The Renaissance Man Uniform Gravitational Acceleration SMACKDOWN

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.

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.

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:

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.

9 Responses to “The Renaissance Man Uniform Gravitational Acceleration SMACKDOWN”

1. Paul Murray Says:

Pehaps Da Vinci was right? Maybe by “the degree of its descent” he meant the distance, rather than the time? Obviously, that makes his velocity wrong … unless he meant total average velocity rather than instantaneous velocity?

And a pyramid is three-dimensional, anyway. The pyramidal numbers increase with the cube of n. Was LdV aware that kinetic energy is proportional to the square of the velocity? Is that what he meant by “the power of it’s motion”?

2. Mark Eichenlaub Says:

According to what I’ve read elsewhere, he was referring to triangular numbers when he mentioned pyramids. But it does seem like there’s a little room for interpretation, based on the tiny bit of text I have.

3. | Quiche Moraine Says:

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4. Shawn Says:

Maybe da Vinci was talking about position and triangular numbers rather than velocity. Suppose your position is given by: 0, 5, 20, 45, 80

Triangular Numbers (And Zero)
0, 1, 3, 6, 10, 15, 21, 28…
Average of successive terms
.5, 2, 4.5, 8, 12.5, 18, 24.5
If g=10, multiply by 10 to get the free fall positions
5, 20, 45, 80, 125, …

5. Peter L. Griffiths Says:

One important aspect of Galileo’s law of falling bodies is that this discovery was also made by Kepler in a reciprocal form v^2=1/r where d+r equal the major axis of the elliptical orbit.

6. Peter L. Griffiths Says:

Further to my comment of 25 April 2011, I hope your readers can tolerate a little bit of algebraic notation, let L equal a small change. From
v^2=d=1/r which is the usual velocity measure, we have
v^2+Lv^2=d+Ld=1/(r-Ld). For the reciprocal velocity measure we have v^2=r=1/d which after the small change will be
v^2+Lv^2=r+Lr=1/(d-Lr). This reflects the two ways of measuring velocity, distance per unit time and time per unit distance, one of which is the reciprocal of the other. Any change of d is equal to the opposite change of r.

7. Mark Eichenlaub Says:

No, that is not correct. The acceleration of a planet in a Keplerian orbit is always changing, since it points towards the sun and is proportional to 1/r^2.

8. Mark Eichenlaub Says:

Peter, you are going to have to study calculus, differential equations, and then mechanics before you can understand the Kepler problem clearly. Ad-hoc algebraic manipulations will not teach you what you want to know.

9. Mark Eichenlaub Says:

Kepler would also have been unable to tell you the acceleration of the planets in their orbits.