v$. It loses speed in the same way it gained speed – the same amount per second. So after a time $t$, the ball loses speed $v$, and is down to moving at speed $2v$. The ball comes to a stop at the height of its trajectory after a time $3t$.

To summarize, if the ball gains and loses the same amount of speed in any moment of time, then if it two balls bounce upward, one three times as fast as the other, the fast one will take three times as long to get reach its apex.

The distance the ball travels just $speed * time$, which is the green area shaded in the previous drawing.

Here is a plot of the speed of the ball as it rises:

The tennis ball's return trip. This time it begins going quickly, three times as fast as before, and slows down. It takes three times as long to reach its peak as it took to fall.

It rises three times as long as it fell, and the distance it rises is purple the area in the above chart. Laying the two plots together, we see that the purple area is nine times as large as the green one – three times taller and three times wider.

The green area represents the distance the tennis ball fell (see first figure). The purple area is the area the tennis ball rises after bouncing off the basketball. The tennis ball rises nine times as high as it was dropped from.

Now we have our first answer to how high a tennis ball can bounce when dropped on top a basketball. It can bounce nine times as high, when we make the following assumptions:

• When things bounce off the ground, they change their direction keeping exactly the same speed and hence bounce back to the same height. (first post).
• A basketball is so much bigger than a tennis ball that it essentially acts as the ground – the tennis ball bounces off just the same as it would bounce off the ground. (third post)
• To understand the way something in motion works, we can imagine we are moving alongside it at the same speed so that it isn’t moving from our point of view, and understand it that way. Then we can imagine going back to the frame in which the thing is moving and translating over our new knowledge over. (third post)
• Gravity pulls an object down such that it gives it the same additional amount of speed in each moment of time. (this post)

My original claim was that I could have understood all these ideas as a child. I think that’s right. I was a pretty bright kid, and if someone had sat down to explain this reasoning to me, and answered my questions, I think I’d have gotten it. But I also hope I’d have realized there’s a problem. When you actually do the experiment, the tennis ball doesn’t bounce nine times as high, or anywhere near that. Three times as high is pretty good for this experiment. So I’d like to think I’d have noticed that, and asked for an explanation of the discrepancy.

We began to discuss this in part two, where we looked at why things bounce to a lower height than they’re dropped from. The assumption about reversing direction and speed when bouncing is simply not correct. It is also not correct to assume that the basketball is so much larger than the tennis ball that it acts like the ground, but this is a smaller source of error. It isn’t true that gravity is completely uniform, either, or that the only influence on the falling ball is from gravity. We’ll look at these things in more detail in a later post.

Before doing that, though, the next post or two will continue looking at the passage from Galileo. This passage isn’t interesting to me simply because it is an early source of someone understanding this fairly simple problem. It’s interesting because it’s an illustration of Galileo laying down a more sophisticated understanding of how we can understand nature. I want to look at what Galileo did and didn’t know, but also at how much he understood about what he did and didn’t know, and how he came to his conclusions.

There’s also a very surprising and egregious logical error in the passage, so we’ll talk about that, too, before returning to the tennis ball a little down the line.