Galilean Freefall Confirmed

It wasn't centuries of successful theories of mechanics, planetary dynamics, and interplanetary travel that confirmed Galilean freefall. Oh no. It was this.

Galileo and the Physics of Skydiving

During a visit to Skydive, PA yesterday, I noticed jumpmaster Jim tightening a corset-like weight belt onto his abs. "Jim, your figure looks so refined," I commented. Jim sneered. "I've got to keep up with the big guy," he said, nodding toward the hulking student he was about to jump with.

True enough. But it sounded funny to me, having just re-read Galileo's Two New Sciences. Galileo doesn't fool around: he clearly states that your speed in freefall does not depend on your mass. His student Viviani famously reported that Galileo discovered this by dropping different weights off the Leaning Tower of Pisa, and observing that they hit the ground at the same time.

And yet, skydivers wear weight belts to fall faster. A slight person like Jim, I myself used to wear such a belt, when I was a student learning to fly with much bigger skydivers than myself. And it really helps. The reason is: even though your body mass doesn't affect your acceleration in a vacuum, it does affect how hard your body pushes against air.

A few seconds after stepping out of a plane, a skydiver's motion will begin to be slowed by the drag force of air -- the same force you feel when pushing your arm through the water of a swimming pool. The weight of the skydiver pushes back against the drag force, until the two forces become equal and a constant (terminal) velocity is reached. The heavier you are, the harder your body pushes back against that drag force -- and the faster your terminal velocity will be. In particular:

where K is a constant, m is your mass, and SA is the surface area that is facing the wind. In short: when you skydive, your surface area slows you down, and your mass speeds you up. That's why small skydivers sometimes wear weights.

Of course, Galileo still had the basic mechanics correct: when air resistance is neglected, mass doesn't matter. However, it does seem unlikely that he would have put much thought into the effect of mass on terminal velocity. After all, accurate measurements involving freefall were hard enough even at earthly speeds. As I've noted before, the difficulty of these experiments gave rise to much confusion about the nature of freefall in the early years of mechanics.

So. Galileo was right, but you may still need weights to keep up with your skydiving partner.

How do you know who's going to fall faster? You can read the basic rules off the equation above. Someone the same height as you but heavier will have increased mass, and fall faster. Someone the same weight as you but taller will have increased surface area, and fall slower.

It's even easier if your partner is of your same proportions (that is, taller or shorter, but no more overweight or underweight than you are). Everyone has roughly the same body density, so mass is roughly proportional to volume. But volume is roughly proportional to surface area (SA) times height (h). So, if you're jumping with someone who's similarly proportioned, above equation reduces to:

For example, 9% taller (about a head) means 3% faster. For a typical terminal velocity of around 200 feet per second, that means: for similarly proportioned people, a head taller means about 6 feet per second faster.

Are skydivers ever in the kind of environment where these rules don't hold? Only in the absence of air. This seems to have happened just once, when Joe Kittinger jumped out of a balloon at 100,000 feet during the (1959) Project Excelsior study. That's well beyond the official edge of the earth's atmosphere, making it the only skydive from space. I'll leave you with the video for today. Blue skies -- I hope to see you up there!

Galileo's Manuscripts: Still Learning the Lesson

When I studied history of science, my teachers -- especially Ken Manders and Paolo Palmieri -- hammered two principles like drill sergeants:

1. Always read the original manuscripts;
2. Always pay attention to the diagrams.

Okay, that seems obvious. We've known that since the mid-20th century. Yes, these lessons are easy to under-estimate -- until you realize how many historians still haven't learned them.

Here's an example. Galileo's personal notebooks on mechanics have survived the last four centuries. I recently took interest in one famous page among them, in which Galileo did very early work on acceleration in freefall. It's distinguished title is Folio 152r of Manuscript 72, written sometime between 1604 and 1610. The Max Planck institute has done us the extraordinary service of putting the whole manuscript online.

I first spent a few days assembling my own interpretation of this manuscript page. Then I took a look at how other people interpreted it. Surprisingly, two features of the manuscript that I thought were significant seem to have been completely ignored by scholars. And no wonder -- it turns out that the only available English reproductions of the document omit a signficant portion of the diagram.

Compare the following. Here's Drake's (1973) English reproduction -- click to enlarge it:

Damerow et al. (1992) later spruced things up with a nicer type-face:

Finally, here's the original. Can you see what's missing?

Two important things. First: the little upside-down triangular diagram in the middle. (The circle visible near it is actually printed on the back of the page.) Both triangular diagrams represent distance along the vertical, and velocity along the horizontal. So, if this is small triangle is a scaled-down copy of the larger one, then the scaling rule will actually indicate which rule of freefall Galileo was considering. (It turns out to be the incorrect "double-distance" rule, which says that velocity in freefall is proportional to the square of the distance fallen.)

Second: the languages used. While the top-left, top-middle, and bottom-most paragraphs are all written in Italian, the rest of the text is written in Latin. Galileo wrote comfortably in both, but his train of thought probably only followed one language at time. That suggests a connection between these three paragraphs, which might not have otherwise been apparent, given their physical separation on the page.

So, here's the way a translation of 152r should look:

Ah, yes. Isn't that better with the diagram restored? Over a hundred years reading Galileo's manuscripts, and we're still learning this lesson!