A fascinating 2022 book by Randall Munroe titled what if? 2

 

 

 

 

 

 

 

 

 

 

 

My friendly local public library got in Randall Munroe’s 2022 book titled what if? 2 (additional serious scientific answers to absurd hypothetical questions. I have enjoyed reading it for his combination of physics, engineering, and droll humor. Here is one example from pages 268 to 271 which is illustrated by seven drawings:

 

“SNOWBALL (#54)

 

What if I tried to roll a snowball from the top of Mount Everest? How big would the snowball be by the time it reached the bottom and how long would it take? – Michaeline Yates

 

When snowballs roll through wet, sticky snow, they grow. For dry snow like what you’d find on Mount Everest, a rolled snowball wouldn’t get bigger; it would just tumble down the mountain like any other object.

 

But even if Mount Everest were covered in the kind of wet snow that made good snowballs, a snowball wouldn’t get that big.

 

A rolling snowball picks up snow and gets bigger, and a bigger snowball picks up more snow. This may sound like a recipe for some kind of exponential growth, but an idealized snowball’s growth actually slows down over time. It keeps getting bigger and wider, but each new meter it rolls adds less to the diameter. The growth slows because the width of the snowball’s track – and thus the amount of snow it picks up - is proportional to its radius, but the surface area the new snow has to cover is proportional to radius squared, which means that each new clump of snow has to be spread out over more area. People use the word ‘snowballed’ to mean ‘grew faster and faster,’ but in a sense the truth is the reverse.

 

Mount Everest is very tall [ 8.85 km], so even with a slowing growth rate, there’s still a lot of room for a snowball to pick up snow. The mountain’s three main faces descend about 5 kilometers before they level off into glacial valleys. In theory, an idealized snowball rolling down a 5-kilometer slope would pass through enough snow to grow to 10 or 20 meters wide by the time it reached the bottom.

 

In practice, it wouldn’t make it more than a few hundred meters, even in perfect wet snow. There’s a limit to how big snowballs can get before they collapse under their own weight. Gravity pulls the edges of a snowball down, so the insides are under tension. If a snowball gets too big, it collapses.

 

Snow has a tensile strength, which means it resists being pulled apart. Its tensile strength isn’t that high – which is why you don’t see a lot of ropes made of snow – but it’s not zero. A typical tensile strength for well-packed snow might be a few kilopascals, which is stronger than wet sand, weaker than most types of cheese, and about 1/10,000 th that of most metals.

 

There’s a number in engineering that measures how long a dangling piece of material can get before snapping under its own weight. It’s called the ‘free-hanging length,’ and it’s a ratio between a material’s tensile strength, density, and gravity.

 

The free-hanging length of a material provides a pretty decent approximation – to within an order of magnitude, at least – of how big a ball of material could get. Its value for snow ranges from less than a meter for fluffy snow to a meter or two for heavy, packed snow.

 

This formula lets us compare different materials. It tells us that the largest snowball would be bigger than the largest ball of sand – which is even weaker than snow and much more dense – but smaller than the largest ball of hard cheese an nowhere near as large as the largest ball of iron. {Sandball 10-15 cm. Snowball 1-2m. Gruyere Cheeseball 8 meters. Ironball 500 meters.}

 

If you look up videos of people rolling large snowballs down hills, you’ll see that they usually break apart when they reach a size of a few meters, just as the formula suggests.

 

But slopes that can support self-growing snowballs are rare, and they’re rare because they can support self-growing snowballs. If a snowball grows while it’s rolling down a hill, it will break apart. A snowball that breaks apart becomes a bunch of little snowballs, which will start to grow, too, just like the original.

 

Congratulations, you’ve invented an avalanche.”

 

On November 2, 2019 I blogged about Randall’s first what if? book in a post titled A thought provoking how to book by Randall Munroe. And on February 7, 2023 I blogged about a comic of his in a post titled An xkcd comic on a size comparison that is unhelpful. 

 

There are a bunch more examples from what if? 2 posted here. One is about how to Catch a Bullet. Another asks How long would it take for a single person to fill up an entire swimming pool with their own saliva?  A third is What if Au Bon Pain lost this lawsuit and had to pay the plaintiff $2 undecillion?

 

An image of a brush rest showing Japanese boys rolling a snowball came from Wikimedia Commons.