Snow Plow Projectile Physics

We got over a foot of snow yesterday and today, so schools are closed. Except Union is a residential college, so we never close, which means I have to dig my car out all the same. Which I did, clearing a path to the unplowed street, then took Emmy for her morning walk. During which, of course, the town snow plows came around and filled in the end of our driveway again...

As I was digging out again, I was struck by just how far the snow had gone up the driveway-- I paced off about six meters from the edge of the road to the farthest clumps thrown up the driveway. As a way to distract myself from being pissed about the extra work, I started thinking about whether you could use this to determine the speed of the plow.

It might seem that there isn't enough information here, but as always, we're really just interested in order-of-magnitude sorts of numbers, here. Well, probably better than that-- maybe a factor of two or so. But nothing hyper-exact.

So, given that, we can approximate our driveway as flat, with snow taking off from and returning to ground level. In that case, there's a really simple formula to give you the range of a projectile launched at some angle, provided you neglect air resistance. That's not a great approximation for snow in general, but there were some good-sized lumps in there, so it's probably not a huge factor. And again, factor of two numbers, here. So, we have the projectile range formula:

$latex d = \frac{v^2 \sin (2 \theta )}{g}$

That requires an angle, so you might think it's hopeless, but what we care about is the farthest lumps of snow, so we can just go for the maximum value, at a launch angle of 45 degrees. There must be some snow thrown at that angle, given the wide range of spray from a moving plow, and that is presumably the stuff that goes the farthest.

With a distance of 6m and an acceleration of 10 m/s/s, that gives a launch speed of just under 8 m/s, which we'll round up, because air resistance will tend to shorten the range, so that's the direction in which to err.

So how does that speed relate to the speed of the plow? Well, the plow is barrelling down the street at some speed, with a blade in front of it at some angle, so stuff on the left edge of the blade gets pushed right, toward the shoulder of the road and the clear end of my driveway. The speed at which stuff moves across the road-- which is what will determine how far it goes up the driveway-- is probably something like the speed of the plow times the sine of the angle the plow makes with the front of the truck. They generally go by pretty fast, and the angle is adjustable, so this is kind of hard to gauge, but I'll pick a nice, friendly number like 30 degrees. In that case, the sine is equal to a half, and the speed of the plow would be twice the speed of the snow, or about 16 m/s.

Converting to units for which I have some intuitive feel, that's about 35mph. Which is technically faster than the speed limit in our neighborhood, but not out of line for the speed people actually drive down our street. And the plows certainly go by at a pretty good clip-- not that you can blame them, given the number of streets they have to clear.

So, there you go: the math I did in my head while digging the end of my driveway out for a second time. Which is the physicist's version of "Serenity now!!!!"

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"God, grant me the serenity to accept the things I cannot change,
The courage to change the things I can,
And the wisdom to make a Fermi problem out of both."

As you can clearly see that some snow is moving forward of the plow, the actual speed of the snow must exceed the speed of the plow. In theory it can have an entirely elastic collision with the plow blade and bounce away at twice the speed of the plow. This puts a lower limit (assuming everything else is the same) of 8 m/s (18 mph) for the plow.

Much depends on the type of snow plow, by which I mean the design of the blade itself.

Standard plows have the top edge parallel to the bottom edge, and a uniform curvature across their entire width, which is a section of a cylinder if viewed from the side. These have to be operated at a steep angle relative to the direction of travel, for example 45 degrees, to prevent snow spilling off the leading edge and into the opposing traffic lane. They are typically used at speeds below 25 miles per hour. Their method of action is to "shove" the show to the side, or transfer the forward vector to a sideways vector.

"Speed plows" have a shape roughly similar to a section of a triangle when viewed from the front, and a shape that is a section of a cone. Their method of action is to cause the snow to take a curved path upward on the plow face and then roll over, transferring the forward momentum of the vehicle to the snow, in a vector that is the output of the upward motion and sideways motion laterally across the plow face. This action results in throwing the snow much further off to the side than with a standard plow.

A speed plow can be operated at slow speed and produce a result similar to a standard plow. But when operated above 20 miles per hour, the rolling and throwing motion becomes predominant over the "shoving" motion. This is particularly useful on highways, to get the snow mound dispersed far off to the side of the traffic lanes.

A video search will bring up many examples of each in action. There may also be blogs and forums where plow operators hang out, and you can ask technical questions.