Drop a cannonball from the Tower of Pisa. Where will it land?

An apocryphal story has Galileo dropping two cannonballs, one large and one small, from the top of the Tower of Pisa in 1589, to show people that small objects fall as fast as large ones.  Whether Galileo in fact performed the experiment, it's been done by many others.  Where did the cannonballs land?  Ignoring the effects of wind, the answer seems obvious; they fell straight down, and landed directly below, just as surely as two plus two equals four.  Indeed that’s a very accurate approximation.  Believe it or not, it’s only an approximation.

I find this weirdly comforting, like when you discovered that you could create many more colors than the six provided at the kindergarten fingerpaint easel. It's an affirmation that the world is full of endless, fascinating details, in every direction and at every scale.  No matter which direction you point your mental lens, no matter what magnification you set it to, the more you look the more patterns, intricate and beautiful, you will see.  Okay, so I like physics.  Sue me.

For the sake of discussion, let's keep it to one cannonball.  Assume there’s no wind resistance, the earth is spherical and of uniform density, the cannonball is as well, and there are no other gravitational forces.  Even in this simplified, imaginary world, calculating the exact landing spot, as opposed to a very accurate approximation, is hard.  (The fact that the Tower of Pisa has a noticeable lean has no bearing on the problem.)

The first complication is that the earth is rotating on its axis, and the tower is rotating along with it.  That means the top of the tower (i.e. the place where you drop the cannonball) is moving laterally faster than the plaza below.  Atop a 100 meter tower standing on the equator, you will travel roughly 628 meters farther every 24 hours (23 hours, 56 minutes, 4.0916 seconds, but who's counting).  This difference in lateral speed means the trajectory of a dropped cannonball will follow a slight curve.

The Tower of Pisa is north of the equator, which means its speed differential relative to the plaza below is smaller.  But it also leads to the second complication: the cannonball's trajectory will curve slightly south, due to Coriolis, an effect which is also responsible for the counter-clockwise rotation of hurricanes in the northern hemisphere.  So the cannonball will land slightly to the east and south of the spot below where it was dropped.

The cannonball's exact trajectory will depend on the forces it is subjected to.  The third complication is the centrifugal force produced by the earth's rotation, which counteracts to some degree the earth's gravity.  When weighing yourself, you will get a smaller number at the equator than at either of the poles (if the earth spun about 16 times faster, those at the equator would be weightless).  When you drop the cannonball, centrifugal force counteracts, somewhat, the downward acceleration due to gravity.

The fourth complication also lowers the downward acceleration at the moment the cannonball is dropped.  Over four centuries ago, in one of the most famous scientific advances, Isaac Newton showed that the force of gravity varies in inverse proportion to the square of the distance from the earth's center.  The top of the tower is farther from the center of the earth, which means the gravitational force is lower.  As the cannonball gets closer to the earth, the gravitational force increases.

The fifth complication has to do with the gravitational force caused by the cannonball itself.  All mass, no matter how small or how distant, exerts a gravitational force. While the cannonball is falling toward the earth, the earth is falling toward the cannonball.

The earth moves!  Two masses - the earth and the cannonball, in this case - accelerate toward each other.  Since the cannonball has less mass, it moves more than the earth does.  But the earth still moves. This universal law of gravitation was discovered by Isaac Newton.

Drop a cannonball from a 100 meter tower, and it will land after traveling slightly less than 100 meters, because the earth has been pulled ever so slightly toward the cannonball.

Einstein's theory of relativity introduces yet more complications, which I'll skip because I don't know enough about them.  And there may be more complications I haven't thought of.

These five complications have small effects, even when aggregated.  In almost any physics class, you can safely ignore them all, and pretend a dropped cannonball will fall straight down.  But rest assured that no matter where you look, if you choose to look further, there is more to find.

The Antidote

"For what?", you ask.  It's an excellent question, and, to be honest, after reading "The Antidote," a Grisham-esquely titled book by Oliver Burkeman, I still can't answer it.  To the best of my knowledge, the word "antidote" appears nowhere else in the book.  Nitpicking aside, I found the book, based on columns the author published in the Guardian.Weekend magazine, well written and well researched.  Like a Michael Lewis story, but with weirder characters - as if Michael Lewis had written a non-fiction version of Harry Potter.  (That analogy occurred to me after the first couple of chapters.  Later, by coincidence, J.K. Rowling makes a brief appearance.)

Here's how the book starts:

The man who claims that he is about to tell me the secret of human happiness is eighty-three years old, with an alarming orange tan that does nothing to enhance his credibility.  It is just after eight o'clock on a December morning, in a darkened basketball stadium on the outskirts of San Antonio in Texas, and - according to the orange man - I am about to learn 'the one thing that will change your life forever.'  I'm skeptical, but not as much as I might normally be, because I am only one of more than fifteen thousand people at Get Motivated!, America's 'most popular business motivational seminar,'  and the enthusiasm of my fellow members is starting to become infectious. ...  'Here's the thing that will change your life forever.'  [Dr Schuller, author of more than 35 books on positive thinking] then barks a single syllable - 'Cut!' - and leaves a dramatic pause before completing the sentence '... the word 'impossible' out of your life!  Cut it out!  Cut it out forever!'  The audience combusts. ...

The book's subtitle is much more specific: "Happiness for people who can't stand positive thinking."   Although I can't remember how the book caught my eye, I imagine this is it. People rarely accuse me of being excessively positive, a characteristic that, according to Burkeman, I share with journalists like himself.

This early passage contains hints of what comes later:

The logic of Schuller's philosophy, which is the doctrine of positive thinking at its most distilled, isn't exactly complex: decide to think happy and successful thoughts - banish the spectres of sadness and failure - and happiness and success will follow.  It could be argued that not every speaker listed in the glossy brochure for today's seminar provides uncontroversial evidence in support of this outlook: the keynote speech is to be delivered, in a few hours' time, by George W. Bush, a president far from universally viewed as successful.  But if you voiced this objection to Dr. Schuller, he would probably dismiss it as 'negativity thinking.'  To criticize the power of positivity is to demonstrate that you haven't really grasped it at all.  If you had, you would stop grumbling about such things, and indeed about anything else.

A brilliant business model: the less effective the philosophy is, the more profitable it becomes, since it can't be falsified, and all 15,000 attendees become prospective customers for Schuller's next visit to San Antonio.  But Burkeman's takedown of Schuller's philosophy turns out to be more perspicacious than merely suggesting that not all positive thinkers are successful and happy (although, I hasten to add, Bush 43 is nearly universally viewed as happy, inexplicably so); he points out that the power of positivity may not even lead to positive thoughts.  This is the realm of ironic process theory, which evolved out of research conducted by Daniel Wegner, a Professor of Psychology at Harvard.  At its core, it suggests that certain things are harder - nee impossible - to do when you try to do them, such as "not thinking about a white bear."  Being happy is one such thing, Burkeman argues.  Paraphrasing Burkeman, we can glimpse it out of the corner of our eye, but can't see it when we look directly at it; "For a civilization so fixated on happiness, we seem remarkably incompetent at the task. ... 'Ask yourself whether you are happy,' observed the philosopher John Stuart Mill, 'and you cease to be so.'"  During a chapter on Buddhism, the astute reader will contemplate that the secret to happiness might be, paradoxically, to not care whether or not we are happy.

Breathtakingly negative topics - Stoicism; a guy who sat on a park bench for two years; failure; insecurity; Mexico's fascination with death; the world's largest pubic louse - succumb to Burkeman's cheerful, engaging prose.  At times I think the book is repetitive, but it doesn't feel repetitive; when he finds several interesting ways to say the same thing, I'll happily read them all.  Analogous to Clinton's "third way" in politics, Burkeman's opus is a third angle on happiness, between "the futile effort to pursue solutions that never seem to work, on the one hand, and just giving up, on the other."  Sounds plausible to me, paradoxes and all.

Time Magazine: Why Medical Bills Are Killing Us

Here is a terrific long read: Time Magazine's fascinating, detailed look at why medical bills are killing us, by Steven Brill.

Update:

Some more links for those interested:

Matt Yglesias comments on the Time Magazine series.

David Goldhill: How American Health Care Killed My Father.

Jonathan Cohn: The Robot Will See You Now.

Atul Gawande: The Cost Conundrum and The Cost Conundrum (*).

Drone 101 - Chapters 1 and 2

I couldn't write a complete overview of drones even if I wanted to.  But the first two chapters often get shortchanged in the current media feeding frenzy over a leaked drone memo.

Chapter 1.  Finding and exploiting an opponent's weakness. On the surface, Chapter 1 has little to do with drones, but everything to do with the broader context in which they get deployed.  The "war on terror," as Bush 43 put it, better framed by Obama as the mission to "disrupt, dismantle and defeat al Qaeda," relies on two sets of rules: the rules of law enforcement and the rules of warfare.  Critics question whether the drone program properly follows those sets of rules.  That's a fair question, and it's good that the press have gone beyond blandly reporting drone strikes as if they were a common weather pattern.  We are great at law enforcement, and we are great at warfare; it stands to reason we should set high standards for both.  But while our gumshoes and GIs have no peer, neither set of rules is well-suited to disrupting, dismantling and defeating al Qaeda, a mission whose merit no one questions.

The rules of law enforcement assume that all countries have the means and the will to enforce laws, and that the focus should be on building cases, not preventing crime.

The rules of warfare assume uniformed soldiers sponsored by countries.  Deadly force gets employed based on split-second decisions by all, even the lowliest private.

Al Qaeda's asymmetric tactics fit neither set of rules.  That is not a coincidence.

A common military strategy is to attack along the seam between two opposing forces, since coordination difficulties will ensure that that is the weakest point along a war front.  In seeking to disrupt, dismantle and defeat al Qaeda, we employ two strategies, a military one and a law enforcement one.  The weakest point is the seam between the two.

Indeed that seam is the weakness al Qaeda seeks to exploit.  They don't wear a uniform, and they're not sponsored by any state.  That would suggest they are common criminals, where the rules of law enforcement apply.  But they gravitate toward countries where the rule of law is tenuous, at best.  In that seam between the two sets of rules, we inherit the disadvantages of both and the advantages of neither.

Chapter 2 - Drone capabilities. Into this foggy milieu enter drones, unmanned planes carrying cameras and missiles, controlled by U.S. pilots in Nevada or another remote location.  Before gnashing teeth over who is a legitimate target, when, under what conditions, and most of all, who gets to decide, one needs to read Chapter 2, which covers the four game-changing capabilities drones bring to the mission.  These capabilities have profound implications for developing new rules that address the threats that have emerged out of the subtle weaknesses of the current two-pronged (law enforcement and warfare) regime.

Section 2.1: Drones are unmanned.  Well, duh.  But this has huge implications for rules of engagement.  The critical problem with deploying troops to trouble spots is that they have to be given the right to defend themselves.  If fired upon, fire back!  Self-defense, however, means that people can get hurt, and it's not always the bad guys.  Innocent civilians and your own troops can get killed.  With drones, self-defense is irrelevant.

Section 2.2: Surveillance.  Lost in the debate about deadly force delivered via drone is the fact that the most powerful "weapon" carried by a drone is a camera.  In fact, most drones, which range in size from hand-held to a take-off weight exceeding a half dozen cars, are unarmed.  The most famous drone strike, which targeted and killed Anwar al-Awlaki, was likely preceded by hundreds if not thousands of hours of drone surveillance.  Regardless of the preferences of hawks as well as doves, we can expect the typical drone strike approval to require more intelligence.

Section 2.3: Time for deliberation.  By providing staggering amounts of data over prolonged time periods, continuous drone monitoring imposes an obligation, moral if not legal, for extended analysis and deliberation.  In World War II General Eisenhower would have rightly objected to having every shoot/don't shoot decision approved by an "informed, high-level official."  No such excuse exists for drones, where the decision to strike can play out over hours if not days and involve gigabytes of data.  Ultimately, if the right decision-making process involves approval from a dozen officials, digital drone technology supports it.

Section 2.4: Small, precise munitions.  Conventional wisdom holds that the U.S. military seeks to build weapons that make bigger and bigger explosions.  In reality, for the most part they are obsessed with building smaller weapons wielding less destructive power that limit collateral damage.  This won't comfort people who follow in the footsteps of Anwar al-Awlaki, who everyone agrees got what he deserved.  But it reduces collateral damage.  Small, precise munitions make it harder to justify collateral damage as a necessary side-effect of hitting a legitimate target.

Given the two discreet sets of rules we can bring to bear, disrupting, dismantling and defeating al Qaeda is a difficult challenge.  The unique capabilities of drones will help us meld the rules of law enforcement and the rules of warfare into a new hybrid set of rules that eliminates the seam al Qaeda is currently exploiting.

Update: Last year an article in the New York Times discussed some advantages of drones.

Update #2: Mark Bowden has a thoughtful article in the Atlantic.

How A/B web testing can unintentionally propagate stereotypes, including racism

Why does Google's ad network think people with names like Latanya and Rasheed have an arrest record?

Imagine the following scenario: An employer is Googling prospective job applicants. Some of those applicants have black-identified names. Due to his or her personal racism, the employer happens to be more likely to click on the InstantCheckmate ads that suggest “Arrested?” next to the black-identified names. And over time, Google’s AdSense algorithm learns that “ads suggestive of an arrest record” work better when associated with black-identified names. Voila.