HOW TO SKI — RANDALL MUNROE
Best-selling author Randall Munroe takes an entertaining slide into the STEM of skiing – including some lateral solutions to our lack of summer snow.
Going off-piste with science and snow
SKIING INVOLVES STRAPPING long, flat objects to your feet and sliding across a surface or down a slope. The surface is usually water, either in frozen or liquid form, but it doesn’t have to be.
You can slide down any slope if it’s steep enough. When an object sits on a slope, gravity partly pulls it downward and partly pulls it along the slope. An object starts to slide when the force pulling it along the surface overcomes the force of friction.
Depending on what material your skis and the surface are made of, you might not start sliding easily. If the skis are made of rubber, and the surface is cement, you’ll need quite a steep slope to ski, which is presumably why rubber-on-cement skiing is so unpopular.1
For any combination of surface material and ski material, you can use a simple physics relation to calculate how steep the slope will have to be to slide.
It seems like it might be a hard problem, but thanks to a convenient coincidence, most of the complicated parts cancel out, and you wind up with this extremely straightforward equation:
coefficient of friction = tan (slope angle)
If you want to know the slope angle, you can reverse the equation:
slope angle = tan-1 (coefficient of friction)
This equation is delightfully uncomplicated, right up there with E=mc2 and F=wma.
Unlike those more famous equations, it’s only useful for this very specific problem, but it’s still neat how simple it is.
Here’s a table of coefficients of friction of different ski/surface materials:
Here’s a table of coefficients of friction and the corresponding minimum slope angle you need to start sliding:
0.01/0.6° (bicycle on wheels) 3
0.05/3º (teflon on steel, ski sliding on snow) 0.1/6º (diamond on diamond)
0.2/11º (plastic shopping bags on steel) 0.3/17º (steel on wood)
0.4/22º (wood on wood)
0.7/35º (rubber on steel)
0.9/42º (Rubber on concrete)
Wooden skis would work on a 16° steel ramp. If the skis were made of rubber, a steel ramp would need to be 35° before you could slide. The coefficient of friction between rubber and concrete is even higher – 0.9 – and you’d need a pretty steep slope of about 42° in order to slide down. This also tells you that a person in rubber-soled sneakers can’t walk up a ramp with a slope steeper than 42°.
In a sense, skiers are really just mountain climbers who are unusually bad at climbing but make up for it with very good balance.
Ice is slippery compared with most surfaces, and snow – which is really just fancy ice – is similarly slick. This makes them a good choice for skiing and similar activities, which is why every sport in the Winter Olympics involves sliding in some way.
The reasons that ice is slippery are actually a little mysterious. For a long time, people believed that the pressure from a skate blade melted the surface of the ice to create a thin, slippery layer of water. Scientists and engineers in the late 1800s demonstrated that the pressure of an ice skate blade could lower the melting point of ice from 0°C to −3.5°C. For decades, pressure melting was accepted as the standard explanation for how ice skates work. For some reason, no one pointed out that it was possible to skate at temperatures colder than −3.5°C. The pressure melting theory suggests it should be impossible, but ice skaters do it all the time.
The actual explanation for why ice is slippery is, surprisingly, still the subject of ongoing physics research. The general explanation seems to be that there’s a layer of liquid water on the surface of ice because the water molecules aren’t firmly locked into the ice crystal lattice. In this way, an ice cube is sort of like a piece of cloth with fraying edges. In the middle of the cloth, threads are locked into a well-organised form, but on the edges, they’re less constrained and more likely to come loose and move around. In the same way, water molecules near the edge of a piece of ice come loose and move around, creating a thin layer of water.
However, the properties of this water layer and how a skate interacts with it aren’t fully understood.
Given how much time modern physics spends on deep and abstract mysteries like searching for gravitational waves or the Higgs boson, it can be surprising how many basic everyday phenomena aren’t well understood. In addition to ice skates, physicists don’t really understand what causes electric charges to build up in thunderstorms, why sand in an hourglass flows at the speed it does, or why your hair gets a static charge when you rub it with a balloon. Fortunately, skiers and skaters can slide on snow and ice without waiting for physicists to finish figuring things out.
Snow is already pretty slippery, but to gain a little extra slickness, skiers add a layer of wax to their skis. The wax serves as a semi-liquid layer, keeping sharp ice crystals from digging into the hard material of the skis and slowing them down.
Waxed skis on snow have a coefficient of friction of about 0.1, which drops to 0.05 once the skis start moving4. This means that you need a slope of 5° to start sliding under your own weight, but once you get moving, you only need a slope of about 3° in order to keep going.
Once you’re sliding down a slope, you’ll continue accelerating until you either run out of snow or you reach a speed at which air resistance is pulling backward harder than gravity is pulling you forward.
Since air resistance doesn’t really start to kick in until higher speeds, even a gentle slope can let a skier or sledder go pretty fast if it’s long enough.
The theoretical top speed of a skier or sledder on a 5° slope of unlimited length is around 48 km/h – 72 km/h if they’re particularly aerodynamic. On a 25° slope, speeds of over 160 km/h should be possible for an aerodynamic skier or sledder.
The world record for top speed achieved on skis is around 250 km/h, but people don’t keep close track of that record, because it turns out not to be a particularly interesting boundary to push. The way to reach higher speed is simply to find a longer, steeper slope.
If you keep doing that, skiing gradually morphs into skydiving – only an even more dangerous version of skydiving, since instead of falling through open air, participants are skimming across ground. Obstacles are very hard to avoid when skiing at
250 km/h, and even if you find what seems like a smooth slope, a small bump or gentle turn could be instantly fatal.
When a competitor’s score in a sport is strongly correlated with their odds of dying, it creates obvious problems for the sport. Speed skiing was briefly featured at the 1992 Olympics, but, after a number of deadly accidents, has been mostly abandoned at the competitive level.
WHEN YOU REACH THE BOTTOM
If you’re skiing down a slope, eventually you’ll reach a point where you can’t go forward any more.
This can happen for a few reasons:
• There are trees, rocks, or hills in the way • You reached the bottom of the mountain • There’s no more snow
If you’re having fun and don’t want to stop skiing, you have a few options.
If there are trees in the way, you can try removing them; for more on how to do this, read my useful book, How To. If there are rocks in the way, read How To for advice on whether you can move them. If you’ve reached the bottom of the mountain, you can try continuing to accelerate yourself forward.
If you’ve run out of snow, read on.
WHAT TO DO IF YOU RUN OUT OF SNOW
From our discussion of friction, we know skis don’t work very well on most non-snow surfaces. There are some artificial ski slopes that use special low-friction polymers, with a bristly hairbrush-like texture that provides some softness and lets the skis dig in when turning. There are also special skis designed for use on grass and other surfaces, but they use wheels or treads rather than sliding.
If you want to keep skiing on snow, but there’s no more snow to slide on, you’ll have to make some yourself.
Almost all of Australia’s ski destinations and about 90% of American ski resorts use artificial snow to ensure that ski slopes are covered as soon as it’s cold enough for snow to stick, and to keep them covered for the whole ski season even if the weather doesn’t cooperate. The artificial snow also helps to replenish snow lost throughout the season due to melting and erosion from skiers.
Snow machines make artificial snow by using compressed air and water to create a stream of tiny ice crystals, and then misting the ice crystals with more water droplets as they float in the air. As the mist drifts down to the ground, the water droplets freeze onto the ice crystals to form snowflakes.
The snowflakes formed in this way are more compact and misshapen than the delicate shape of natural snowflakes. Natural snowflakes have much more time to grow slowly in a cloud, one water molecule at a time, which allows intricate and symmetrical shapes to form.
Artificial snow forms quickly, in the short time it takes water to descend from the nozzle to the ground, from a handful of drops clumsily jumbled together.
Suppose you need a 1.5-metre-wide path to ski on, and you’re going to descend at a speed of around 30 km/h. Natural snow might be 10% water and
90% air by volume, although this ratio varies quite a bit depending on how light and fluffy the snow is. For simplicity’s sake, let’s also assume you want about 20 centimetres of somewhat heavy snow to ski on, with snow that’s ⅛ as dense as water, equivalent in mass to a layer of water an inch thick. The total amount of water you’ll need is therefore:
1.5 metres x 20 cm x 1/8 x 20 km/hour = 310 litres/second = 1125 m3/hour
Skiing the length of a football field will require more than 30,000 litres of water, along with the equipment to turn it to snow.
You’ll have a hard time finding equipment that can produce snow fast enough for you. The biggest snowmaking machines might produce snow at rates of 100 cubic metres per hour. That’s just 10% of what you need, so you may need a lot of them.
Snow from typical snowmaking equipment needs a lot of time to drift down to the ground, which means you’ll have to produce the snow far ahead of your current position to give it time to settle, and the movement of air currents may make it hard to concentrate enough of it along a narrow path.
The long, slow descent is necessary because it takes a long time for the water droplets to lose heat to the air through evaporation to attach to the ice crystals. There are ways to cool the water droplets down more quickly – but they have some drawbacks.
If you inject low-temperature substances like liquid nitrogen into the air/water stream, they can reduce the temperature and cause almost-instant freezing. These techniques can produce snow quickly, and are used by some snowmaking companies for special events in areas where the air temperature is too high for normal artificial snow to be produced.
Cryogenic freezing techniques are generally not used by ski resorts – it’s far too expensive and energy intensive to freeze water this way compared to letting it freeze on its own in the air.
For your small, very narrow ski slope, liquid nitrogen just might be affordable. If you buy the liquid nitrogen in the form of small tanks, your ride could cost $50 per second, but industrial suppliers can get you a much better price if you buy in bulk.
You don’t have to use liquid nitrogen – you could also try other cryogenic gases. Liquid oxygen is similar to liquid nitrogen and just as easy to produce, and could in theory be used for snowmaking. However, this is not recommended. Liquid nitrogen is a popular cryogenic fluid in part because it’s so inert and nonreactive. Liquid oxygen is neither of those things.
MAKE THE PROCESS MORE EFFICIENT
You could reduce the snow consumption if you could somehow scoop up the snow behind you and reuse it, rather than producing more snow as you go.
If you lay down some kind of a tarp under the snow, you can pick the whole sheet of snow back up and reuse it with minimal losses.
The tighter you make the snow-transfer loop, the less snow you’ll need.
You can even make the loops smaller than your body if you pass the stream of snow around your legs, rather than over your head. . . . . .at which point you’ll realise you’ve effectively reinvented roller skates.
RANDALL MUNROE studied physics before working for a short time at NASA Langley Research Centre building robots. He created the webcomic xkcd in 2006, and now draws and writes full-time. He has been nominated for a Hugo Award three times and has had an asteroid named after him – 4942 Munroe is big enough to cause mass extinction if it ever hits a planet like Earth. This is an extract from How To …, his third book.
Would you like to ask Randall a puzzling or entertaining question? Email us at contribute@cosmosmagazine.com