## Monday, August 28, 2006

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### The math of hailstones

At the risk of invading my colleague Mark C-C's territory (see Good Math, Bad Math), I've a small comment on one of NPR's items about last Friday's hailstorm in Iowa.

Jim Keeney, of the National Weather Service, gives us a hailstone scale that's meant to represent hail of increasing size:

pea, marble, dime, penny, nickel, quarter, half dollar, walnut, golf ball, hen egg, tennis ball, baseball, tea cup, grapefruit, softball
Hm. I suppose it depends upon the size of your marbles and of your grapefruits, but my experience has it that marbles are larger than dimes, and I wouldn't put grapefruit in there as clearly smaller than softballs. And tea cup? Um....

That aside, though, we have this statement from Mr Keeney, commenting on the Iowa "grapefruit-sized stones" not quite making it to the top of the scale:

Right, just a liiiiitle bit bigger, but when we're talking half an inch when it's that big, that's not that different.

Let's over-analyze the physics for a moment. A softball has a diameter of about 4.7 inches, or a radius of about 2.35 inches. If a grapefruit is a half inch smaller, 4.2 inches, that makes its diameterradius 2.1 inches. What difference does that make for hail? Well, it's true that you'd likely be hard pressed to see the difference in size. But....

The wallop (force) that a hailstone packs is proportional to its mass, which is, in turn, proportional to its volume[1]. Its volume is proportional to the cube of its radius. So the ratio of the volume of the "softball" to that of the "grapefruit" is (2.35/2.1)3, which is about 1.4. In other words, while the "softball" has only 12% more diameter, it has 40% more volume than the "grapefruit". That means it has 40% more mass, and hits your car (or your head) with 40% more force.

That's a lot more than a mere "half an inch when it's that big" makes you think! It's very different.

[1] Reference formulas:

force = mass * acceleration
We assume that the acceleration is purely due to gravity, which is a constant 32 ft/sec/sec.
mass = density * volume
We assume the density of the matter composing the hail is constant, regardless of the size of the hailstone.
volume = 4/3 * π * radius3
We assume a hailstone is a sphere. It's close enough to that for the purposes of this on-a-napkin calculation.

Ross said...

"If a grapefruit is a half inch smaller, 4.2 inches, that makes its diameter 2.1 inches."

Obviously you meant "that makes its radius 2.1 inches".

Ross Patterson

Ross said...

That reminds me of an old physics joke:

The king of a small country, having decided that the key to prosperity is to improve the production of its primary industry, dairy farming, asks the council of the department heads of the state university, a biologist, a sociologist, and a physicist.

After a brief moment of contemplation, the biologist says "we should undertake a program of selective breeding to enhance the milk production traits of our cows."

Next, the sociologist says "we should study the behavior of our cows, how they interact with their keepers, surroundings, and each other, to determine how to minimize sources of stress, which, as we all know, reduces milk production."

Finally, the physicist, moving toward the blackboard, begins "first, assume a cow is a sphere..."

Ross Patterson

Barry Leiba said...

Yes, "radius", obviously. Typo corrected, and thanks.