Anatomy of a Monster Aerial

When snowboarding's living legend Terje Haakonsen launched off of the big quarterpipe at The Arctic Challenge last March, he sailed 9.8 mind-blowing meters (32 feet) in the air. He set the world record for big air on a snowboard and gave us some wicked footage to boot. Transworld's video is my favorite clip of Terje's air, but the YouTube video below is pretty entertaining too.



I thought it might be interesting to run some of the numbers to see what sort of physics was involved when Terje shattered the record. Here's what I've come up with so far . . .


-Terje's top speed as he approached the bottom of the giant quarterpipe was at least 70.92 kilometers per hour (44.1 mph).

-The g-forces he experienced as he rode up the ramp peaked out at around 4 to 5 times the force of gravity, which means his legs were briefly supporting the equivalent of about 337 kilograms (743 pounds) or more.

-Terje's trip above the top of the ramp lasted just about 2.8 seconds, although I'm sure it seemed a lot longer to him.

-As you may recall from my post about the FMX backflip limit, it takes energy to rotate as well as catch air. Terje set the record with a massive 360 aerial. If he hadn't been spinning he could have gone just a little higher. But it turns out that doing an air-to-fakie instead of a 360 would have only boosted him another centimeter or so. It looks like the Arctic Challenge judges couldn't have measured such a slight difference, so he still would have ended up with the same 9.8 meter record.

What if Terje had approached the hill at world record downhill snowboarding speeds instead?

-At an approach speed of 201 kilometers per hour (124 mph), the current world record for snowboarding, Terje would have sailed about 79 meters (259 feet)in the air.

-He would have experienced a crushing g-force 32 times gravity, the equivalent of about 2400 kilograms (5291 pounds), as he rode up the ramp.

-His total hang time would have been about 8 seconds.

In case you want to check the numbers yourself, I've listed the equations and other information I used to make these estimates below.

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The Mathy Bits

Some of the things you need to know to analyze Terje's monster air are

Terje's mass - roughly 75 kilograms


The radius of the Arctic Challenge quarterpipe's transition - about 10 meters


Terje's moment of inertia when he reaches down to grab the board is about 5 kilogram meters^2. (I got that number from page 313 of a book called "The Physics of Sports", edited by Angelo Armenti, Jr.)


The equation for gravitational potential energy, E = m g h

where,

E = energy

m = mass

h = height


The kinetic energy equation, E = (1/2) m v^2

where v is velocity, and v^2 means velocity squared


The centripetal force equation F = (m v^2)/r

where r is the radius of the quarterpipe's transition.


The equation for motion of an object under constant acceleration x = x0 + v0 t + (1/2)g t^2

where g is the acceleration due to gravity

t is time and t^2 is time squared


The equation for rotational energy is E = (1/2) I w^2

where I is moment of inertia

w is angular velocity


Read the rest of the post . . .

eXtreme Sports Physics Gift List, Part 1

The holiday season is rapidly approaching. It won't be long before your folks will start asking what you want this year, and you're going to have to start thinking about what to give others. That means there's trouble a brewin'.

Those of us who practice eXtreme sports tend to be pretty picky about equipment. It can be so awkward when Uncle Bob beams and hands a skateboard with wheels that barely turn and a deck made out of something that looks like high grade cardboard (lovingly selected from the rack at a big box department store) to his skater nephew or niece. The same goes for Aunt Esther's choice of inline skates, or Cousin Frank's best guess in plywood skimboards.

So if you're reading this, Bob and Esther and Frank: don't even try to buy eXtreme sports equipment for someone unless you happen to participate that particular sport too. Instead, take a look at the list below to get some ideas for gear that most eXtreme sports folks could really use, regardless of their particular choice of discipline.

On the other hand, if you've been a victim of a well-meaning relative's generosity in the past, just send them a link to this post (and the forthcoming eXtreme Sports Physics Gift List, Part 2). Or print out the stuff that you like and leave copies lying around the house at the family's Thanksgiving dinner. They'll get the idea.

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Protect Your Head


The physics of what happens to your skull and brain on sudden impact is fascinating in theory, but the experiments are harsh. Believe me, head protection is vital in just about any sport worthy of the eXtreme category.

Unfortunately, it can be tough convincing teens in particular to wear helmets. Besides, the kids are sometimes right: full-on head protection is a little more than necessary when it come to casual cruising on a bike, board, or blades.

The Swiss have come up with a physics-based solution for times when you need a little head protection without looking like a dork. Ribcap is a company that makes hats with a material that's soft and flexible until impact, at which point it rapidly changes into a rigid structure to dissipate the blow.

The magic material in Ribcaps is a non-newtonian fluid. That means its viscosity (the fluid's thickness) changes depending on how much stress you put on it. You can experiment with your own non-newtonian fluid by mixing cornstarch into water to make a goop that's runny and nasty until you thump it, then it becomes rock hard.

To show you how it works, here's a video of a couple of guys running on top of a pool of cornstarch goop.



Ribcap is new to the states and doesn't seem to have a US distributor, but their hats are available on Amazon for about $90 to $125. Now that you know the secret, I guess you could make your own hats filled with non-newtonian fluid, but I think it's worth a c-note to get one that won't start dribbling goop down your neck as you try to look cool in your stealthy head protection.

Ribcap - protect your head without losing your steeze
$89.99 to $124.99

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Rugged Phone

I know . . . you want an iPhone for Christmas. What are you gonna do with it when you hit the water for some skimboarding, surfing, or whitewater kayaking? Hide it in your shoe and leave it on the shore with your towel? That'll keep it safe.

Seriously, the iPhone is the hot wireless device of the moment, but the true eXtreme athlete needs a phone that can take the water, mud, and impacts of hardcore fun. That's why I'm going to drop my wireless provider and pick up the G'zOne (despite the lousy name).

The G'zOne is a military spec rugged phone that can take serious abuse and handle dunking in water down to a meter. That's good enough for most eXtreme water sports and great for those warmer days on the slopes when you get soaked with sweat and slush. It's just at the hairy edge of what you need in order to slip one into your surf trunks and paddle out to the line up. (I'll give it a shot next summer and let you know how it holds up at the beach.)

Even better, it's an older cell phone so it's comparatively cheap ($99 from Verizon with a two year contract), especially for the G'zOne type-V edition. You could pay a bit more for the newer type-S, if you really need Bluetooth and a few other bells and whistles, but you will have to make do with a low resolution camera.

I prefer the type-V (just in case you're reading this, mom) because I'll gladly trade gimmicks that come with the type-S for the 2-megapixel snapshots of the type-V.

G'zOne - a rugged phone for rugged sports
$99 through Verizon

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Tune Your Senses with a Balance Board

When it's too cold to surf, or too warm to hit the slopes, or too wet to skate, or too . . . you get my drift . . . you gotta find something to do to keep your panther-like reflexes in tip top shape.

That's when it's time to break out the Indo Board. There are other balance boards on the market, but Indo Boards are my favorite. They come in lots of sizes and have flat bottoms so you can do more than just balance. (Second-rate balance boards come with a rail that runs down the middle, which does nothing but limit your options.) My son pulls shove-its and ollies on ours, and I can do a kick turn (sometimes).

This video could use some music, but the tricks are cool and the toy bunny seems pretty impressed.


Balance boards may look dangerous, but they're actually pretty safe so long as no one is dimwitted enough to stand next to one in use (a lesson our cat hasn't picked up on just yet). They're also great for developing your intuitive feel for the physics of balance and the mechanical feedback that keeps you stable.

My favorites are the small, skateboard-like versions. Surfers, wakeboarders, skimboarders and snowboarders might get more out of the longer ones.

Once you've used one for a while, it becomes second nature. I like to ride my Indo Board when I watch TV or talk on the phone, then I try learning a new move or two until the cat gets in the way.

Indo Board - keep those reflexes sharp even when you're stuck inside - $59 and up

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Document Your Secret Spot

You should thank Einstein every time you fire up a GPS device. Both special relativity and general relativity are vital to calculating your precise position based on signals your GPS receiver collects from an orbiting phalanx of satellites. So the next time you hear someone doubt Albert's brilliance, just wave your Magellan Triton 2000 in their face.

Of course, you won't be carrying the Triton around just to verify modern astrophysics, you'll have it handy so that you can make a record of that new skate/surf/bike/free climbing spot you just found.

Any GPS receiver can help you find your way there and back. Only the Triton includes both a built-in voice recorder and a 2-megapixel camera that lets you save geo-stamped proof and memories of your eXtreme sports adventures.

It has other nice features (it's waterproof, includes a flashlight, and touch screen, lots of maps, etc.), but you can get all that with cheaper GPS receivers too.

Magellan Triton 2000 - saving memories in all four dimensions of spacetime
$499.95

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Give the Gift of Skate

It's the season of giving, and who's more deserving that those up-and-coming eXtreme athletes in your life?

The Yo Baby Action Board is a plastic skateboard without trucks and wheels. It's designed for the young 'uns who lack developed motor skills just yet, but want to follow in the footsteps of their ultra-cool sisters, brothers, uncles, aunts, and parents.

Lots of skaters and other board-sport types give their kids plain old skateboard decks to play with on the carpet. Leaving off the trucks and wheels makes it safe for the tots and gets them used to standing on a board (perhaps the most important lesson of their young lives).

But when they slip and shoot one of those things across the room, the wooden decks are murder on table legs, sliding glass doors, your ankles, and the neighbor kid you're supposed to be keeping an eye on for the afternoon.

The Yo Baby solves all the nagging little safety issues that come with a wooden board, and at $10 each they're way cheaper than buying a real deck for your baby bro or sis to drool on.

Check out the Yo Baby promo on YouTube for the rest of the story.


Yo Baby Action Board - for the eXtreme athletes of tomorrow
$10 from GaragecoToys.com

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That's it for now. I should have the eXtreme Sports Physics Gift List, Part 2 done in a few days, so stop by again for some more holiday gift ideas for eXtreme sports addicts.

If you know of any cool stuff that I should add to the list, drop me a note at buzzskyline@gmail.com

Read the rest of the post . . .

eXtreme Sports Physics in Print

A few magazines have written about some of the posts in this blog. You can read about my quad backflip limit calculation in a story by Corey Binns in the magazine Science World. There's also an article about it in this month's the November issue of Popular Science magazine (one of my favorites mags), but as far as I can tell it's only available for now in the print edition or through iPhone subscriptions.

The website Lat34.com supposedly mocked up a faked version of Travis Pastrana doing a quad backflip, but I can't get it to run on my computer. Man, do I ever wanna see that.

I'm hoping these and other stories about the backflip limit will get some up-and-coming FMX competitor to try at least the triple backflip soon. How cool would that be?

Also coming up in November is a story about skateboarding in the Weekly Reader. The reporter called me to ask about physics of ollies, 540 aerials, 180 ollies, ramp and pool skating, gravity, and a bunch of other stuff. The story is about twin skaters who are big news lately, so I don't expect to see much about physics in it, but I hope I helped the reporter get the science straight.
Read the rest of the post . . .

The Wheel Deal Part 2: Wheel Size (again)

Bigger wheels are faster on rough surfaces, as I pointed out in my last post, because more of the forces that they experience as they roll over bumps pushes the wheel up rather than pushing backwards and slowing the wheel and rider down.

So, why not just ride the largest wheels you can find? Unfortunately, when it comes to skateboarding and inline skating, larger wheels simply don't roll as well on smooth surfaces. The problem is that the urethane they're made of changes shape when you ride.

A simplified model of a skate wheel might look like this, where the jagged lines are meant to be springs that represent the compressible urethane. That's not to say that there are actual springs inside the wheels. It's just that the urethane acts much like springs, and it's simpler to understand the way wheels work if you pretend that they're are made of springs.

As you can see in this sketch (I've exaggerated things a lot to make it easier to visualize), a wheel deforms as you ride along, resulting in a flat spot where the wheel touches the ground.

Most skate wheels are solid cylinders of material, except for the hole running through the middle where the bearings sit and the axle passes through. The springy urethane compresses when your weight pushes down on it. If you look at two wheels that are identical in shape except that one is large and the other is small, and both are made of exactly the same urethane, the larger wheel will deform more under the same weight.

This is where we can use physics to understand what's going on a little more precisely. Physicists think about springs in terms of something called the spring constant (usually symbolized with the letter k). The higher the spring constant, the more force you need to stretch or compress the spring.

If you connect two springs end to end, the total spring constant goes down (from k to k/2). In other words, it will be easier to stretch two springs connected in a row than it is to stretch just one. (You can test this yourself by tying rubber bands end to end.)As you can see in this sketch, attaching two springs side by side increases the total spring constant (from k to 2k), making them harder to stretch.

If you cut a spring in half, it will double the spring constant. It's like taking the two springs connected end-to-end and getting rid of one, which doubles the spring constant from k/2 to k.

This is relevant to skate wheels because you could always make a small wheel by shaving down a big wheel. If you were to do that with the model of a wheel that I drew above, you're essentially shortening the springs. This makes the wheel stiffer (which is to say, less springy). Your weight pressing down on a small wheel will not deform the wheel as much because it's effective spring constant is much higher than it would be for a wheel that's identical in very way except for its larger size.

This makes larger wheels slower because compressing and stretching springs, or springy urethane, takes energy. With a perfect spring, you get all the energy back as it springs back to its natural shape. But no springs are perfect, and urethane is usually far from perfect.

Urethane is fairly resilient, which means that once it's deformed it bounces back into shape and gives back some of the energy that deformed it, just like stretching and releasing a spring. Depending on the exact formula of the urethane, a portion of the energy is always lost. Most of the lost energy turns into heat that warms the wheel and escapes into the air. If you squish a skate wheel you can expect to get back no more than 75% of the energy you put into it, and usually you get back a lot less. As a rough estimate, the deformation that comes with rolling on a urethane wheel will cause a large wheel to lose twice as much energy as one half its size. That's what makes larger wheels slower.

There are several ways to reduce the amount of energy lost due to the squishing of skate wheels. One common solution is to replace some of the urethane with a rigid core, like this Spitfire wheel.


Another possibility is to make the wheels wider, like these old school wheels.

Take a look at the cutaway sketch below that shows why wider wheels are less squishy. By widening the wheels, you're adding more springs (well, springy urethane anyway) in parallel. If you recall the diagram above, adding more springs side-by-side increases the total spring constant and makes it harder to stretch or compress the springs.

The wheel on the right is three times wider, and should be three times more rigid than the wheel on the left.

So, if you want bigger wheels that will roll as fast as smaller wheels, you have to make them wider. That leads to other problems. For one thing, wheels that are wide and have big diameters are heavy. That's not so good for all the ollie-based street moves, but fine for ramp, bowl and downhill skaters.

Another problem, which can be bad for all sorts of situations, is that the wider you make a wheel the harder it is to corner. If you try to ride in a circle, the outer edge of the wheel travels farther than the inner edge. Because both parts have to roll at the same speed, either the outer part of the wheel ends up turning too slowly or the inner part turns to quickly. That can lead to lots of wear and tear on the wheels, as well as extra friction that will slow you down whenever you change direction.

I've already explained why you need large wheels for riding on rough surfaces. Now you can see why smaller wheels are better for skating park concrete and obstacles. So, why not get REALLY tiny wheels for skating on smooth surfaces? Unfortunately, when wheels get too small other problems start to crop up. This post is already long enough, so I'll tell you about those issues some other time.

Read the rest of the post . . .

The Wheel Deal Part 1: Wheel Size

Many extreme sports rely on wheels of one type or another, including skateboarding, mountain boarding, inline skating, street luge, BMX and FMX. Different situations require different types of wheels, depending on the terrain and the types of riding you're doing.

My favorite extreme sport is skateboarding, so this post focuses primarily on the options available in skate wheels. But the physics involved applies to any wheeled sport.

If you skate, you know that there are lots of wheels designs on the market from tiny, rock-hard wheels for street skating to giant, gummy wheels for old school cruising. Why are some wheels better for certain uses and not so good for others? As you probably guessed - it all comes down to physics.



Wheel Size

Among the many things you need to consider in choosing the best wheel for your riding is size. Most pro street skaters opt for small wheels. It's a good choice. Small wheels are fast on smooth surfaces such as skate park concrete, wood ramps, and most of the boxes, benches and banks you're likely to hit. But on asphalt or chewed up concrete, little wheels are much slower than big wheels. Just about every skater has at one time or another had the unpleasant experience of running across a pebble or crack that stops their board dead in it's path, sending the rider for a rough tumble. Those sorts of sudden stops are more likely if you ride tiny wheels.

So, what's size got to do with it? Well, here's a little sketch to show you what's going on. The small red cirle represents a wheel with a diameter of about 50 millimeters, typical of lots of street and park wheels. The big black circle is like a large (95 millimeter) cruising wheel. This sketch shows the wheels just as they hit the edge of a 20 millimeter step (in this picture, I'm imagining the wheels rolling to the right), which is the sort of thing you might run across as you ride over the joints between sections of a typical sidewalk.

The arrows show the direction of the force that results from the wheels hitting the obstacle. As you can see, the black arrow points up and to the left. That means some of the force pushes the wheel upward and some of it pushes back.

The red arrow is mostly pointed to the left and just a bit up, which means most of the force exerted by hitting the step goes into slowing the wheel, and the board it's attached to.

Of course, most of the bumps and cracks you'll run across in real life are a lot smaller than this. Even for smaller obstacles, though, more force will go into slowing a small wheel down than would go into slowing a larger wheel. You'll still get a force pushing the larger wheel upward, which makes for a rough ride, but at least it doesn't do as much to sap your speed (or stop you in your tracks).

If you race down a big hill made of asphalt, you end running over lots of little bumps that seriously slow small wheels, but aren't such a problem for big ones.

Are big wheels always better than small ones? Not at all. In fact, small wheels are usually MUCH faster than large wheels on smooth surfaces. Want to know why? Check out The Wheel Deal Part 2 in my next post to find out one reason that small wheels are better (sometimes).

Read the rest of the post . . .

Line Rider eXtreme Sledding PC Game

For those surfers, skaters, free climbers and other summertime athletes who are sadly watching the season change, and for the ice climbers, snowboarders and extreme skiers who can't wait for winter to arrive (readers who live on the other side of the equator should reverse that), here's a game to keep your mind occupied for at least a few minutes.

Line Rider is a simple toy that lets you design the terrain for a little sledding person. With a bit of practice and editing, you can have the sledder catching air, doing flips, riding full pipes - anything that you can imagine as long as it's in two dimensions.

I love this game in part because it uses real physics equations to guide the sledder's motion. Line Rider takes into account gravity, friction, drag, momentum, inertia, acceleration, and perhaps other things I haven't noticed.

If you get tired of making sled runs, check out the movies that the game's creator Boštjan Cadež and fans of the game have made. They should inspire you to come up with more terrain designs.

Read the rest of the post . . .

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Mogul's Happen

(Oops, I made a calculation error. It turns out that moguls should form at much shallower slopes than I initially estimated. See the Mathy Bits section for an explanation. Thanks for catching the mistake, MC!)

People seem to have a love-hate relationship with moguls - they're lots of fun if your timing and knees are good. They can be a real pain if you prefer a more casual descent down the slopes and you happen to get stuck above a gnarly mogul field.

I've hit the moguls on occasion, and regretted it more than once. I can recall sitting on the snow nursing my hyper-extended knee and wondering why on earth they make these things.

After looking into the science behind them, I found out that no one actually makes moguls at all, at least not on purpose. Moguls happen all by themselves, with a little unintentional help from snowboarders and skiers.

Moguls are an example of something called a self-organizing structure. All you need is a snow-covered slope and some folks to slide down it, and soon they will push the snow into lumps that form stunningly organized moguls. There's no way to stop it from happening. All you can do is run over the slope with a groomer to cut the things down from time to time.

Typically, self-organizing structures arise whenever patterns are formed as a result of many simple parts that follow a few basic rules. Common examples you've probably seen include crystals (salt, diamond, and snowflakes to name just a few), sand dunes, ocean waves, flocking birds, schooling fish, and the patterns in bubbles rising through a glass of champagne.


It's hard to predict in advance how the rules might lead to patterns. The best we can do, in most cases, is to look at the pretty structures and then work backward to figure out what rules might have made it happen. Even then, it's not always clear what the underlying rules really are, so physicists sometimes make up rules that seem to explain things reasonably well. This is what we call empirical science.

As far as I can tell from researching the scientific journals, nobody truly understands moguls. That hasn't stopped them from coming up with theoretical models that work adequately, even if they're probably not completely true.

One of the scientists who has looked into moguls is Regis University computer science professor and extreme skier Dave Bahr. He's made time lapse movies of moguls and shown - now get this - that moguls slowly migrate up hill over the course of a winter. Check out his videos. They're eerie. It's like the moguls are alive.

Bahr explains how this happens with a little cartoon, and has promised to write a formal paper about it eventually. Basically, as you make your way through the moguls, you end up scraping some of the snow from the downhill side of one bump and depositing it on the uphill side of the next one down. So even though you're pushing snow downhill, the moguls themselves move up.

There isn't much else in the scientific literature about moguls, but a recent physics paper analyzing the washboard ripples that often form on dirt roads seems to offer some insight into bumps on the slopes. The physicists who wrote the paper filled a tray with sand, and then ran a wheel across it to see when and how ripples would form. And ripples almost always developed - unless they rotated the tray very, very slowly. They concluded that all dirt and gravel roads would develop washboard ripples if traffic moved any faster than a few miles an hour. That is, you would have to keep traffic to speeds slower than the rate at which most people walk if you want to stop ripples from forming on your dirt driveway. Check out this short video showing their experiment in action.

The thing that makes this research relevant to snowboarding and skiing is that they also tried dragging a non-rotating square block along the sand. That too caused ripples to form. When you think about it, it's a lot like what you're doing when you slide down the slopes on a board or skis. Mogul fields are essentially giant washboard roads that are created by snowboard and ski traffic rather than cars and trucks.

If you apply the theory behind road ripples to moguls on snow, it turns out that moguls won't form if the slope is mellow enough. It has to be VERY mellow though. If I did the math right (you can check it in the mathy bits below), only slopes with less than 7 degrees 2.3 degrees incline (about a 12% 4% grade) will stay mogul-free on their own - that's bunny slope territory darned near flat! Anything steeper will have to be groomed to keep the moguls down.

The bottom line is there's no one to blame for moguls but ourselves and physics. So you either have to learn to ride them, or stick to groomed trails, or (best and hardest of all) find a fresh back bowl where the snow is still untouched and nobody's had a chance to pile up those bumps.

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The Mathy Bits

(As you will see from the portions crossed out, I initially calculated that moguls form on slopes steeper than 7 degrees. But after someone pointed out that I seemed to have made a math error, I found that the actual answer is probably closer to 2.3 degrees. It turns out that I put the factor for the snowboard width in the numerator rather than the denominator at one point when I made the substitution for v^2. I don't show all these details because it's hard to do math in the Blogger interface. But if you want to talk about the details, drop me a note at "buzzskyline at gmail.com".)

In their Physical Review Letters paper about washboard roads, physicists Nicolas Taberlet, Stephen W. Morris, and Jim N. McElwaine suggested that there's something called the Froude number, which predicts when ripples will form on a dirt or gravel road. They write the Froude number equation essentially like this

Fr=(v^2/g)*(p*w/m)

v is velocity
g is the acceleration due to gravity
p is the density of the sand (or snow)
m is the mass of the wheel (or snowboarder)
w is the width of the wheel (or snowboard)

Whenever Fr is greater than one, washboard ripples form. Of course, the only thing that isn't constant in the equation is v, which is just the speed of the wheel over the sand. All the rest of the components are fixed.

It's pretty clear that all you need to do to keep Fr below one is to slow down.

To apply this to snow, I estimated that the density of packed snow is about half the density of liquid water (500 kg per cubic meter), the mass of the average snowboarder is about 70 kg, and the width of the average snowboard is about 20 centimeters.

If you recall from my post about Speed Snowboarding, you can solve for the terminal velocity of a snowboarder pretty easily by rearranging this equation

m*g*sin(theta)= u*m*g*cos(theta) + (p*A*Cd*v^2)/2

The terminal velocity is the speed limit on the hill. Not that it's illegal to speed at the slopes, it's just that physics won't let you go faster than the terminal velocity.

If I plug the terminal velocity from the Speed Snowboarding post in the equation for the Froude number, I can see that Fr<1 whenever the angle theta is less than about 7 degrees 2.3 degrees.

You should bear in mind that the washboard road paper uses some pretty shaky logic to come up with their Froude number, and that I am just blindly applying their calculation to snow, so take this all with a grain of salt.

Still, it seems reasonable that there should be some slope that's too mild for moguls to form. Although I don't know if anyone has tested it, I bet you'd never see moguls on a bunny slope with a 7 degree 2.3 degree or less drop, even if it were never groomed.

On the other hand, even a gentle beginner slope would be steep enough to develop moguls if left ungroomed for long.

Read the rest of the post . . .

Speed Snowboarding - 125 mph and Beyond

As a continuation of my last post, I thought I'd do a few calculations to figure out just how fast a snowboarder should be able to go.

Obviously, the steeper the slope the higher speeds that are possible. Because my last post was motivated by the discussion of avalanches, I decided to do the calculations using a slope of 38 degrees. That's the incline that Wikipedia says is the most likely to have avalanches induced by human activity (snowboarding, skiing, hiking, etc.)

When you bomb a hill on a snowboard (that's what my little brother calls it when you head full tilt down a slope), you have basically three forces to deal with.

- Gravity pulling you down. Of course, the ground below you gets in the way, preventing you from falling straight to the center of the earth, but some of the force of gravity ends up getting directed along the face of the hill, pulling you along the surface and down to the lifts so you can start back up again.

- Air Drag pushing back against you. The thing about drag is it increases much more as you speed up than you might expect. If you double your speed, the drag quadruples.

- Friction. Unlike drag, friction stays relatively constant regardless of your speed down the hill. Although, this isn't exactly true, it's a good approximation. All kinds of things affect the amount of friction you encounter on snow, from the temperature of the snow to it's texture. It's likely that sliding friction changes at least a little as you speed up, but the change is probably pretty small, at least on the firmly packed snow where speed records are set.

Gravity pulls you down the hill, while drag and friction are directed back up the hill. As you build up speed down a slope, the forces of gravity and friction stay constant (provided the slope doesn't change). The drag, on the other hand increases until the forces pulling you down and the forces pushing back up balance out. At that point, you've reached terminal velocity for the hill.

For those of you who like equations, terminal velocity occurs when

Force of gravity = Drag Force + Frictional Force

Skip to the bottom of the post to see it in even more mathy terms.

To cut to the chase, if you assume a 70 kilogram (154 Lbs) snowboarder standing straight up on their way down the slope (giving them a frontal area I estimate to be about 0.5 square meters), with a drag coefficient of about 1 (which Wikipedia tells me is typical of skiers, and is probably about right for snowboarders), a friction coefficient of 0.04 (typical for waxed surfaces on packed snow), and a 38 degree slope, you get a top speed of about . . . 80 mph (129 kph).

Go into a racing tuck, put on a racing helmet and some slick racing clothes, and you can pretty easily push that up over 125 mph (200 kph), which is coincidentally just about the top speed record on a snowboard.

In fact, the fastest speed possible on a 38 degree slope is probably much higher. I made some pretty conservative guesses in doing these calculations, and didn't consider aerodynamics much at all. In fact, the skiing speed record of about 156 mph (252.4 kph), which was set under similar conditions that I used in my calculations, shows that there's lots of room to improve snowboard speed records.

So why haven't snowboard speed records equaled skiing speeds already? That's a subject for another post. I'll tell you what I think about it some other day.

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The mathy bits.

Here's the equation I started with to calculate a snowboarder's terminal velocity

m*g*sin(38 degrees)= u*m*g*cos(38 degrees) + (p*A*Cd*v^2)/2

where

'*' is the multiplication symbol and '/' is the division symbol

'm' is the mass of the snowboarder

'g' is the acceleration due to gravity

'p' is the density of air

'Cd' is the drag coefficient

'u' is the friction coefficient

'A' is the frontal area of the snowboarder

And 'v^2' means velocity squared, or v*v

You can go through and rearrange this equation with a little algebra to solve for the velocity.

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Read the rest of the post . . .

PopSci's Extreme Sports Error

I love the magazine Popular Science, but nobody's perfect, and this time PopSci slipped up.

In an otherwise excellent article by John Mahoney about physics in the movies, he criticizes the Vin Diesel adventure xXx by saying that there's no way anyone can outrace an avalanche on a snowboard.

Sorry John, you got that wrong. There's a good chance you wouldn't survive, and it's unwise to try, but it's at least theoretically possible to snowboard to safety in front of an avalanche.

The problem is that Mahoney is under the mistaken impression that the downhill speed record on a snowboard is only a piddling 50 miles per hour. In fact, the record is closer to 125 miles per hour (201 kilometers per hour). Even that's a bit slow for my comfort, considering that avalanches typically move (according to Mahoney) at minimum of 130 miles per hour. But with a slight head start, you could certainly stay out of trouble for a little while.

Considering that a slow avalanche would overtake a world class speed snowboarder (which I am assuming Vin's character must have been) at a relative speed of about 5 miles per hour, the gradual approach of the avalanche on Vin's heels would have been a very dramatic (and entirely possible) moment.

Check out a these videos of legendary snowboarder Big Mountain Jeremy Jones racing some avalanches (Jeremy actually performed Vin Diesel's snowboarding stunts in xXx). It seems like a really stupid thing to do. Thank goodness he survived. But as you can see, he snowboards (and falls) at about the same rate as the avalanche.

Mahoney should probably have realized his error by considering the fact that both snowboarders and avalanches descend the mountain with very little friction. In other words, the only force affecting the descent is gravity (to a pretty good approximation anyway).

Galileo showed that when friction and drag are low, all things accelerate (and slide down hills) at the same rate under the pull of gravity, regardless of size or mass. So whether you're a snowboarder, a boulder, or an avalanche, your fastest trip down a mountain is going to be about the same.

Why do some avalanches fall at speeds that top out around 200 mph (as Mahoney notes)? Because they are sliding down VERY steep slopes. If you were to snowboard down those sorts of slopes, you could theoretically go just as fast as the avalanche. Snowboard speed records are generally set on slopes that descend at a little over 30 degrees. If the slopes were steeper, the speed records would be higher.

Read the rest of the post . . .