Special Guest Natalie Angier

Today we're pleased to have a question from special guest Natalie Angier.  Ms. Angier is a Pulitzer-prize winning science journalist for the New York Times.  She has authored several books, most recently, The Canon: A Whirligig Tour of the Beautiful Basics of Science.  She writes, "How many leaves are raked up nationwide on an average weekend afternoon in October?"


Before I begin, I have leaf-raking riddle for you.  Without adding or rearranging the words, add punctuation to the following sentence to make it grammatically correct: "A boy raking leaves." The answer is below.

If you're going to rake leaves, you need leaves to rake.  Some states like Arizona are desert-y and won't have many leaves to rake, but most places in the U.S. will have trees that shed.  Even if you live in the right climate, you still need a yard with at least one tree in it.  I'll assume that 1 out of 10 people owns a yard with a tree in it, since it's very likely that the actual number is greater than 1 out of 100 and less than 1 out of 1.  Of these people, some will rake, but many will use a leaf blower or just let the leaves lie.  Using a similar order-of-magnitude argument to the previous one, I'll assume 1 out of 10 people who have leaves rake them.  The average leaf raker might rake his/her leaves once in October, and there are about 8 good leaf-raking October weekend afternoons each year.  There are 3.1×10⁸ people in the United States.   Combining these assumptions, we can estimate that,

# of people raking = (3.1×10⁸ people) · (0.1 tree owners per person)
· (0.1 raker per tree owner)· (0.125 chance of raking now)

= 390,000 people raking leaves each weekend afternoon in October.

But the question specifically asked for the number of leaves raked.  This will clearly depend on the number of leaves a person has in his/her backyard.  According to at least one source, a mature tree can have up to 200,000 leaves.  To confirm this, I looked at leaves strewn across Tappan Square in Oberlin.  The mean separation was about 6 inches between leaves.  If you spread them out over a reasonably sized lawn (about 1/5 of an acre), you get about 200,000 leaves.  If each raker rakes this many leaves one a weekend, there will be,

# of leaves = (200,000 leaves per raker) · (390,000 raker)

= 7.8×10¹⁰ leaves raked.

That's 78 billion leaves raked nationwide each afternoon in October.  Thanks for the great question, Ms. Angier! 

For those wondering about my earlier grammatical riddle, the correct answer is "A boy, raking, leaves."

A Relatively Good Calculation

In 1905, Einstein published his special theory of relativity.  The most well-known part of this theory is almost certainly the famous E=mc² equation that predicted a future with nuclear bombs and atomic energy, but this is not the only surprising prediction.  The theory also predicted that objects shrink when they move really fast.¹  After hearing a professor describe this strange and fascinating phenomenon, I wondered two things.   First, what was Einstein smoking?  Second, if I ran really fast, would I be able to see atoms?  How fast does a person have to run to be atom sized?

According to special relativity, the length of a moving object is equal to its original length times an extra factor

L' = L [1 – (v/c)²]^0.5.

Here, c is the speed of light, L' is the length of the object when it's moving, L is the length of the object when it's not moving, and v is the velocity at which it's moving.  Our new length L' will be about 10^-10 m or roughly the size of an atom.  Our original length will be about 1.5 m.  We can solve for v

v = c [1 – (L'/L)² ]^0.5

 = (3.0×10⁸ m/s)[1 – (10^-10 m / 1.5 m)² ]^0.5

= 0.9999999999999999999977778 c.

You would need to move very close to the speed of light to be atom sized.

[1] This phenomenon is called "length contraction".

Lieutenant Commander Banana Clip

I saw this on Totally Looks Like the other day, and it certainly brought back a few repressed memories.  When I was a kid, I definitely wore a banana clip pretending to be Geordi LaForge, and my family wasn't even big Star Trek fans.  How many people in the U.S. wore banana clips over their eyes pretending to be Lieutenant Commander Geordi LaForge?

Apparently, I was not alone.  It strikes me that role playing as Geordi takes three things: a certain level of maturity, Star Trek knowledge, and banana clips.  I chose the phrase "a certain level of maturity" carefully.  I'm pretty sure most 2-year olds and 90-year olds didn't do this.  You need just the right amount of maturity.  With too much, you just think it's silly.  With too little, you can't see how genius the banana clip Geordi visor really is.  I suspect about 1/3 of the population has the appropriate maturity level. 

After maturity, you need Star Trek knowledge.  As I said earlier, I wasn't even that big of a Star Trek fan, but I still rocked the Geordi visor.  That said, I suspect at least 1 out of 10 people had my level of Star Trek knowledge. 

The final ingredient is, of course, banana clips.  These were fairly ubiquitous in the late 80s and early 90s when Star Trek the Next Generation was popular.  As a reasonable guess, I would say at least at least 1 out of 10 people had access to banana clips at this time.

Using the assumptions above, we can estimate that about 1 out of every 300 Americans, or about one-million people pulled off the Geordi look.  LeVar Burton, you deserve a commission from the banana clip companies.

[1] Doh!  "Jordi LaForge" should be spelled "Geordi La Forge."  Thank you, to my student Kara Kundert for the correction!

Safety in Numbers

It’s commonly said that airplanes are the safest mode of transportation. It’s true that more people die in car crashes than plane crashes each year, but most people also drive more often than fly.  On a "per trip" basis, which mode of transportation is safer?

The question did not specify whether we were considering just crashes in the U.S. or in the entire world.  Assuming crashes are equally likely in all parts of the world, the “fatalities to trips” ratio should be about the same in both cases.  Just make sure you are consistent (i.e. don’t divide the number of U.S. plane crash victims by the total number of flights in the world.)

According to “Ask a Scientist” ¹, in the U.S. there are

“…about 40,000 deaths per year in automobile accidents vs. about 200 in air transport.”

You can check these numbers against other references to make sure they are accurate.  To answer the question, we need to estimate how often a person flies and how often a person drives.  On average, we might guess that an American flies about once per year.  This is reasonable since you’d certainly expect people to fly more than once every 10 years and less than once per month.  In contrast, most of us drive (or ride in) a car about twice per day, even if it’s just to get to and from work or school.  Since there are 3.1×10⁸ Americans, this means there are

# flights per year = (1 flight per American per year) × (3.1×10⁸ Americans)

= 3.1×10⁸ flights per year,

and

# car trips per year= (2×365 car trips per Amer. per year) × (3.1×10⁸ Americans)

= 2.2×10¹¹ car trips per year.

The fraction of deaths is then just,

Fraction = (# deaths per year) / (# trips per year)

= (200 deaths per year) / (3.1×10⁸ plane trips per year)

6.5×10^-7 deaths per trip

or

= (40,000 deaths per year) / (2.2×10¹¹ car trips per year)

1.8×10^-6 deaths per trip.

You are 2-3 times more likely to die in a car trip than a plane trip, so according to our numbers plane travel is still safer.  Given the precision of the estimation, it’s possible that other reasonable assumptions would come up with the opposite result, since both figures are within an order of magnitude of each other.

[1] You can find this stat and a nice discussion of the topic at here.  Their numbers are different partly because they’re talking about fatalities per mile not fatalities per trip.

I Can Haz Estimation?

If you’re like me, there’s nothing you like better after a long work day than kicking back and looking at some grammatically incorrect cats.  And who better to provide all your LOLcat needs than The Cheezburger Network?  In honor of my favorite intertubes diversion, how many pictures of cats are on the entire Internet?

This is a difficult problem.  That being the case, we expect our estimate will likely be very different from the actual number.  In cases like these, it is helpful to determine what the wrong answers are.  To do this, you need to calculate upper and lower bounds.  To start, let’s estimate how many people are cat owners.  You might guess that 10% of people own cats.  Is this reasonable?  Well the actual number is certainly less than 100% of people, and very likely greater than 1% of people.  If you asked a 100 of your friends and family, at least one of them probably has a cat.  The next question you could reasonably ask is how many cats do cat owners typically have.  Some people have 10 cats, while others only have one.  A reasonable guess is 2 cats per owner. 

Now comes the tricky part.  What fraction of people put pictures of their cat on the web?  Many of you are tech savvy and have accounts on Facebook, Flickr, etc.  If you have a cat, you very likely have at least one picture of it on the web somewhere.  But what about your parents, grandparents, and friends that may not be as tech savvy as you?  What about people in developing nations that may not have Internet access?  If you average over everyone including people in other countries, what percentage of cat owners will post pictures on the web?  To be safe, lets say 10% again.  Is this reasonable?  As before, the actual number will certainly be less than 100%.  Will it be great than 1%?  Possibly, but there are a lot of people that don’t like putting their information up on the Internet and the ones that do might not put their cat on the Internet.  To be safe, let’s put the lower bound at 0.1%.  We can summarize what we know in a chart like the one below:

	Upper	Guess	Lower
% of people that own cats	100%	10%	1%
Cats per owner	10	2	1
% of owners that put cat on web	100%	10%	0.1%
Pictures per cat	100	10	1
World population	6.7x109	6.7x109	6.7x109
# of cat pictures	6.7x1013	1.3x109	6.0x104

Our upper and lower bounds are very far apart (i.e. 9 orders of magnitude).  The answer certainly lies between them, but is there any way we can tighten these bounds?

Let’s try calculating the answer in a different way.  According to Netcraft, there are 2.5x10¹⁰ web pages as of a few years ago. As an upper bound, we might say that at most, every page has 10 cat pictures on it.  Even in this extreme case, there would only be 2.5x10¹¹ cat pictures, so we’ve effectively reduced our upper bound by a factor of 100. 

Now lets see if we can adjust our lower bound.  A quick Google image search for the word “cat” pulls up about 1.5x10⁸ results.  Looking at the pictures on the first page, we can see that the vast majority of the images are of cats, but occasionally you get one that’s not a cat.  This is very rare.  In fact, it seems much more likely that Google is missing cat pictures since it doesn’t have access to private photos on sites like Facebook and Flickr.    For this reason, we could probably take the Google result as a lower bound.  Just to be extra careful, let’s say the lower bound is 1/10^th the number of “cat” images found by Google.

Our final results:

Upper Bound -- 2.5x10¹¹ cat pictures

Actual Estimate – 1.3x10⁹

Lower Bound -- 1.5x10⁷

My Eyes Are Buggin’

I had an interesting conversation with one of my physics students recently.  She had a dance class right before she came to lab.

Her: I hate dancing with glasses.

Me: Because they fall off when you’re doing the turn-y things?

Her: Yes, it’s so much easier with contacts.

Me: You know, you can calculate how fast you’d have to turn to get the contacts to pop out.

How fast would a dancer have to spin to get her contacts to pop out?

Contact lenses weigh about 0.05 g and are about 1.0 cm² in area.  They are held on mostly by suction.  Suction occurs because there is a partial vacuum created under the contact as it's pulled away from the eye.  To pop out, a contact needs to overcome air pressure, which for our atmosphere is about 100 kPa.  As the dancer spins, her contacts move in a circle with roughly a 10 cm radius.  The force to keep an object in uniform circular motion is given by

force = (mass) · (velocity)²/ (radius).

The maximum this force can be before the contact pops out is determine by atmospheric pressure

force = (pressure) · (area).

Solving for the velocity, we get

velocity = [(pressure) · (area) · (radius) / (mass)]^1/2

= [(100 kPa) · (1 cm²) · (10 cm) / (0.05 g)]^1/2

= 140 m/s.

A dancer would have to rotate about 1400 per second to get her contacts to pop out.

Smile! You’re the Monocle Smile Contest Winner!

Hello, Ladies.  How are you?  Fantastic.  We have a winner!

The Question: If he worked nonstop for the rest of his life, how many personalized Youtube videos could Mr. Mustafa make?

Born February 11, 1974, Isaiah Mustafa is currently 36 years sexy.  He’s young and he’s in shape, the latter of which makes longevity more likely.  As best I can tell, he only has one major thing going against him: he played NFL football.  However, his career was fairly short and he was a wide receiver, so he’s not as likely to have the same health issues as a 12-year 350 lb lineman.  One simple way to solve this is to subtract his current age from his life expectancy.  According to Wikipedia, Americans are 38^th in the world for life expectancy with and average lifespan of about 78.2 years.  Using Wolfram Alpha, we know that Mr. Mustafa was 36.6 years old on the contest deadline.  From this we can compute how many years he has left,

78.2 years – 36.6 years = 41.7 years.

Each personalized video appears to be about 30 seconds in length.  If he worked nonstop¹ making 30-second videos for 41.7 years, he would have a total of about 44 million videos.

Congratulations to our winner, Joey Reichert!

[1] Some might argue that we haven’t taken into account sleeping.  True.  If he did sleep, our final number would only be one third as big.  But the original question did say “nonstop.”  “But wouldn’t he die younger if he never slept?” you ask, suspiciously.  Also true.  But we could always film him while he slept, and which of you ladies out there wouldn’t appreciate that?