The Physics of Snowflakes & the Spirit of Christmas

We shovel snow from our door steps because although no two snowflakes are alike, far more than two land in the same place. Rigorous shovelling helps beat the cold. and once warmed up, we can think about falling snowflakes. And if we want to be even more captivated, we can observe them.

First the thinking part. If we want to merely predict the velocity of falling snowflakes, we already run into a complication. At least raindrops begin as spheres, and then as they grow larger, their shape approximates that of a burger bun. That affects their area and drag coefficient —numbers needed in assessing to what extent air slows down the rate of falling drops. But snowflakes are formed in a countless variety of shapes and sizes. There is far more averaging out to do.

So assume that it’s been done. We subsequently write an expression for the product of air density, the flakes’ average area, their average total drag coefficient and square of their velocity. Then we subtract that expression from the force of gravity. The difference will equal to the so-called net force, which is the product of mass and rate of change of velocity with respect to time—Newton’s Second Law.

In our differential equation, velocity appears on the equation’s two sides, one of which also has the variable of time. Isolating the variables and using appropriate substitutions allows us to integrate and solve for velocity. As the time that the snowflake falls increases, exponential terms drop out of the equation, and the flake’s terminal velocity seems to depend only on the  the snowflake’s mass and the shape -influenced and gravitational constants we mentioned earlier.

8051109Now we observe. As we stated at the onset, many snowflakes land in the same place. But only a few meters above any given spot, it is apparent that many paths lead to a common destination. Some flakes tumble; some abandon the terminal velocity we took so long to calculate, and they yield themselves to whimsical eddies. How they arrive is influenced not only by shape, mass and gravity but by sheer luck—luck due to the random, pinpoint fluctuations in temperature and pressure that affect their air space.

And these unpredictable*, forgotten, dance-like movements of deviant snowflakes open our eyes and widen our mouths. They drain our minds of thoughts of shovelling and of future slush and social conflicts. For a few moments the destinies of snowflakes is all that matters, and then we are reminded of a beautiful, non-mathematical expression in which snow is equated with Christmas.

*N.B. In reality the larger snowflakes may behave like sheets of falling paper which experience aerodynamic lift, a lift dominated by the product of linear and angular velocities. Those of you interested in computer simulations of falling snow might find this link interesting: https://www.cs.rpi.edu/~cutler/classes/advancedgraphics/S08/final_projects/fermeglia_willmore.pdf

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