Stopping power of snow - physics question

I was on a ski trip this week with a friend who fell on some hardpacked snow (basically ice) and broke his forearm. Unfortunately at this particular ski area, it has not snowed in over a week so there is no new powder covering the hardpack area, making the impact of falls much greater.

Naturally this prompted us to consider the question of how deep would a covering of new snow have to be to have prevented the broken arm. By way of explanation, if there were only a few fresh snowflakes, they would not have cushioned my friend’s fall at all and he would still have a broken arm; by contrast, if there were say 10 feet of fresh powder, he would have safely landed in a soft pillowy mass of snow.

Unable to solve the physics of the matter, we decided to turn to you, the teeming millions, for help resolving the question.

So take the following facts as a starting point:

  1. The arm was just barely fractured from the actual fall. Therefore we think that a decrease in impact of maybe 50% would have safely prevented the break.
  2. The typical powder snow that falls around here is about 7% water. So, pretty light, powdery snow.

Just how much snow is needed to safely prevent a broken arm in this type of accident? The more involved and intricate the answer, the better!

Boy lots of unknown variables.

The power to slow down is going to be the volume of snow being displaced times the density of the snow. Arm straight on cutting through the snow will displace little volume, arm and body landing sideways more (essentially the same floatation principle that snowshoes rely on). The force needing to be offset depends on mass and velocity.

The density of water is 62.42 pound/cubic foot. 7% of that is 4.36 lb/cf. Outstretched gloved hand maybe 4x4 SA inches wide so figure per 4 inches of depth 1/27th of that as floatation. So 0.16 lbs floatation roughly per 4 inches outstretched arm cutting through the snow straight on. You’d have to have more surface area hitting to slow you down much.

Assume deep snow and it gets denser under you so it gets more complex, more than I can handle, but by that point I think the physics would be that either it would rapidly get dense enough to stop rapidly enough to cause the break, or more likely to serve as a fixed point as the body continued forward and have a break occur due to that leverage.

Someone PLEASE check my math!

How about forearm on its side and assume some bulky ski clothes?

6 inches wide, 18 inches long, to make it easy. That’s then 1.45 lb of floatation per 4 inches of depth. So again not much help for the first few inches. And again, after a few inches the issue gets more complex with the density changing due to the compression.

Sorry to triplepost but this bit more: at several feet of depth of fresh snow there are multiple other factors. I think the biggest is the surface area issue: the whole body will get involved at some point even with an outstretched arm. Figure for simplicity a trunk 5x2 ft then the first 4 inches is 14.5 of floatation, ignoring the change in density from compression.

That compression bit is the bit we need the real math whizzes for. I am sure that even before the impact that the density of fresh fallen snow is higher 4 inches down than in the first 2 inches just because of the compression from the snow above. Add in the dynamic effect of compression from the falling body and the stopping power per additional inch of depth has to increase much more than linearly.

Anyone with a good way to model that?

Okay. Here’s an article that discusses how model it, but it is beyond my ken to apply it. One of their points is that nature of the snow beyond its density matters (granular, particulate, etc.). Still it cites field data that shows the force begins to increase dramatically at about 0.25 meters of depth. So I’d guess that a foot of snow would do ya.

Anyone else want to take a stab with that as a resource?

I have nothing to add but I’ll think you’ll find that snow is 100% water no matter where it falls :wink:

Also a factor would be the time the snow has laid there since falling.

Snow packs harder as it sits. Let it sit long enough, and the bottom layers are compressed into ice – that’s how glaciers & icebergs are formed.