Ask Santos: Goofy Edition

Now we go from the seriousness of Ebola to something a little more goofy. Today's question comes from Kaylee.  She writes,

Since I was little and watched A Goofy Movie quite often, I always wondered what the answer would be to Goofy's question he asks Max while sleeping, trapped in their car due to the presence of Bigfoot: How many cups of sugar does it take to get to the moon? 

I'm not sure if it's even possible to calculate, or what exactly the question is specifically asking (whether it is meaning to use the sugar as fuel, or to build a bridge from sugar), but I'm just curious. Call it a question that's plagued me since childhood.

Why is it that Goofy can talk, but Pluto can't? Seriously, what mutation lead to this genetic monstrosity?
I'd hate to see the Punnett square on that one!  (Image from Wikipedia.)

As you point out, the answer to your question depends on context. Unfortunately, I've never seen A Goofy Movie (I'll have to check it out when I get a chance), so I'll need to use your interpretations:

(1) Building a bridge of sugar. There are obvious practical problems with this (e.g. structural integrity of the sugar, aligning the bridge with the constantly moving moon, not being able to breathe in space, the base of the bridge dissolving when its humid out, etc.) However, from a pure volume perspective, we might imagine building a ladder bridge that was two feet wide, half a foot thick, and long enough to reach to the moon. The moon is 240,000 miles away. If we use the equation 

volume = base*width*height, 

we can get a rough estimate of the volume of sugar ladder and use the density of sugar (1.6 grams per cubic centimeter) to find the total mass of sugar needed. By my estimate, you'd need about 60 million tons (~57 billion kilograms) of sugar to build a bridge to the moon.

(2) Using sugar as a fuel. Food is just like gasoline: both act as fuel. "Calories" are a unit of energy. Just like we talk about the miles per gallon you get out of a good car, you could talk about miles per gram of sugar you get out of a fit person. However, rather than power people, we're using the sugar energy to power a rocket. To escape Earth's gravitational pull, you need an "escape speed" of roughly 12 km/s.¹ Rockets typically have a mass of around 5 million pounds, which means they'd require about 40 billion food calories to reach the moon. Carbohydrates have about 4 calories per gram, meaning you'd need about 5000 tons of sugar to reach the moon.²

Just to compare, it would take 5000 tons of sugar to power a rocket to the moon, and 60,000,000 tons to build a ladder to the moon. That's about 10,000 times more sugar required for the ladder.

Thanks for a great question, Kaylee!

Do you have a question you'd like to ask Santos? Try Tweeting him at @aarontsantos for a chance to have your question appear on Diary of Numbers.

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.

[1] This ignores important effects like air resistance.

[2] This weighs more than the rocket itself, meaning you'd need even more energy to reach the moon. For comparison, you can get a rough idea how much rocket fuel is needed here:

http://www.space.com/18422-apollo-saturn-v-moon-rocket-nasa-infographic.html

Another Reddit AMA

My last Reddit AMA was fun. I'm going to try to do another "Ask Me to Calculate Anything" next Tuesday, November 4. Tell your friends...or your enemies, I'm not picky.

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.

Ask Santos: Ebola Edition

Today's question comes from my buddy Karen at the University of Michigan:

What are the odds of an individual being killed in an auto accident in Michigan (~680 deaths in 2014 so far) vs. ANY individual in MI contracting Ebola?

Not sure if ebola or modern art. Possibly both. (Image from Wikipedia)

At present, there are no reported cases of Ebola in Michigan. The only way to contract the disease is through direct contact with the bodily fluids of someone who already has the disease. For this reason, the only way of infecting someone in Michigan is by bringing an infected person to Michigan. Roughly 2.8 million people fly internationally into Detroit Metropolitan Airport each year.¹ That's out of a total of 800 million passengers flying to or from the United States.² According to the news, no more than 10 people have gotten on a plane with Ebola in their system. From these numbers, we can compute

probability of a random person on a plane having Ebola = 0.00000125%

probability of that plane flying to Detroit = 0.35%

This means the probability of someone with Ebola making it to Michigan is 0.0000000044%. Even if we assume the probability of contracting the disease when exposed is 100%, that's still a minuscule probability. You're about 2 million times more likely to die in a car crash in Ann Arbor than you are to catch Ebola. Thanks for a great question, Karen!

Do you have a question you'd like to ask Santos? Try Tweeting him at @aarontsantos for a chance to have your question appear on Diary of Numbers.

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.

[1] http://en.wikipedia.org/wiki/Detroit_Metropolitan_Wayne_County_Airport

[2] http://www.rita.dot.gov/bts/press_releases/bts016_13

***Update***  After reading this post, a friend of mine commented on the inaccuracy of using data from relatively new phenomena to make predictions. In his words, "using prior cases over a few months for an event without precedent is bad modeling." I agree wholeheartedly. The exact numbers should be taken with a grain of salt. Statistics like the "likelihood of someone getting on a plane with Ebola" are clearly complex, time-dependent phenomena and need more than just an order of magnitude estimate to make useful predictions. In a more professional setting, I would model it as such. However, that's not really the point of this blog. My main goal is to get people interested in working out the numbers on their own, and using very complex models would quickly alienate readers who might already be math-phobic. If the blog inspires these people to learn more about math, and they eventually realize I'm pulling some numbers out of my ass, my mission will complete. However, even for the math-phobic, my description could have been more precise.  In my friend's words, 

It is better to say: 

"Contracting Ebola is non-quantifiably improbable and if you're living anywhere in the developed world, you, as an individual, have bigger problems.  

On the other hand, West Africans could use your government's directed attention. Perhaps you should be focusing your attention and advocacy on Ebola in West Africa because it is both morally good and in your enlightened self-interest. "

I couldn't have put better myself.

Ask Santos: Scrooge McDuck Diving

This one is for my buddy Carolyn and her wife Heather.  They ask,

How much money would you have to have to be able to pull off swimming around in it Scrooge McDuck-style?

An Olympic-size swimming pool is 50 meters long, 25 meters wide, and one to three meters deep, giving a total volume of roughly 2500 cubic meters. Swimming in gold coins is a lot more expensive than swimming in pennies. To make this more cost effective, I'm first going to assume he's swimming in pennies rather than gold. A penny has a volume of 0.349 cubic centimeters, meaning you'd need roughly 7.2 million dollars to fill an Olympic-sized swimming pool with pennies.¹ Even a smaller non-Olympic pool would require over one million dollars.

If money is no option, you could instead fill the pool with gold coins. Given that one cubic centimeter of gold costs about 80 dollars, a gold-filled swimming pool would require roughly 2.0 trillion dollars, or about 12% of the national debt. This is certainly a hefty amount of coin, but I would contend the sheer dollar amount is not the main obstacle to swimming around in a swimming pool Scrooge McDuck-style. Once again, Family Guy does a pretty good job illustrating the practical difficulties of the matter,

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.

[1] You need slightly less money if you take into account the packing fraction of pennies.

I Ain't Dead Yet

In the words of the great Richard Pryor, "I ain't dead yet."

If you've checked the blog over the past six months, this fact might comes as a great surprise. I haven't exactly been lighting up the Blogosphere lately. There's a reason for this. Apparently starting a new tenure track position is quite time consuming.

"Good new, Lieutenant, you've just been promoted!"

Between applying for grants, research, prepping classes, grading, and working with students, I've been super busy.

Lack of free time aside, life at Simpson has been very good. When you ask new physics teachers about their students, they often respond by saying...

Fortunately, Simpson has many talented and intellectually curious students, so I rarely feel like the offensive ethnic stereotype in the video above. That said, I'm older now, and I don't always know how to relate to students. Why do they think Christian Bale is the best Batman? What is the obsession with One Direction? Why do my student evaluations say I need more cowbell? And, perhaps most importantly, why did did they give me the nickname (and website) Moon Jesus?¹

Needless to say, with my students keeping me busy, I haven't had a lot of time for writing (either blogs or books), which is probably just as well...

"Piss off, Stewie!"

That said, I have been keeping somewhat busy. I wrote an article for the Naked Scientists. I made some holiday estimations for the always fabulous Desiree Schell at Science for the People.² That lead to Kyle Munson's nice write-up in the Des Moines Register. Also, I got this kinda fun email:

Now, if I can only round up Dentist Aaron Santos, Photographer Aaron Santos, and Baseball Player Aaron Santos, I can fulfill my dream of starting "The Legion of Super Heroes Named Aaron Santos".³

Anyway, that's where I stand. Hopefully, I can carve out some free time soon so I can start writing more consistently again.

Stay well, Internet.

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.

[1] I still have no idea whether or not this is a compliment.
[2] Also, my apologies to Desiree.  I just realized that I—idiot that I am—have been misspelling her name for years, and she's been too nice to correct me on it.
[3] Admittedly, our super powers are less than inspiring.

Special Guest: Rick Lombardo

Today's special guest is San Jose Repertory Theatre's artistic director, Rick Lombardo. A prolific regional and off-Broadway director, Mr. Lombardo has received numerous awards for his work both at San Jose Rep and New Repertory Theatre in Boston.

Many artists are concerned with the current trends in climate change. Mr. Lombardo is no exception.  He writes,

I've been thinking about climate change a lot, and while this question doesn't directly correlate to a warming planet, it was inspired by the problem. If all the water molecules present in the atmosphere at any one moment fell to the surface of the earth, what would happen to average sea levels around the world?

If you've paid attention to science news, you know that melting glaciers are contributing to rising sea level with an average rise of 3.3±0.4 mm over the past twenty years. This doesn't seem like very much, but over several decades it can have a profound impact on coastal communities. How does it compare with the rise that would occur if all atmospheric water rained down at once?

Wikipedia's entry for "Atmosphere of Earth" states the following:

According to the American National Center for Atmospheric Research, "The total mean mass of the atmosphere is 5.1480×10¹⁸ kg with an annual range due to water vapor of 1.2 or 1.5×10¹⁵ kg depending on whether surface pressure or water vapor data are used; somewhat smaller than the previous estimate. The mean mass of water vapor is estimated as 1.27×10¹⁶ kg and the dry air mass as 5.1352 ±0.0003×10¹⁸ kg."

From these figures, we can see there are, on average, roughly ten trillion tons of water in the atmosphere. If it all fell to the Earth at once, would it produce floods of literally biblical proportions?

"Get your cubit stick ready!"

Liquid water has a density of one gram per cubic centimeter. If you simultaneously condensed all the water in the atmosphere into liquid form, you'd have about 10 billion cubic meters of water. The oceans cover about 400 million square kilometers of the Earth's surface. Spread out over this area, all the water from the atmosphere would cause the oceans to rise a grand total of 25 microns, roughly one-fortieth of a millimeter. Needless to say, things wouldn't change very much.

"OK, never mind...everybody off the boat."

Thanks for a great question, Rick!

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.

Special Guest: George Goodfellow

To celebrate the start of a new school year, we have a question from a very special guest. In addition to being Rhode Island's 2008 teacher of the year, George Goodfellow was also my high school chemistry teacher and one of the main reasons I became a scientist.¹  Mr. Goodfellow writes,

At what point in the equilibrium that is a balance of living plant organisms, living animal species and the total available energy on Earth will the ratio of Animal/Plant Species become so large as to create a collapse of the human population?

Leave it to Mr. G to start us off with a light topic. The question reminds me of Trantor, the fictional city-world in Isaac Asimov's Foundation Trilogy. To support Trantor, tens of thousands of ships from twenty agricultural worlds had to be flown in just to supply enough food.

Trantor would look a lot like Star Wars's Coruscant if you got rid of all those pesky Jedi.

If we ignore help from other worlds, the collapse should happen much more quickly. At present, the world population sits around seven billion and is constantly growing. Given Earth's land area is roughly 150 million square kilometers, each person would own about 5.3 acres if the land were divided equally. How much land does one person need to survive? This homesteading infographic provides a good starting point:

How much land is enough to live off? (Click to expand) 

According to the infographic, you need at least 0.5 acres of land per person to survive. This would imply Earth could support, at the very most, about 70 billion people before collapse would occur. Note that we haven't accounted for the fact that not all land is farmable. Given the large swaths of land in desert, mountain, and other inhospitable regions, we're probably significantly closer to carrying capacity. If only half the land were farmable, we could support 35 billion people, meaning we'd already be at 20% of the maximum carrying capacity.

Are there any ways to expand this limit? I've written previously about skyscraper farms. While the maximum number of people that could be fed by one of these farms is greatly exaggerated by the farms' proponents, the farms may still significantly increase Earth's maximum carrying capacity. Furthermore, food scientists are constantly finding ways to feed the growing population...

...food scientists like Norman Borlaug. Note: Never try to be as cool as Norman Borlaug. Unless you can save over a billion people from starvation, you're not going to come anywhere close. And to think, this probably the first time you've heard of the man.

Short of coming up with more efficient ways to develop food, our most realistic solution seems to be pumping NASA full of money so they can supply us with tens of thousands of ships that will travel back and forth between twenty terraformed agricultural worlds in order to supply Earth with its daily food needs. Or, you know, people could start using birth control and have fewer kids. Either way would work.

Thanks for a great question, Mr. G!

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.

[1] Admittedly, there were a few nights when I cursed him for bestowing this fate on me, but for the most part it's been pretty good.

Wave at Saturn

WHO HAST WOKEN ME FROM MY SLUMBER?!?!?!?!

Oh, it's Rob Sparks, author of The Half Astrophysicist Blog:

In case you missed it, there's a cool picture of the Earth and Moon taken from Saturn. Apparently a bunch of us earthly homebodies decided to wave at the Cassini spacecraft as it took pictures of us.

"Hi, Mom!"

On Earth, we're hit with a flux of about 1400 W/m² of sunlight. Our bodies have an area of about one square meter, and I'll assume only 1/10 of the light that hits each of our bodies gets reflected out of the atmosphere. Since visible photons carry a few electron volts,¹ we can estimate that 10²⁰ photons leave each body every second.  These photons will be distributed over a sphere with a radius equal to the distance to Saturn, approximately 1.2 billion kilometers. A small fraction of this sphere coincides with the Cassini's camera lens. Assuming it's like most digital cameras, the area of Cassini's lens should be roughly 10 square centimeters. That's about one part in 10²⁸ of the total area covered by the photons. Assuming a shutter speed of one second, the probability that one of your photons will appear in the "Wave at Saturn" picture is about one in one-hundred million. Since the world contains 7 billion people, there's a pretty good chance at least one of humanity's photons is in the picture.

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.

[1] One electron volt is equivalent to 1.6×10^-19 Joules.

101 Things I Learned in Engineering School

I just got John Kuprenas and (friend of the blog) Matt Frederick's 101 Things I Learned in Engineering School, and I absolutely love it. It concisely and elegantly summarizes the essential lessons you'll discover as an undergraduate science or engineering major. I plan to reference it frequently in my physics classes at Simpson next year. I highly recommend it as a gift for any recent engineering graduate. Also, check out the other books in Matt's 101 Things I Learned... series.

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.

Special Guest: Damon Brown

Today's special guest is Damon Brown. Damon has quite an eclectic collection of interests. He's written for a diverse group of audiences, with articles appearing in everything from Playboy to Family Circle, while covering an equally diverse array of topics: pop culture, technology, video games, music, human sexuality, etc. He's written 14 books, his most recent being Our Virtual Shadow: Why We Are Obsessed With Documenting Our Lives Online.

Damon asks,

The latest data says the average American spends about an hour using his or her smartphone every day, usually doing some non-phone activity like using an app. So, assuming we get a phone when we become teenagers, how much of the average American life will be spent using the phone?

Admittedly, I'm an odd choice to do a calculation like this.¹ I still (somewhat proudly) have what I consider to be a state-of-the-art flip phone. Despite being seven years old and having been dropped so many times it's developed the phone equivalent of cerebral palsy, it still serves my purposes quite well. For a luddite like me, Damon was kind enough to provide me with this reference breaking down the various uses of smart phones:

My phone can do exactly two of these things.

The lifespan of a person living in a technologically advanced society is about 80 years. If everyone receives a phone upon becoming a teenager, then, on average, people will each have about 70 years to stare at tiny screens. One hour per day is 1/24th of your total time, which means you'll spend a total of roughly 3 years on your smartphone. Breaking this down, that'd be 300 days of talking, 220 days of texting, 96 days of gaming,² and 340 days worth of visiting the Internet and social networking.

Thanks for a great question, Damon! Find out more about Damon on his website or follow him on at @browndamon.

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.

[1] At least, I would be an odd choice if there were other silly physicists out there doing calculations for people on their blog.

[2] FIVE GOLDEN RIIIIIINGS!!!!!!

A Treatise on Jose Iglesias

Last week I posted a note on Facebook about Red Sox shortstop Jose Iglesias. According to Baseball Reference, the light-hitting Iglesias has a career WAR¹ over three times larger than slugging third baseman Will Middlebrooks, despite having less than half as many at bats. This season has been particularly rough on Middlebrooks who sits at a WAR of minus 0.7 with a batting average hovering around the Mendoza line. In contrast, Iglesias has been (relatively speaking) tearing the cover off the ball by hitting well over .400 with a WAR of +1.8, all while playing stellar defense. To put this in context, Iglesias's stats, if extended over a 162-game season, would give him a WAR of 10.4, which is about 50% better than Miguel Cabrera's WAR during his MVP Triple Crown season last year. This inspired my buddy Adam to ask me about sample sizes, so I figured I should address this on the blog. In baseball, when should a sample size be considered significant?

First off, it should be stated that there are no exact cutoffs in probability and statistics. As I explain in Ballparking, even a career .200 hitter like Mario Mendoza has a (small) chance of hitting .400 over the course of an entire season. There's no magic number above which we can definitively say, "These results are statistically significant." Fluctuations happen in any sample size no matter how large. That said, if we have a random sampling of statistically independent events, we can make definitive statements like the following:²

There's a 95% chance that Jose Iglesias's average over his next 92 at bats will be between X and Y.

Here, X and Y define what's called a confidence interval. We have limited data, but given the information we do have, we're 95% percent certain that Iglesias's batting average over the next 92 at bats will be between two numbers X and Y. What are those two numbers? Wikipedia's entry for sample size determination gives a good description of how to calculate them. The width W of the confidence interval is given by

where n is the sample size, i.e. the number of at bats. Since Iglesias has had 92 at bats so far, we have W = 0.045. Iglesias is currently batting .435. If we believe Iglesias's stats represent a random unbiased sample, then we would expect there to be a 95% chance Iglesias's next 92 at bats will give a batting average between .412 and .457.

Did I make a math mistake? Is Jose Iglesias the next Ted Williams? The problem lies in the fact that Iglesias's 92 at bats were not selected at random. I noticed Iglesias's batting average because it was incredibly large. If I took any 750 baseball players and gave them each 92 at bats, there's a good chance some of them would, by shear dumb luck, hit over .435. Even at a 95% confidence ratio, you'll still have 5% of players with batting averages that lie outside the confidence interval. Given 750 Major League players, this means roughly 38 players will be outside the confidence interval. Iglesias's .435 average is almost certainly an outlier due to random fluctuations in the large population of baseball players. The only reason I singled him out is because he randomly (and luckily) happened to have one of the largest and therefore most attention-drawing fluctuations.

Despite the fact that Iglesias's average is almost certainly a random fluctuation, the shortstop still shows a lot of promise. If we look at his 2012 season during which he hit an abysmal .118 in 25 games, we notice he still nets a positive 0.3 WAR. Extended over a 162-game season, he would get a not terrible WAR of 1.9. Why? His defensive capabilities more than adequately compensate for poor hitting. Over his career, he's averaging a 5.4 WAR per 162 games, which is more than double the 2.6 WAR averaged by current starting shortstop Stephen Drew and over seven times greater than 0.7 WAR averaged by current starting third baseman Will Middlebrooks. Even given the small sample size, it's tough to argue that Iglesias doesn't deserve a spot in the starting lineup.

If you like math and sports or know someone who does, make sure to check out my book Ballparking: Practical Math for Impractical Sports Questions.

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.

[1] "WAR" is one of those newfangled stats that sabermetricians like to throw around. It stands for "wins above replacement" and is supposed to represent the number of extra wins a player is expected to contribute compared to a standard replacement player.
[2] Strictly speaking, it's a bit more complicated if we're talking about actual baseball players rather than mathematical probability distributions. For example, a player's theoretical batting average is not constant over time. It can increase or decrease depending on the player's age or health.

Father's Day Contest Winner

We have a winner! If you followed the blog for any length of time, you know I love quirky questions. This one is pretty quirky.  Carl writes, "How many beers did it cost me to give my kid flugelhorn lessons?"


Ah, the flugelhorn: bagpipe of brass instruments. I suspect Carl paid in more than just beer:

It's OK, Carl. Little Timmy has to get better sometime.

If flugelhorns are anything like pianos, then a single lesson likely costs somewhere between $10 and $50. Some kids take lessons for years, while others take one lesson and never go back. I'll assume that the average flugelhornist takes lessons once a week for a year at a cost of $20 per lesson. That gives a total cost of about $1000 for flugelhorn lessons. I'll assume Carl is a Dos Equis man.

At $7.99 for a six-pack, it would cost 750 beers to pay for Carl's kid's flugelhorn lessons.

Thanks for great question, Carl! You'll be receiving a free signed copy of Ballparking in the mail shortly.

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.

Happy Father's Day!!!

Happy Father's Day!  I'm travelling today, but will post the winner of the Father's Day contest soon.

-AS

Bad Ass Astronauts

Given my disappointment with the new Star Trek movie, it's nice to know there are still some legitimate (and non-fictional) bad ass space travelers out there. Canadian astronaut Chris Hadfield recently made headlines for an awesome video where he sings David Bowie's Space Oddity.

In a karaoke battle of the Chrises, Hadfield would totally destroy Pine.¹

While that video is awesome in its own way, I personally prefer how Hadfield explains cool science to the masses:

All this talk about astronauts and cool science reminds me of a question my physics buddy Kendall asked me to do:  How much extra time do astronauts gain by being in orbit?

According to Einstein's theory of relativity, time slows down as you move faster. Since the International Space Station (ISS) travels about 4.8 miles per second, Hadfield and the other astronauts on board should age somewhat slower than the rest of us.

Using the time dilation formula, we find that a person traveling at the speed of the ISS ages at a rate 0.000000033% slower than the rest of us. As our paper of the week demonstrates, even though this effect is tiny, it's still measurable if you've got a precise atomic clock.  After one year on the ISS,² an astronaut would age 10 milliseconds less than a person at rest on the Earth because of special relativity.

As amazing as our above result is, it's not quite correct. The problem above illustrates the principle of special relativity, which Einstein discovered was the correct way to describe fast moving objects in the absence of heavy masses. Eleven years later, Einstein published his general theory of relativity, which explains how time dilates in a gravitational field. According to general relativity, time slows down as you move closer to heavy masses (i.e. people on Earth would age more quickly than people far away from its gravitational pull). At a height of 230 miles above Earth's surface, astronauts age 0.000000098% slower than objects without any heavy masses in the vicinity. After one year on the ISS, astronauts would age 31 milliseconds less than a person far away from any masses, but 1.8 milliseconds more than a person on the surface of the Earth.

[1] However, in a battle of the Kirks, I'm pretty sure this means Pine could take Shatner.
[2] Valeri Polyakov holds the record for the longest time on the space station with a time of 437.7 days.


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.

Father's Day Contest Update

Per popular request, I'm gonna say it's OK if you want to mail me your Father's Day questions rather than tweet them. Send them to "aaron at aaronsantos period com".

Scientific Paper of the Week: Slowing Down Time

Today's paper of the week was published in 1972. To my knowledge, it's the first direct measurement of time dilation as predicted by Einstein's special theory of relativity. The authors used atomic clocks to measure time slowing down in fast moving commercial jets. Good stuff!

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.

Special Guest: Jim Ottaviani

Today's special guest is author Jim Ottaviani. Jim has written several excellent comic books on science history including Dignifying Science: Stories About Women Scientists, Feynman, and Primates: The Fearless Science of Jane Goodall, Dian Fossey, and Biruté Galdikas. Jim writes:

I live close to Detroit so I always have to hear about horsepower this, horsepower that. I prefer dinosaurs, though, so my question is this: How many Dryptosauruspower is under the hood of a beautiful, split-windowed, 1963 Corvette Sting Ray Coupe?

The whole point of "units" is to provide a convenient means of comparing quantities. Despite most scientists' insistence that the United States convert to metric units, there's nothing fundamentally special about meters and kilograms. You're perfectly welcome to continue measuring distance with a dead king's foot provided you don't mind remembering more complicated unit conversions. Still, perhaps it's better to have units we can all relate to. With that in mind, Mr. Ottaviani's question is a particularly useful one because the term horsepower is, at least for those of us in urban areas, a bit of an anachronism.¹ According to Wikipedia,

The development of the steam engine provided a reason to compare the output of horses with that of the engines that could replace them. In 1702, Thomas Savery wrote in The Miner's Friend: "So that an engine which will raise as much water as two horses, working together at one time in such a work, can do, and for which there must be constantly kept ten or twelve horses for doing the same. Then I say, such an engine may be made large enough to do the work required in employing eight, ten, fifteen, or twenty horses to be constantly maintained and kept for doing such a work…"

In context, horsepower is a perfectly legitimate unit. If today an infomercial claims you'll lose weight three times faster with Bowflex than with a Shake Weight, than you could define

1 bowflex = 3 shake weights,

and in 100 years "shake weight" will be a legitimate unit of measure, not an utterly ridiculous punchline. 

The Shake Weight: laughing stock of the exercise community

Much like with the shake weight, we could easily standardize dryptosauruspower as a unit provided we can find the correct conversion factor between it and horsepower. A dryptosaurus is thought to have weighed about 1.5 metric tons or roughly 3.3 times the mass of a horse. This would put it closer to the weight of a giraffe or hippopotamus. Giraffes eat about 75 pounds (34 kilograms) of food per day, while a hippo eats about 88 pounds (40 kilograms) of food per day. Since a dryptosaurus is about the same size, it likely eats about the same amount of food.² Let's assume the typical dryptosaurus takes in about 80 pounds of food each day.

It's bigger than you, so it eats a lot more.

A 1000 pound horse takes in about 30 pounds of food per day, roughly 2.7 times less than the dryptosaurus. Food is a source of energy. Power is the amount of energy used in a given amount of time. Since power is proportional to energy which is proportional to the amount of food, it should be true that the dryptosaurus is roughly 2.7 times more powerful than a horse.³ We could use the following equation to convert from horsepower to dryptosauruspower:

1 dryptosauruspower  = 2.7 horsepower

A 1963 Corvette Sting Ray Coupe with 100-dryptosauruspower

Depending on what kind of engine is installed in your Corvette, you could have a variety of different horsepowers. According to Wikipedia, typical values range from 250- to 340-horsepower. Using our conversion equation above, the horsepower of a 1963 Sting Ray Corvette ranges from 93- to 130-dryptosauruspower.

Thanks for a great question, Jim! You can find out more about Jim Ottaviani and his books at G. T. Labs or on Twitter.

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.

[1] There are those who might ask why we would want to convert to dryptosaurus power since the dryptosaurus, having lived over 60 million years ago, is an even greater anachronism than the horse. A fair point, Dear Reader, to which I have two responses. (1) Dinosaurs are awesome, and (2) horses suck.
[2] I am, of course, making a big assumption here by assuming food intake is related to body size. The difference between dryptosaurus and a giraffe could be substantial given that one is a carnivorous reptile and the other is a vegetarian mammal. Even within one species, you can have a pretty wide range of values for caloric intake. For example, I'm pretty sure I eat about twice as many calories each day as my wife does. That said, the caloric need for similarly-sized animals should be roughly the same to within an order of magnitude. We can see this by comparing the giraffe and hippo, two very different but similarly-massed creatures that take in roughly the same amount of food each day.
[3] Here, we're talking about the average power output as opposed to the peak power output, which is theoretically what we're quantifying when we talk about horsepower. These could be somewhat different, but I suspect they're still relatively close.

Star Trek Review and Estimation

***WARNING: Spoiler Alert and long rant ahead***

There's not too much I'll be able to add to what other people have already said more eloquently, but I'm going to put in my two cents in anyway. I just saw Star Trek: Into Darkness, and I have to admit I was disappointed. It wasn't Catwoman bad or even Phantom Menace bad, but it was quite possibly "Phantom Menace minus the scenes with Jar Jar" bad.

Don't get me wrong, Benedict Cumberbatch was pretty good as Khan¹, but I found the editing to be downright dreadful. In one scene, Spock just barely pulls himself onto the edge of a flying car and in the next shot Khan kicks him back 15 feet from the middle of the car. Now, I could easily overlook one or two poorly edited shots, but I can't ignore the scene with Chekov holding up the much larger Kirk and Scottie, who are dangling over the side of a bridge. Mind you, Kirk couldn't hold Scottie by himself, but somehow the 130 pound Chekov has no problem lifting them both. I can only assume some amazing and dramatic miracle feat of strength occurred while the cameras were off, because a second later they're all happy and running through the Enterprise without any explanation. Whatever Chekov did must have been amazing to see, but apparently J.J. Abrams just wants us to fill in the details by ourselves (more on that in a moment).

While we're on the subject, when did J. J. Abrams decide to become Michael Bay?² Seriously, I'd much rather see how that bridge thing got solved than watch another giant explosion that I'm just gonna tune out. And, no, Mr. Abrams, you can't just lazily copy highly emotional moments from the old movies and expect to elicit the same emotional response in your audience. Kirk's "death" didn't make me sad or even nostalgic. It made me think, "Hmm...they've got a cash cow of a franchise here and there's no way they're killing off Kirk, so I guess the dead tribble's coming back to life." That said, having Spock tear up did make me feel some genuine emotion, so I guess there's that.

Still, the main source of my disappointment has little to do with these small quibbles³ and more about the general direction this franchise has taken. The best part about old Star Trek is that it was actual science fiction, not cheesy action-adventure set in space. With that in mind, you can't just freeze a volcano and call it cold fusion without Gene Roddenbury and Isaac Asimov rolling over in their graves so fast we could use them as a renewable energy source.⁴  I'm used to Star Trek having a bigger message, and if there was one here I totally missed it.

Since this is Diary of Numbers, I can't justify ending this rant without at least some bit of calculation. With that in mind, my friend John had the best explanation I've heard for how Chekov could suddenly gain superhuman strength. According to John, "Russian Special Forces kettlebell workouts gives you strength of two men..."  Where would this put Chekov in the pantheon of great Russian weightlifters?

There's nothing the Muscleless Wonder and I take seriously if not for science and weightlifting, and this estimate combines both. If Chekov is as small as I think he is, he'd be in just about the lightest weight class of lifters. To get both Kirk and Scottie back on the bridge, I imagine him doing a motion similar to a snatch. I'd put Chris Pine and Simon Pegg at about 175 pounds each, meaning their combined weight would be 160 kilograms. With a 138-kilogram snatch, Halil Mutlu of Turkey holds the world record in the 56-kg division. Chekov would easily smash this record. Unless he's the next Pocket Hercules, there's no way he's pulling Kirk and Scottie back on the bridge.

Seriously, get it together, J.J. Abrams. I'm willing to give you a pass on this one under the assumption that you're distracted putting together a kick ass Star Wars movie. No second chances after that.⁵ Help me, J.J. Abrams.  You're my only hope.

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.


[1] My buddy Matt was not so glowing in his endorsement of Cumberbatch: "I don't feel like I can call him 'Khan' because he isn't. He's British. And white. Khan Noonian Singh was of Indian descent. Hence the reason he took the title of Khan."

[2] QOTD from a friend on Facebook: "I'm pretty sure J.J. Abrams is the non-union Mexican equivalent of Christopher Nolan..."
[3] Let's call them "tribble quibbles"!
[4] Yes, I'm stealing that joke from somewhere, but I can't remember where, so I can't cite it.
[5] Though if I'm completely honest, I'm still probably going to spill out full price for whatever creatively emaciated junk they're going to throw at us. I can't help it....I need my cheesy movie fix.

Father's Day Contest

Father's Day is in a month, and I know the perfect present for the sport-loving dad:

In fact, I'm holding a Father's Day contest. It's a little different than my usual contest. Normally, I give you guys a question to answer. This time, I want you to give me a question.  What's a good estimation question that every father would get a kick out of?

To enter, tweet your question to @aarontsantos by June 14, 2013. I promise not to spam your inbox or sell your email to evil corporate overlords. On the 14th, I'll select what I think is the most interesting question and answer it on the blog for Father's Day. The winner will receive a free signed copy of Ballparking to give to your dad (or anyone else you like).

***Edit*** Per popular request, I'm gonna say it's OK if you want to mail me your questions rather than tweet them. Send them to aaron at aaronsantos period com.

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.

Scientific Paper of the Week: Alpher, Bethe, Gamow

Oh, the things physicists do for a lark. Today's paper, titled "The Origin of Chemical Elements", describes how the Big Bang explains the relative abundance of hydrogen and helium in the universe.

The universe expanding after the Big Bang.

While tremendously interesting in its own right, the content of the paper is not the reason for its selection. In my General Physics class, we just covered alpha, beta, and gamma decay. The authors of this week's paper are Ralph Alpher, Hans Bethe, and George Gamow.

"Well, that's a fun coincidence!" you say.

Well, not quite...

Left to right: Ralph Alpher, George Gamow, and Hans Bethe, who's totally riding their coattails.

It turns out Bethe played no actual role in writing the paper. Gamow added him because he liked play on words with "alpha-beta-gamma". And they say physicists don't know how to have fun!

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.