Wednesday, June 27, 2012

Matthew Frederick on Sinking Ships

Today's question comes from Matthew Frederick .  Matt is an architect, urban designer, and author of the bestselling 101 Things I Learned in Architecture School.  He writes:

Monel metal is a very hard alloy of nickel, copper, and iron. It is extremely corrosion resistant and is excellent for many wet applications. However, if not isolated from other metals in salt water environments, it can cause corrosion. In 1915, a ship was built with a hull entirely of Monel, with the expectation of exceptional durability. However, the 215-foot-long, 34 foot wide Sea Call had to be scrapped after just six weeks of use. While its monel hull was fully intact, the steel frame of the ship deteriorated beyond use from electrolytic interaction with the monel in the salt water environment.
Question: How much of an electrical current did the corrosion produce? Could it have lit a light bulb? Many light bulbs?

If you've ever built a lemon battery, you know that two different metals can pass electric current between each other when connected by a salt bridge.  This works because of a chemical reaction that occurs when the metals are exposed to the ions in the salt bridge.  In the case of the lemon battery, the metals are usually a copper penny and a zinc-coated nail, while the salt bridge is the lemon.1  In high school chemistry, you may have learned that this type of interaction is known as an oxidation-reduction or redox reaction.  

In principle, I could look up the oxidation-reduction potentials for Monel and other common metals that appear in the sea, and from this I could calculate the number light bulbs one could light up.  However, I'm in the middle of packing and moving to a new state, so, in the spirit of order of magnitude estimates, I'm gonna wing this a bit.

At 215 feet long and 34 feet wide, we have a surface area of at least 680 m2. A typical atom has a radius on the order of angstroms (~10-10 m), so there are roughly 1023 atoms directly in contact with the water.  The corrosion must have been visibly noticeable, so I'm betting it ate through about a millimeter of the hull.2   This thickness is equivalent to roughly 107 atoms giving a total of about 1030 oxidized/reduced atoms during a six week time span.  If each atom gave up one electron, you would have a total current of

(1030 electrons) · (1.6×10-19 coulombs/electron) / (6 weeks)  
= 44 kA.

That's 44 kiloamperes of current.  A normal 100 W bulb plugged into a standard 120V North American wall socket receives roughly 0.8 amperes of current.  Distributed evenly, this could have lit about 55,000 light bulbs.  

Thanks for a great question, Matt!  You can find out more about Matthew Frederick on his website and blog.  You can also check out the 101 Things I Learned series here.

[1]  Fun Fact: Alessandro Volta, after whom the electrical unit the "volt" is named, is credited with inventing the first battery after allegedly using his tongue as the first salt bridge.  I'm not sure how one makes a discovery like this.  I can just imagine Volta saying in a Homer Simpson-esque voice, "Mmm...metal alloys.  Ooooh."
[2] Note: This is a big assumption, which I'm not entirely sure is correct.  The final answer can change dramatically depending on this number.  For example, if the corrosion was only a micron deep, the final result will be about 55 bulbs.  Feel free to play around with different thicknesses to see how many bulbs you could light.

EDIT: Matt just informed me that, "The steel frame corroded; the Monel was intact!"  Doh!

Wednesday, June 20, 2012

Holy Flaming Burritos, Batman!

Another Redditor made me this....

It's been over a week since I did my Reddit AMA, and I want to say thank you to every one who asked questions. As evidenced by the giant spike in page views, it had a bit of an effect on my blog viewership.

So thank you to everyone who asked questions (even I didn't get a chance to answer yours.)

There were, however, a few questions that I said I'd get back to people on, and I wanted to address those.

(1) One of the questions I got and couldn't answer at the time was the following:
A physicist! I've been waiting for one. I've been wondering this for a while, but can't come up with a solid answer. 
If I was in space and I attached an LED light to one corner of a cube, is it possible for me to push/toss/throw/rotate the cube in such a way along a linear path that the LED light's pattern would never repeat itself (aka, there would never be a period)?

Sadly, I still don't have a good answer to this one. The best I can do for now is say that another Redditor provided me with this link to his work on the chaotic motion of rigid bodies. It seems pretty cool, but I'm too bogged down in summer research (and moving to a new state!), so I can't devote a whole lot of time to this one.

(2) Yum_Krill (an appropriate username if I ever heard one) asked,
How large would a creature have to be to make a sound like The Bloop? The largest creature on earth, the blue whale, would not be able to.
I had to look up the Bloop.1  Apparently, it's some weird animal-like deep ocean sound that is both loud and low-frequency:

According to at least one source, the Bloop reaches about 246 decibels. For context, normal human conversation occurs around 60-70 decibels, the loudest rock music reaches about 150 decibels, and blue whales can reach about 188 decibels.

Decibels are a logarithmic scale, which means that a human conversation at 60 decibels would have a power ratio of 106 and a blue whale singing at 190 decibels would have a power ratio of 1019. The Bloop would have a power ratio of about 1025, or roughly a million times bigger than a blue whale.

Whatever the Bloop is, it presumable has some object that mechanically oscillates to produce sound. The amplitude of this oscillation grows as the square root of the power. Since the Bloop's power is 10times bigger than the blue whale's, its amplitude of oscillation would be about 1000 times larger than a blue whale's. If the rest of the Bloop remains proportional in size, it should be about 30 kilometers long and weigh close to 180 billion metric tons.

(3) In response to a message about a flying Pegasus, I replied
We need to consider two things here: wing area and wing flapping rate. I did a similar problem for Mothra's wingspan. Horses weigh about 500 kg, which gives a downward gravitational force of about 5000 N. If you assume her wings flap 2 meters down and do so once every second, she'd need winds that were about 1000 m2 in area. A 2 meter wide wing would need to be about 5 football fields long.
After posting the problem, I got a message that read as follows:
You say Horses couldn't possibly fly unless they had unrealistic, impossibly long wings (up to 5 football fields long), then how come Pterosaurs weighted up to 400 pounds could fly at blazing fast speeds with reasonably sized wings?
I understand the difference in weight, but what was the main difference here to allow this other heavy animal to fly easily, is it wing-flapping speed?
I was unaware of overweight pterosaurs when I was doing the problem, but I still had reason to be concerned with my result because of the earlier problem I did on Mothra. However, in an effort to keep up with the questions, I was working pretty fast and (stupidly) had my normal skeptical filter on silent. That said, I think I know what wrong here.

A horse weighs about 500 kilograms, which means it feels a downward gravitational force of about F = 5000 Newtons. You need at least this much force to have liftoff.

To generate this force, Pegasus pushes air down. The mass of the air pushed depends of the density of the air (ρ =1.2 kg/m3) and the volume V of air swept out with each downward thrust:

m = ρ V = 2 ρ L w h

where L is the length of the wing, w is the width of the wing, and h is the vertical height the wing is thrusted downward. The factor of 2 comes in because there are two wings.

The factors affecting thrusting force are the frequency of flapping f (in Hertz), the distance h the wings are thrust down (in meters), and the mass m of the air pushed (in kilograms). Using dimensional analysis, we can estimate the magnitude of the upward force as

F = 2 ρ L w h2 2.

This can be solved for the length,

L = F / (2 ρ w h2 2).

If we use, as I did previously, the values F = 5000 N, ρ =1.2 kg/m3, w = 2 m, h = 2 m, and f = 1 Hz, you find that the wings must must be 260 meters, or roughly two and half football fields long. I had forgotten the two, but that's not the only problem.

There's two other potential problems I foresee with this. First, dimensional analysis is great for figuring out how certain physical quantities scale with other physical quantities, but if you're looking for actual numbers you can be off by quite a bit. Putting that aside, there's still a second problem. Whenever you do an estimation, you should go back and check to see if your original assumptions make sense. Now if your wings are five footballs long, it doesn't make sense that they're thrusting only two meters downward. In the video below, they appear to be thrusting about 4 meters, which reduce the result to roughly a 65 meter wingspan.

Moreover, if I up Pegasus's wing flapping rate to 3 flaps per second, we further reduce the wingspan to a much more believable 7 meters. I apologize if I mislead any Pegasus fans out there.

I'm not sure what this says about pterosaurs. I know very little about them except that they varied in size from 10 inches (~25 centimeters) to over 40 feet (~12 meters). Just from physical considerations, I would have guessed the largest of these were more like ostriches (i.e. flightless), but maybe there's some evidence to the contrary.

There's an important take home point here. Whenever you reach a conclusion, whether it be for an estimation or a general conclusion about life, you always need to go back and ask what assumptions you made to get there. You've got to make sure the assumptions you made still make sense by the time you get to the end!


Thanks once again, Reddit. If there are more questions you want answered, message me. I can't guarantee a response (I'm still reading some from last week), but I can guarantee that if I see your question and it intrigues me, I'll answer it on the blog and give you credit.

[1] Fun Fact: Previous Diary of Numbers guest Chris Moore's book, Fluke, or, I Know Why the Winged Whale Sings, features the Bloop.

Aaron Santos is a physicist and author of the books How Many Licks? Or How to Estimate Damn Near Anything and Ballparking: Practical Math for Impractical Sports Questions.   Follow him on Twitter at @aarontsantos.

Tuesday, June 19, 2012

Dan Ariely on the Cost of Justice

We're getting the cognitive scientist hat trick today.  Today's guest is Duke professor Dan Ariely.  Dr. Ariely is a renowned author, psychologist, and behavioral economist.  He is the author of two New York Times bestsellers: Predictably Irrational and The Upside of Irrationality.  His most recent book, The Honest Truth About Dishonesty: How We Lie to Everyone--Especially Ourselves, came out earlier this month.  His TED talk on human irrationality has over one million views.

Dan asks,

How much do blue-collar criminals steal per year compared to white-collar criminals? How much do blue-collar criminals make per year on average?  How much money do we spend on the justice system in totality? How much economic gain do we get from it?  In other words, how much are we spending on the system and how much crime in economic terms are we preventing?

According to Wikipedia, white-collar crime

...was defined by sociologist Edwin Sutherland in 1939 as "a crime committed by a person of respectability and high social status in the course of his occupation."

In contrast, blue-collar crime is committed by individuals of lower social status. Using these definitions, there's not a sharp cutoff between blue- and white-collar crimes. Even if these terms were well-defined, getting precise statistics on how many of these crimes are committed is difficult since many of them go unreported. For this reason, I'll need to make a lot of assumptions in this problem.

According to at least one source, there are roughly 2.3 million people presently held in American prisons. If we use data from the Bureau of Prisons, we find the relative percentage of prisoners incarcerated for various crimes:

As mentioned above, there's some arbitrariness surrounding the definitions of blue- and white-collar crimes. Given the context of Dr. Ariely's question, I'm only going to consider crimes that have an inherent financial component.1 I'll further assume that drug, robbery, burglary, larceny, and property offenses all fall under blue-collar crime, while banking, insurance, counterfeit, and embezzlement offenses fall under white-collar crime.2 Using this categorization, over 50% of crimes would be considered blue-collar, which is in stark contrast to the 0.4% of crimes that are considered white-collar. According to Sutherland, "less than two percent of the persons committed to prisons in a year belong to the upper class," which is consistent with my assumptions. It's worth noting that these figures only count crimes where the crook was convicted and sentenced to serve time. There are undoubtedly a large number of crimes where the perpetrator has been either not apprehended, acquitted, or given a sentence that did not involve incarceration. Below is a chart illustrating the clearance rate (i.e., the fraction of time charges are filed for reported crimes) for various crimes:

Even these figures are fraught with peril. For example, robberies are more likely to be reported than money laundering, if for no other reason than the fact that a victim is immediately aware that a robbery is taking place when there's a gun pointed at his face.

For simplicity, I'm going to pull some numbers from Wikipedia and make some very basic assumptions.  According to the Wikipedia entry for "Crime in the United States", in 2009 roughly 3466 crimes were committed for every 100,000 people.3 Spread over a population of 300 million people, this means about ten million (~1×107) crimes are committed each year. Roughly 50% of these (~5×106) can be considered financially-motivated blue-collar crimes, while roughly 0.5% of these (~5×104) can be considered financially-motivated white-collar crimes.

I suspect the amount a blue-collar criminal steals each year varies quite a bit from person to person. A professional crook might be hitting a new house every day, but shoplifters and other petty thieves are less likely to be raking in the dough. You could steal anything from one dollar to over a million, but the average take, at least for professional crooks, is much more likely to be in the thousands. For concreteness, let's say $1000 per crime. If only to avoid being caught, I suspect the average criminal is not likely to rob more than 10 times a year (certainly fewer than 100). From these numbers we can conclude that the average blue-collar criminal steals at most $10,000 a year, with one-time offenders stealing much less and Ocean's 11-esque pros looting much more. Using this average as an upper bound, we can compute the maximum total amount our five million blue-collar criminals steal each year to be about $50 billion. Since this is an upper bound, we can reasonably argue that a realistic number is more likely in the one billion to $20 billion range.

White collar criminals are a different story. While fewer in number, they can use their fortunate disposition to gain control of much more money. A list of the top ten white-collar crimes shows thefts ranging from Martin Frankel's $200 million (~$2×108) swindling of insurance companies to Bernie Madoff's $65 billion (~$6.5×1010) ponzi scheme looting, but these are extreme cases.4 A more typical example might be Martha Stewart, who "avoided a loss of $45,673 by selling all 3,928 shares of her ImClone Systems stock...." If we assume a $100,000 (~$1×105) take, then our 50,000 white-collar criminals would rake in about $5 billion (~$5×109) each year.

There's a problem with this estimate. As one can easily see, Madoff alone took in more than ten times my estimate.5 What gives? While crimes like Madoff's are rare, they're significant because of the sheer volume of money involved. Significant rare events like these can be kryptonite to Fermi estimators. Moreover, as I pointed out earlier, blue-collar crimes are less likely to be reported. As such, my estimate is probably only good as a lower bound, so I'm going to need to up it a bit. Since we're talking order-of-magnitude estimates, it makes sense to bump my number up by a power of ten. This would give about $50 billion (~$5×1010) stolen each year by white-collar criminals. This is within an order of magnitude of the same numbers listed by the Wikipedia entry for "White-collar crime":
While the true extent and cost of white-collar crime are unknown, the FBI and the Association of Certified Fraud Examiners estimate the annual cost to the United States to fall between $300 and $660 billion.
At least one other source gives similar numbers:
It is estimated that white-collar crime cost the United States from $200 to over $300 billion every year. This is staggering compared to the estimated $15 billion to $20 billion in damages that blue-collar crime inflicts.
As you can see, my numbers are within a power of ten of these other estimates, but there is, admittedly, a huge error range associated with these figures.  

Let's try to put these numbers in context. The 2013 budget for the Department of Justice is set at $27.1 billion. Roughly $8.5 billion (~$8.5×109) of this is spent on prisons/detention facilities. Federal prisoners make up about 6% of the total incarcerated population, or about 138,000 people. As such, each federal prisoner would cost

($8.5 billion per year) / (138,000 people)
= $62,000 per person per year.

State prisons are a bit more efficient. According to the Wikipedia entry for "Incarceration in the United States"
In 2007, around $74 billion was spent on corrections. The total number of inmates in 2007 in federal, state, and local lockups was 2,419,241. That comes to around $30,600 per inmate.
In 2005, it cost an average of $23,876 dollars per state prisoner. State prison spending varied widely, from $45,000 a year in Rhode Island to $13,000 in Louisiana.
Even if we're building fairly efficient prisons at a net cost of $30,000 per prisoner per year, imprisoning blue-collar criminals costs the American taxpayer anywhere from 3 to 30 times more money than they would lose by letting these criminals run free. In contrast, letting 50,000 white-collar criminals run free would cost taxpayers 30 times more than it does to them lock up. This number skyrockets to 440 times as much money if we assume our white-collar criminals are stealing $660 billion.

A more efficient justice system would prosecute fewer of these guys...

...and more of these guys.

Economically, it seems drastically more efficient to lock up the high-class crooks and leave the petty thieves to themselves. However, this doesn't address Dr. Ariely's last question. Taken by itself, incarcerating blue-collar criminals is not cost-effective, unless you assume the mere threat of incarceration has prevented anywhere from 3 to 30 times more crimes. More on this in a moment. For now, let's naively assume the rate of crimes committed doesn't depend on the type of punishment doled out. If that's the case, taxpayers lose on average between $20,000 and $29,000 for every incarcerated blue-collar criminal. In contrast, taxpayers save between $70,000 and $13 million for every incarcerated white-collar criminal. Given five million (~5×106) blue-collar criminals and 50,000 white-collar criminals, the taxpayers would lose at most

($29,000) × (5×106) − ($70,000) × (5×104
= $140 billion

and potentially save

($29,000) × (5×106) − ($70,000) × (5×104
= $500 billion.

Given the huge error ranges, it's a little bit of a wash to say whether or not the justice system is cost effective for financial crimes. One thing that might shift the balance toward being a good system is the notion that the threat of punishment has prevented crimes from happening. Sadly, I don't really have a good way of estimating the number of crimes that would have occurred without a justice system. All I can really say is that you'd need to have prevented about five million crimes to break even.

So what have we learned from this? First off, whoever said crime doesn't pay was grossly misinformed. Second, the Department of Justice would do well to focus less on petty thieves and more on the white-collar criminals who got us into the financial mess we're currently in.

Dr. Ariely, thank you for a very challenging question. I feel like I learned a lot doing this one.

[1] For example, rape, murder, and jaywalking are all crimes, but there's no money exchanged, so I'm excluding these.
[2] Drug related offenses can be either possession (which does not have a financial component) or selling (which does have a financial component).
[3] It's unclear if this number represents the number of crimes committed or the number of crimes reported to police. I'm assuming the former for simplicity, but if it is the latter, one would need accurate clearance rates to compute the number of crimes committed.
[4] Seriously, who trusts a guy named Bernie Madoff? As in, "Bernie made off with my life savings!"
[5] Even if we take the $50 billion upper bound figure for the amount that all blue-collar criminals made in a year, it's still less than the amount that was stolen by one Bernie Madoff.

Aaron Santos is a physicist and author of the books How Many Licks? Or How to Estimate Damn Near Anything and Ballparking: Practical Math for Impractical Sports Questions.   Follow him on Twitter at @aarontsantos.

Thursday, June 14, 2012

Bruce Hood on Darwin and Redwoods

We're rockin' the cognitive science circuit this week. Today's guest is Dr. Bruce Hood.1  Dr. Hood is the director of the Bristol Cognitive Development Centre at the University of Bristol.  He's authored three popular science books: SuperSense: Why We Believe in the UnbelievableThe Science of Superstition: How the Developing Brain Creates Supernatural Beliefs, and his most recent Self Illusion: Why There Is No 'You' Inside Your Head.

Dr. Hood writes,

How many copies of Darwin's On the Origin of Species could be made from the largest Californian Redwood?

In addition to being some of the world's largest trees, sequoia or redwoods can support vast ecosystems of their own, so relating them to Darwin's Origin of Species is particularly appropriate.

Redwoods can grow to over 110 meters tall with widths broad enough to drive a car through. From the linked image, we can estimate the width as roughly 6 meters.  This would give redwoods a volume of roughly 4000 cubic meters.  

Paper making is a fairly complex process.  According to at least one source,
A cord of wood is approximately 8 feet wide, 4 feet deep, and 4 feet high...It has been estimated that one cord of this wood will yield...1,000-2,000 pounds of paper (depending on the process)....
From this, we can estimate the number of number of pounds of paper produced

(2000 lbs) × (4000 m3) / ( 8 ft × 4 ft × 4 ft )
= 1100 tons.

Amazon lists the The Origin of Species shipping weight as 1.5 pounds, which means you could create roughly 1.5 million copies.  That's enough copies to send to half the people in Kansas.

Thanks for a great question, Dr. Hood.

[1] You can check out some of Dr. Hood's cool talks here and here.

Wednesday, June 13, 2012

Steven Pinker on the Living and the Dead

Today's special guest is Harvard professor Steven Pinker.  Dr. Pinker, an experimental cognitive scientist and linguist, has written several best-selling popular science books including The Language InstinctThe Blank Slate: The Modern Denial of Human Nature, and most recently The Better Angels of Our Nature: Why Violence Has Declined.  In addition to all these accomplishments, he has, quite possibly, the greatest scientist/rockstar hair one can imagine and was deservingly chosen as the original member of the Luxuriant Flowing Hair Club for Scientists.1  

Dr. Pinker writes,

How many people lived during the 20th century?
How many people died during the 20th century?

More generally:

If the world's population was X at time t, and Y at time v, how many distinct individuals were alive between t and v? This question obviously presupposes another one, namely, what other variables does one need to compute an estimate (birth rate? Average age of death?)

All of us have, at one time or another, suffered the misfortune of being born.2  As such, we can answer the first of Dr. Pinker's questions by knowing what the population was at midnight on New Year's Eve 1899 and adding to that any births that occurred during the next one hundred years.  The first of these is easy to look up (about 1.6 billion), and I'll discuss the latter shortly.  Before getting to that though, I should point out that once you have this answer, the answer to the second question is easy to find.  Since another unfortunate consequence of birth is, of course, the inevitable death that follows, all you have to do is subtract the world population at midnight on New Year's Eve 1999 (about 6 billion) from the number of people who "lived in the 20th century" and you'll have found the number of people who croaked during this time.  The only tricky step is figuring out how many people were born in the meantime.

A nice "world population vs. time" plot courtesy of Wikipedia.

There's an old statistics joke about the average family having 2.3 kids.  While it's fun to imagine the extra three-tenths of child as a torso-less pair of legs constantly running around and bumping into walls, the statistic itself will be useful here.3  Let's say people reproduce on average once every 25 years.  Assuming no one dies, you should have 4.3/2 = 2.15 times the original population (about 6.9 billion people) by the time you reach 1925.4  Since there will be about four generations, you would have to repeat this procedure four times to get roughly the total number of people who lived during the 20th century.  We can write this using a formula that contains only three variables:

[(2 + average number of children) / 2](number of generations) × (original world population) 
= 2.154 × (1.6 billion) 
= 34 billion people.

That's about 34 billion people who lived at some point during the 20th century.  Since there were only about 6 billion people living at the end of the century, it stands to reason that roughly 28 billion deaths occurred during the 20th century.5

Dr. Pinker, it has been an honor having you and your awesome hair on the blog today.  Thanks for a great question!

[1] To quote the Improbably Research website, "From that lone, Pinkerian seed, there has grown a spreading chestnut, black, blond, and red-haired membership tree."
[2] Birth is, of course, a vital prerequisite for living, but that fact does not make the matter any more pleasant or less messy.
[3] The exact number might be closer to 2.1 kids or 3.1 kids.  Fortunately, most of us aren't the Duggars, so we should be good to at least an order of magnitude here.
[4] For those who have a hard time seeing why it's 2.15 and not just 2.3, consider a world with only 20 people.  If those people reproduce at the assumed rate, they will have 23 kids giving a total population of 43 people, or roughly 2.15 times the original population.
[5] This means that roughly 18% of the people who were alive at some point during the 20th century were alive at the end of it.  Avid readers of this blog will note that one of the first calculations I did was determining what fraction of people that have ever lived are still living.  I got about 30%, but a more reasonable number is 6%.  I'll leave it as a challenge for you to figure out why that's still not a bad estimate in light of today's result. 

Tuesday, June 12, 2012

New Contest: Look Out!

To celebrate the Kings winning their first Stanley Cup in 45 years, I'm gonna have a hockey estimation contest.

Here's how it works. I’m posting a Fermi question below. To enter, estimate an answer and send it to “aaron at aaronsantos period com.” If your answer is closest to mine, I'll mail you a free signed copy of Ballparking.1 Second prize receives a signed copy of my other book, How Many Licks? Or How to Estimate Damn Near Anything. Submit your entry on or before July 31, 2012.  Don't worry…I won't spam you or share your email with any third parties. Here's the question:

The last time I went to an NHL game, I saw this:

"That can't be safe!" I thought to myself.  As it turns out, I was right.  There's a small but finite chance of the puck hitting the camera:

The question: How many pucks have broken cameras in NHL history?

[1] NOTE: I make no pretenses that my answer is correct or even close. Your answer may very well be a better estimate than mine. In fact, your estimate may even be exactly right and you still may not win the contest if somebody else's answer is closer to mine. Sorry about that. This is the best way I could come up with to pick a winner and I'm not changing it now. Like any good game, there's an element of luck required even if you do have great skill. With that disclaimer out of the way, good luck and happy calculatings!

Monday, June 11, 2012


Reddit likes verification, so here's me...

Here's me with bacon.

This is me pretending to be Geordi La Forge with bacon.

Saturday, June 9, 2012

Reddit AMA On Monday

I'm going to do a Reddit AMA on Monday morning next week.  If you have anything you want calculated, stop by Reddit and ask a question.

Father's Day Shopping

Looking for a Father's Day gift?  Is your dad a sports fan?  If so, I have a pretty good suggestion.  Try Ballparking: Practical Math for Impractical Sports Questions!

Sunday, June 3, 2012

Ballparking: We Have a Winner!

Ladies and gentlemen, we have a winner!  To mix things up a bit, I decided to simulate this one.1  I used Mathematica for the simulation.  There is a gravitational force

F = m g

on the tennis ball.  Here, m = 57 g is the mass of the ball and g = 9.8 m/sis the acceleration due to gravity.  The wind can be represented by a drag force

F = C A ρ v2 / 2,

where C = 0.3 is the drag coefficient, A = 0.0036 m2 is the cross-sectional area of the ball, ρ = 1.2 kg/m3 is the density of air, and v is the velocity of the ball relative to the air.  I made an animation of the simulation below.


It turns out you need a wind of about 45 mph.  Congratulations to our winner!

[1] I know, I's supposed to be an estimation.  You know that scene in Raiders of the Lost Ark when Indiana Jones shoots the sword-wielding guy?  Well, it turns out that was supposed to be an elaborately choreographed fight scene, but Harrison Ford had severe diarrhea that day so he called an audible.  Spielberg loved it, so it stayed in the movie.  Why do I bring this up?  Do I have severe diarrhea?  No, but I was tired, lazy, and in the mood to simulate/animate something.  Plus, we had a lot of good estimates and I wanted to make sure my number was halfway decent.  Besides, I thought this one deserved a stick-figure animation.  Long story short, this was my "shooting the swordsman" moment.

Aaron Santos is a physicist and author of the books How Many Licks? Or How to Estimate Damn Near Anything and Ballparking: Practical Math for Impractical Sports Questions.   Follow him on Twitter at @aarontsantos.