Roller Coaster Winner!

We have a winner for the End of Summer Contest!  The question: how many people get stuck at the top of the Cedar Park roller coaster each year?

A stuck roller coaster is likely held in place by friction.  From the looks of it I would guess a 2 pound weight added to the front of a stuck roller coaster wouldn't do much to budge it, but a 200 pound man hanging off the front seems like more than enough to get it rolling again.  As such, I'll assume a 20 pound force is needed start a stopped roller coaster.  This force is equivalent to the frictional force holding it in place. 

There's a narrow range of heights near the peak at which the cars could stop and not accelerate because friction would be too strong.  It looks like the radius of curvature for the track is about 10 meters give or take.  The range of heights over which the carts would get stuck is only a small fraction of this.  I'll guess 5 centimeters since it's likely bigger than 5 millimeter and smaller than half a meter.

The initial velocity of the cart will correlate with how far it makes it up the hill.  Carts with speeds greater than some cutoff velocity will all make it over the hill.  Carts with speeds less than some other cutoff velocity will all roll back down the hill.  Carts with speeds between these two cutoffs will get stuck.  Using conservation of energy (perhaps a dubious move since we're relying on friction to stop us) we can relate the initial velocity to the height of the cart will climb up the hill:

m v² / 2 = m g h

Here, m is the mass of the carts, g is the acceleration of gravity, and h is the height climbed.  The roller coaster might be 100 meters tall.  Carts stopping at heights anywhere between 99.95 meters and 100 meters will get stuck.  Using the energy equation, we can solve for the range of initial velocities for which this will happen.  Leaving out the messy math, you'll find that this occurs for velocities between 70.009 mph and 70.025 mph.

As you can see, there's a narrow range of velocities over which the roller coaster will get stuck.  What causes fluctuations in the initial velocity?  If we assume each ride gets pushed with the same impulse, then the only fluctuating variable would be the mass of the riders.  A roller coaster full of fatties is less likely to make to make it to the top.  I'll assume the average person weighs 170 pounds with a standard deviation of about 40 pounds.  If there are 50 riders, the standard deviation in weight would be about 3% of the mean total weight.  I'll assume the deviation in the initial velocity is the same as that of the weight, i.e. 3% of the mean.¹

We still need to find the mean velocity.  For the sake of keeping their customers happy, I'm going to assume that roll backs happen only 5% percent of the time.  We can assume a normal distribution to find the mean for which this occurs.  Skipping the messy details once more, I get about 73.6 mph.

Using this mean and standard deviation, we can then find the fraction of times a cart will get stuck at the top of the hill.  Again assuming a normal distribution, I get about 0.1% of the time, which seems like a fairly reasonable estimate.

If the roller coaster operates 8 hours per day and each ride take 2 minutes, you'd have roughly 240 rides per day.  Over a 100 day season, you'd have 24,000 rides.  Roughly 24 of these would get stuck at the top.

Congratulations to our winners!

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'm assuming the carts are of negligible weight, which is probably stupid.  I suspect it won't throw the number off too much as long as it's comparable to the weight of people.

A Sticky Situation

10 million pounds of maple syrup!  Am I the only one who thinks this number can't be right? If the thieves loaded up their truck with 3 tons on each trip, they would need to make 1700 trips!


On an unrelated note, contest winners will be chosen later this week.

My New Job at Gustavus Adolphus

I'd been at my new Gustavus office only a couple hours when someone stopped by and tacked my nameplate up.  I could get used to this.  Good schools really know how to treat their faculty with respect.

Give It a Rest, Fermi

I can't get away. I went to a "Night of Unbelieveable Fun" hosted by the Saint Paul Saints. Who do I find in the bathroom?  Enrico Fermi watching me pee:

A Question from Ninja Brian

Today's question comes from physicist Brian Wecht. Brian is co-founder of Story Collider, a podcast that shares people's personal stories about how science has affected their lives. Brian is also a member of Ninja Sex Party.¹

Here's a picture of Brian in ninja form:

Brian's a very good ninja.

Brian asks,

What is the weight of all the facial hair grown by the world population of men on a given day? If you concentrated all that hair into one curly, villainous moustache, how long would that moustache be?

If I go a week without shaving, I'll have about half a centimeter of facial hair, which means my hair grows about 0.1 centimeters per day. Follicles are separated by about 1.0 millimeter, and they span an area of roughly 100 square inches.  Assuming each hair has a thickness of 0.1 millimeters and the same density of water at 1.0 grams per cubic centimeter, I would grow about 60 milligrams of facial hair each day.

For simplicity, I'll consider myself to be a typical man. This is not necessarilly a safe assumption however, since, follically speaking, I'm much more Wolverine than Bieber. Still, it should be a safe assumption for an order of magnitude estimate. If that's the case and we assume 20 percent of the world population (~1.4 billion people) are facial-hair growing men, then the total weight of facial hair grown on a given day would be about 90 tons.

This brings us to the second part of Brian's question: How long of a villainous mustache can we make?

Ninja Brian will save us from this dastardly villain.

Mustaches take up much less area than beards. If you glue all the beard hair grown in one day to the ends of the mustache hair grown in one day, it would be as if the mustache grew about four times faster than normal, which amounts to roughly 0.4 millimeters of growth each day. Spread out over the 1.4 billion facial-hair-growing men, that would give a villainous mustache that's about 350 miles long. Snidely Whiplash would be proud.

Thanks for a great question, Brian!

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] Somehow how I'm super happy that the blog has gone from a Nobel Laureate to a member of Ninja Sex Party.

A Question from Nobel Laureate Bill Phillips

When I was an undergrad, I had this delusional desire to become an actor. As a male thespian with no experience and mediocre talent, I quickly discovered there's one sure way to get decent theatrical roles: audition for male parts at all girl schools. And so, I auditioned for student productions at Wellesley College where I met a variety of wonderful people. One of said people is my buddy Christine. Christine was (and is) super cool, and not only because she used to get me free food at the Wellesley cafeteria. Christine was also cool because she was taking physics.¹ I distinctly remember one conversation I had with Christine about her physics class:

Christine: I'm taking physics at Wellesley.

Me: You should really see if you can take it at MIT instead.

Christine: [trying to be nice] Um...my dad thinks they do a better job teaching physics at Wellesley.

Me: [offended] What the hell does your dad know about physics?

Christine: Well, he did get his doctorate from MIT, and he just won the Nobel Prize in physics.

Me at this point in the conversation.

This week's question comes from Christine's dad (aka Nobel Laureate Bill Phillips.) He asks, "How many grains of sand are there on the world's beaches?"

Judging from a map, the Eastern Seaboard of the United States appears to be about 2000 miles long. By comparing with other coastlines, we can estimate the total length of coastlines in the world to be about 50 times this.² I'll assume one-third of the world's coastlines are sandy beaches.

Assuming the Cape Cod beaches I grew up near are fairly typical, they might extend about 200 feet up from the water. The depth of sand varies quite a bit from place to place. I've been on beaches where you'll hit rock before finishing the moat around your sandcastle, but many beaches have sand that extends much deeper. I'll assume the sand extends 10 feet deep on average since the actual number is likely to lie between 1 foot and 100 feet. From this, we can calculate the total volume of sand on the beach to be about 10¹⁰ cubic meters.

One type of sand.

Like beach depths, sand grains too vary over a size range that's greater than one order of magnitude. I'll assume 0.3 mm for the width of a sand grain since even the largest sand grains are each only a few millimeters in size. This gives a total volume of 0.03 cubic millimeters. From this and the total beach volume above, we can estimate that there are 10²⁰ grains of sand on all the beaches in the world.

Thanks for the great question, Dr. Phillips!

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] Taking physics automatically makes you cool.
[2] It will certainly be between 5 and 50 times the Eastern Seaboard.

ABC Radio National Interview

Check out my interview with Robyn Williams on ABC Radio National.