Sunday, January 31, 2010
Friday, January 29, 2010
= 2 · (100 m) / (4.07 m/s2)
= 7.01 s.
Realistically, Johnson would probably reach his top speed somewhere between 10 and 40 yds and then maintain a roughly constant speed throughout the rest of the race. This being the case, he would almost certainly finish somewhere between these two bounds. Logic would seem to indicate that Bolt, typically regarded as the fastest man in the world, would still win handily. Unfortunately, our simple estimation is not precise enough to say who would win with any certainty. This is an important point. Sometimes we don't have enough background information to calculate definitive answers, and we can only calculate upper and lower bounds. Sadly, we'll have to deal with ESPN pundits endlessly debating who the faster man is.
Thursday, January 28, 2010
Thursday, January 21, 2010
Here's part II of the problems I solved for Skeptically Speaking:
Physicists say it's possible for all the air molecules in a room to spontaneously slide over into one corner, suffocating everyone in the room. What are the odds of that actually happening?
Assuming an ideal gas*, the probability of a molecule being on any side of the room is ½. The probability of n molecules all being on the same side is then (½)n. The density of air is about 1.2 kg/m3 at sea level and 20° C. A reasonably large room might be 10 m by 10 m by 3 m giving a total volume of 300 m3. Multiplying the density of air by this volume gives 360 kg of air in total. Most of the air is made up of nitrogen with a molecular weight of 0.028 kg/mol. Dividing this into the total mass of air gives about 13,000 moles or n = 7.7x1027 molecules.
Therefore, the probability of finding all the gas molecules on one side of the room is ½ raised to the 7.7x1027th power. I’m not going to even attempt to write out this number. It is extraordinarily small. “How small?” you say. If you were to try to write out all the zeros, you would cover the surface of 10 million Earths.
*A real gas is non-ideal. If you want to do this problem rigorously, you need to account for the fact that each molecule takes up a finite amount of space.
If the Skeptically Speaking team got locked in the on-air booth, how long would it take for them to run out of air? We have 5 people in the booth every show, and the booth is about 10 ft by 6 ft.
Technically, you’d never run out of air, but you would run out of breathable oxygen, which gets converted to CO2. Assume you take a breath once every 3 seconds. According to Hypertextbook.com, human lungs have a volume of about 5000 cm3. Oxygen makes up about 20% of air by weight. A quick web search shows that about 25% of the oxygen we breath in gets converted to CO2. Using the density of air from the previous problem, we can compute the amount of oxygen lost per second:
*(5000 cm3 per breath per person)
*(1/3 breaths per second)
*(1.2 kg of air per m3)
*(10-6 m3 per cm3)
*(0.20 kg of O2 per kg of air)
*(0.25 kg of O2 lost per kg O2 breathed in)
5.0x10-4 kg of O2 lost per second
Assuming the room has10 ft ceilings, the total volume is then ~17m3. From this, the total starting mass of O2 can also be calculated:
*(1.2 kg of air per m3)
*(0.20 kg of O2 per kg of air)
4.1 kg of O2
Dividing the amount of O2 lost per second into the total amount of O2, we get ~8200 seconds or roughly 2.5 hours.
How likely is it that we inhale a molecule breathed by Carl Sagan?
This is a pretty hard problem for a number of reasons. For starters, air molecules are constantly reacting to form new molecules. Second, it's not clear how long it would take Carl Sagan's air molecules to mix evenly across the globe. Third, Carl Sagan may have breathed in the same air molecules more than once.
To simplify, let's assume that air molecules get evenly distributed around the world very quickly, that Carl Sagan never breathed the same molecule twice, and that air molecules don't react. Strictly speaking, only noble gases won't react, but this assumption might be approximately true for nitrogen since it's fairly inert. Dr.Sagan lived 62 years and there's roughly 32 million seconds in a year. From this and the numbers used in the previous problems, we can estimate the number of molecules breathed by Dr. Sagan:
(5000 cm3 per breath)
*(1/3 breaths per second)
*(3.2x107 seconds per year)
*(62 years per lifetime)
*(1.2 kg of air per m3)
*(10-6 m3 per cm3)
*(1 mole per 0.028 kg of air)
*(6.022x1023 molecules per mole)
~8.5x1031 molecules breathed in a lifetime
According to Hypertextbook.com, the mass of the whole atmosphere is 5.3x1018 kg. If we assume that air molecules have an average molecular weight equivalent to nitrogen's 0.028 kg/mol, then there should be about 3.2x1042 molecules in the atmosphere. This means that the probability P of a molecule being "Saganized" is then,
P = (8.5x1031) / (3.2x1042) ~2.7x10-12
Here's where it gets a little tricky. The probability of a single molecule not being breathed by Sagan is (1-P). This is very close to 1, so you might be tempted to think it's very unlikely that to share any of your molecules with Sagan. However, the probability that none of the molecules you breath were ever breathed by Dr. Sagan is given by (1-P)n, where n ~ 8.5x1031 is extremely large. Since the exponent is so large, the probability of never breathing a Saganized molecule will be very close to zero. In short, you breathe a lot of molecules breathed by Carl Sagan!
I met my girlfriend online, and she's totally the one, but she lives on the other side of the continent. I've been wondering about the odds: not just that we would meet, but that we'd both be born at this time in history when the Internet exists and allows us to interact over such a large distance.
I did a similar problem in How Many Licks? In it, I estimated the probability of finding Mr./Mrs. Right is about 1 in 200,000, but this is assumed that special someone was born at the same time in history. What if the love of your life was born in a different time period, like in the Christopher Reeve/Jane Seymour movie Somewhere in Time?
To start, I looked up the world population as a function of time. If you assume an entire population dies off after about 70 years (1 lifespan) and is replaced by an entirely new population, you can trace back about 1000 different to 70,000 B.C. With this assumption, you can calculate the total number of humans who have ever lived on the planet and then calculate the fraction of people living today. Surprisingly, I computed about 30% of people who have ever lived are still living. I was a little skeptical of this, so I decided to look it up and found that other people have done this problem as well and they estimated 5.8%. Since these numbers are easily within an order of magnitude of each other, they're likely close to the actual result. As such, the chances of both lovers being born at the same time in history are much better than the chances that they are born in the same geographical location. This is very surprising, but it shows just how fast exponential growth is. In any event, the probability of finding that special someone is 1 in about 600,000.
That's all for now. Thanks again to Desiree, the Skeptically Speaking crew, and the audience for coming up with some great questions!
Wednesday, January 20, 2010
Thanks to Desiree and all the people at Skeptically Speaking who invited me on their show a few weeks back. In case anyone is wondering how I came up with the answers to the audience’s questions, I’m posting my estimations below. There are a quite a few, so I’m posting this in two parts. Enjoy!
How much mass does Bruce Banner gain when turning into the Hulk?
At first, I was thinking of the old Hulk TV show starring Bill Bixby as Bruce Banner. According to IMDB, Mr. Bixby actually turned into a green Lou Ferrigno (~275 lbs), so the mass gain is fairly small. Even if he turned into Shaq (~325 lbs), the most he would gain is about 90 kg (~200 lbs), but even that’s being generous. According to the Marvel Comics website, there are several different Hulks with weights ranging anywhere from 900-1400 lbs. Bruce Banner is listed as a slight 128 lbs. Doing the math, Bruce Banner will gain anywhere from 350-580 kg (~770-1270 lbs).
How long would it take to find a needle in a haystack?
This clearly depends on how big your haystack is. When I visit my wife’s family in Nebraska, we often see huge hay bales. They’re cylindrical and look about 1 m (~3.3 ft) in radius and about 3 m (~9.8 ft) in length. You can calculate the volume of a cylinder using the formula,
V = Pi R2 L,
where R is the radius, L is the length, and Pi=3.1415926... Using this formula, you’ll get about 10 m3 for the volume of a haystack. I figure you can sift through about one handful of hay every 30 seconds. If each handful is a cube 5 cm (~2 in) on a side, then the volume per handful is 1.25x10-4 m3. You can then calculate the total time it would take:
(10 m3 per haystack)
*(1 handful per 1.25x10-4 m3)
*(30 s per handful)
~2.4 million seconds or about 1 month
How many gallons of ketchup get wasted every year in those little foil packets that get thrown in the garbage when you empty your tray?
First off, there are much more creative things you can do with ketchup packets. However, since people are going to throw them out anyway, we should probably figure out how much environmental damage they cause. I assumed the average American gets fast food at least once per week. (Sadly, it may be a lot higher.) Whenever I go, it seems like they always give me two ketchup packets per tray. There are about 5 cm3 ~ 0.001 gallons per packet. There are about 300 million people in the U.S.
(2 packets per week per person)
*(52 weeks per year)
*(0.001 gallons per packet
~15 million gallons of ketchup
That’s enough ketchup to fill a football field 40 ft high.
How much wood could a woodchuck chuck if a woodchuck could chuck wood?
I hate this question. If I answer it, will you guys please stop asking it? OK, here goes…
A quick search of the Interwebs shows that woodchucks (a.k.a. groundhogs) consume up to 0.5 kg per day, and they can live up to 6 years in the wild or 10 years in captivity. I’m assuming we’re talking about wild woodchucks because if I had a captive woodchuck, I’d like to think I’d feed him something tastier than wood. From these numbers, we can calculate the amount of wood chucked:
(0.5 kg per day)
*(365 days per year)
~1100 kg (~2400 lbs)
That’s over a ton of wood chucked.*
* Note: I’ve assumed “chucking” means “eating.” Google also lists “to throw away” and “to vomit” as alternate definitions.
During the "Great Flood," how much rain would have to fall each day, over 40 days, to reach the top of Mount Sinai?
According to Wikipedia, Mount Sinai is 2285 m (~7500 ft) high. Dividing the height by 40 days, you find that you'd need an average of 60 m of rain per day to reach the top of the mountain. As point of reference, Seattle, which is known as “the rainy city," gets about 3.6 m per year. Even if none of it seeped into the ground, it would take 15-20 years to reach the top of Mount Sinai.
Check out tomorrow's blog for the rest of the Skeptically Speaking calculations.
Monday, January 18, 2010
Thank you to everyone who entered our Fermi Estimation Contest this past January. Congratulations to winner Alex D, who received a free copy of How Many Licks? Several contestants asked for my solution, so I’m posting my results below:
How many snowflakes are in this snowman?
V = (4Pi / 3)*(R13 + R23+R33) ~ 1.6m3
From Hypertextbook.com, there are roughly 100 crystals per snowflake and 1018 molecules per crystal. One water molecule weighs 2.992x10-26 kg. From these facts, we can calculate the mass of a flake:
(100 crystals per flake)
*(1018 molecules per crystal)
*(2.992x10-26 kg per molecule)
3.0x106 kg per snowflake
According to Wikipedia, the density of snow varies considerably from 8% the weight of water for newly fallen snow, to 30% after it settles, to 50% in late spring. Since the density of water is about 1000kg/m3 and the snowman is pretty densely packed, I chose 500 kg/m3.
Using these numbers, we can calculate the number of snowflakes:
(1.6 m3)*(500 kg per m3) / (3.0x106 kg per snowflake) ~ 2.7x108 snowflakes
Sunday, January 17, 2010
- How many calories are in the Stay Puft Marshmallow Man?
- How many times can you wash your favorite t-shirt before it turns entirely into dryer lint?
- How much would it cost to run the entire U.S. with solar energy?
- It’s quicker to write a blog than it is to publish a book (much, much quicker).
- The capitalist in me says it’ll help sell more copies of my already published book.
- Hopefully, people will read it, get really excited about doing math and science, and the world become a more enlightened place.