Posts Tagged ‘falling bodies’

Air Resistance as an Analog Relativity Computer

October 25, 2010

A Good High School Science Fair Idea

Suppose you drop a hamster off a tall building. It will fall, initially with an acceleration of g, about 10 m/s^2. But after the hamster has swept out a volume of air whose mass equals the hamster’s own, the effects of air resistance dominate and the acceleration is small.

A hamster is about as dense as water, which is about a thousand times as dense as air. So air resistance becomes important when a hamster falls 1000 times its own thickness. The thickness of a spread-eagled hamster is 5 cm, so air resistance dominates falls over 50m.

(Compare this to a sheet of paper. A sheet of paper is as dense as a hamster, but 1000 sheets of paper are only as thick as a 1000-page book, so for the sheet of paper, air resistance is important even if you fall just a few centimeters.)

50m is about 20 stories, so if we want the speed of the hamster when dropping it off a building at least that tall, we had better include air resistance.

A reasonable model for air resistance is that its force is proportional to the square of the hamster’s velocity, because if the hamster falls twice as fast, it strikes the air twice as hard, but also strikes twice as much air per second. Thus

F_{air} = -kv^2

for some constant k. To use Newton’s second law, include the force of gravity and set the force equal to ma.

F_{air} + F_{grav} = m a

-kv^2 + gm = m a

Let k/m = b and you get

-bv^2 + g = a = \frac{\textrm{d}v}{\textrm{d}t}

This is a separable equation. We get

\textrm{d}t = \frac{\textrm{d}v}{-b v^2 + g}

which integrates to

t+c = \frac{1}{\sqrt{bg}}\textrm{arctanh}\left(\sqrt{\frac{b}{g}} v\right).

Solving for v yields

v = \sqrt{\frac{g}{b}}\tanh\left(\sqrt{bg}(t+c)\right)

If the hamster is dropped from rest, the initial condition gives c = 0 so

v = \sqrt{\frac{g}{b}}\tanh\left(\sqrt{bg}t\right)

The hamster speeds up, approaching its terminal velocity

v_{ter} = \sqrt{\frac{g}{b}}

To estimate the terminal velocity, go back to the heuristic that air resistance dominates when the hamster sweeps out its own mass worth of air. That happened at 50m, and in a 50m free-fall you accelerate to about 30m/s, or 70mph (the hamster should survive, I’m told).

If we measure time in units of \sqrt{gb} and then choose units of length so that the velocity \sqrt{g/b} = 1, the hamster’s velocity during free fall is just

v = \tanh(t)

An Impractical High School Science Fair Idea

Suppose the hamster survives, so we put it in a rocket blasting into deep space. We stay behind on Earth. The hamster has constant acceleration g, as viewed from its own reference frame.

The effects of relativity will be important when gt \approx c, which happens after about a year. If the hamster’s trip will be longer than that, we better include the effects of relativity.

At a given moment, the hamster is going along at some speed v. To transform from the reference frame of Earth to the hamster, we use the Lorentz transformation

\left( \begin{array}{cc} \cosh\theta & \sinh\theta\\ \sinh\theta & \cosh\theta \end{array} \right) \left(\begin{array}{c} x \\ t \end{array} \right) = \left( \begin{array}{c} x' \\ t' \end{array}\right)

where \theta is the rapidity, defined by

\tanh\theta = v/c. We’ll choose the proportion of units of length and time such that c = 1.

The hamster accelerates, so this transformation depends on time. Let’s try to find \theta(\tau), the rapidity as a function of the proper time experienced by the hamster.

In a small proper time \textrm{d}\tau, the hamster accelerates by g \textrm{d}\tau. The rapidity of this transformation is given by

\tanh\theta' = g \textrm{d}\tau.

For small \theta', \tanh\theta' = \theta', so

\theta' = g \textrm{d}\tau.

We can measure time in units of 1/g. Then the transformation from the hamster’s comoving frame from one moment to the next is

\left( \begin{array}{cc} \cosh \textrm{d}\tau & \sinh \textrm{d}\tau \\ \sinh \textrm{d}\tau & \cosh \textrm{d}\tau \end{array} \right) \left(\begin{array}{c} x' \\ t' \end{array} \right) = \left( \begin{array}{c} x'' \\ t'' \end{array}\right)

To find the hamster’s new velocity in Earth’s frame, we simply compose the transformations

\left( \begin{array}{cc} \cosh \textrm{d}\tau & \sinh \textrm{d}\tau \\ \sinh \textrm{d}\tau & \cosh \textrm{d}\tau \end{array} \right) \left( \begin{array}{cc} \cosh\theta & \sinh\theta\\ \sinh\theta & \cosh\theta \end{array} \right) \left(\begin{array}{c} x \\ t \end{array} \right) = \left( \begin{array}{c} x'' \\ t'' \end{array}\right)

That’s an easy matrix multiplication – it’s almost a rotation matrix. We get

\left( \begin{array}{cc} \cosh(\theta + \textrm{d}\tau)& \sinh(\theta + \textrm{d}\tau)\\ \sinh(\theta + \textrm{d}\tau) & \cosh(\theta + \textrm{d}\tau) \end{array} \right) \left(\begin{array}{c} x \\ t \end{array} \right) = \left( \begin{array}{c} x'' \\ t'' \end{array}\right)

We see that \theta(\tau + \textrm{d}\tau) = \theta(\tau) + \textrm{d}\tau

Since \theta(0) = 0 (i.e. the hamster has 0 speed when t = 0), this integrates to

\theta(\tau) = \tau.

Returning to the velocity v = \tanh(\theta) gives

v = \tanh(\tau).

The velocity of the hamster dropped from the tall building, as a function of time, follows the same equation as the velocity of the hamster accelerating in the spaceship, as a function of proper time. All we have to do is substitute c = v_{ter} and, choosing the correct units of length (depending on the constant in the equation for the hamster’s drag), the analogy is complete.


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.

The Renaissance Man Uniform Gravitational Acceleration SMACKDOWN

October 9, 2009

Matt at Built On Facts posts about coriolis forces, and points out that a falling body is deflected by them one eighth as much as one tossed from the ground to the same height, and that they’re deflected in opposite directions. Here’s my attempt to explain this intuitively.

This makes me think of the competing da Vinci – Galileo laws for bodies (not their own I hope) falling freely under gravity. They stated their rules in the same basic way. I remembered these laws from watching The Mechanical Universe in high school – before taking physics from the real life version of David Goodstein three years later.

da Vinci said (or so I hear, I never met the guy) that if you fall one unit of distance in the first unit of time, you’ll fall two in the next unit, three in the one after that, then four, etc. So if you fall 5 meters in one second, in the next you’ll fall another 10 for 15 meters total.

Galileo said almost the same thing, but with odd numbers. If you fall one unit of distance in the first second, then in the second you fall three, then five more, then seven, etc. So if you again fall 5 meters in one second, in the next you’ll fall another 15, for 20 total.

Galileo was right; da Vinci wrong. But let’s not screw over our primitive-flying-device-making friend with such a cursory examination. They’re both awesome dudes, as Leonardo’s testudine counterpart would say.

Galileo was right because acceleration is constant, so the distance fallen is proportional to the square of the time. Adding Galileo’s odd numbers gives a square number. 1+3+5 = 9, for example. This is easy to see from a picture.

Each new section adds the next odd number worth of dots, and takes you to the next bigger square number when counted as a whole.

Each new section adds the next odd number worth of dots, and takes you to the next bigger square number when counted as a whole.

da Vinci, instead of the square numbers for total distance fallen, gave the triangular numbers. 1 + 2 + 3 = 6, which is triangular. This has its own picture.

According to da Vinci, each new row is how much you fall in one additional second.

According to da Vinci, each new row is how much you fall in one additional second.

da Vinci’s fub may have been in misunderstanding the relationship between speed and distance. If da Vinci’s rule had been giving the speed at the end on each second, rather than the incremental distance fallen, he’d have been right. If you’re going 10m/s after one second, you go 20m/s after two, and 30 m/s after three, etc. The problem is that you can’t find the distance traveled in a second by taking the speed at the end of that second and multiplying by time. If you do that, you get only an approximation to the correct integral, like this:

Dont worry about the numerical details.  I stole this from the internet somewhere.  da Vincis law overestimates distance fallen every second by assuming your speed at the end of the second was you speed for the entire second.

Don't worry about the numerical details. I stole this from the internet somewhere. da Vinci's law overestimates distance fallen every second by assuming your speed at the end of the second was you speed for the entire second.

It’s possible that da Vinci was actually right on about the kinematics, but that he made a mathematical error in reporting his result. I wanted to follow up on this, so I checked online to see precisely what Leonardo said. I did not succeed. Fritjof Capra’s book quotes da Vinci:

The natural motion of heavy things at each degree of its descent acquires a degree of velocity. And for this reason, such motion, as it acquires power, is represented by the figure of a pyramid.

But when I search online texts of Da Vinci’s notebooks, I can’t find this passage. I can’t find the relevant passages in my Dover copy of Richter’s translation, either. In fact, I can’t find this passage anywhere else on the entire internet, except one book that doesn’t cite the source. So I’m not sure what to make of this. da Vinci’s writings on falling bodies must be somewhere, if we know about them. But as of now I’m still uncertain. Based on the preface to my translation of the notebooks, it looks like they decided to omit some of Leonardo’s physics, since that is obviously unimportant and uninteresting to readers of his notebooks.

Let’s assume Leo had the right idea, but brain farted on the integration thing. Considering how clever Da Vinci was, his mistake is very surprising, because his law is not only empirically wrong, it is logically impossible.

To see what I mean, let’s carry out Da Vinci’s argument a little further. According to his rule, in four units of time you fall 1+2+3+4 = 10 units of distance. But the choice of how long a unit of time is was arbitrary. So let’s do it again, but consider the unit of time to be twice as long as it was previously. We’ll call these “shmunits” of time. In one shmunit of time, you have to fall three units of distance to be consistent with the first calculation. Then you fall six units of distance in the second shmunit of time, because the second has you falling twice as far as the first. After two shmunits of time, you fall a total of nine units of distance. But we already said that with the same law you fall ten units of distance! Surely if Leonardo had considered his law carefully he’d have seen this error, right?

Unless it’s not an error. What if Leonardo actually meant that you have to take the limit as your unit of time becomes infinitely short? In that case, Leonardo’s law

distance \propto t(t+1)

can simply be reduced to the correct law

distance \propto t^2.

Could this really have been what Leonardo had in mind? I think it’s possible, but not likely. The Greeks explored the basic ideas here. They approximated \pi using the method of exhaustion, and Archimedes is said to have been doing what amounted to integral calculus. If Leonardo was aware of this research, he might have stated such a law accurately. But it seems far-fetched.