Now, to measure the time. Looking at the timestamp on the video, I see that the volcano let out the plume at 13:23:38. Depending on what part of the debris you want to consider, it reaches the treeline somewhere between 13:24:22 and 13:24:46. So it took somewhere between 44 to 68 seconds to reach that point.

Divide the two numbers, and we get our speed. **The slow estimate puts it at 49 meters/second (109 mph), and the fast estimate puts it at 75 meters/second (168 mph). Taking the average, we get 62 meters/second or 139 mph.**

Since we got a pretty big variation, I decided to check my calculation by putting another marker half way down the mountain, and timing how long it took the plume to travel this new distance.

Here's what the mountain looks like with the new marker on it (it's a little hard to see, but there's a fourth pin halfway along the red line):

The distance to the halfway marker was 1.64 km, and the time was 26 seconds. I think these numbers are a bit more reliable than before. Divide distance by time, and you get a speed of 63 meters/second or 140 mph.

*Hmm*, that's basically the same number from before, which is a bit odd. Given the huge uncertainty in our speed calculation, and the various factors like gravity and friction that can change the speed, this is just a coincidence. Nonetheless, **I'd conclude that 140 mph is a pretty good estimate of the speed of debris flow down the volcano.**

So now we've got the speed. What can we do with that? **Surprisingly, we can actually use the speed of the mud to estimate the pressure inside the volcano.** What follows is what physicists call an order-of-magnitude estimate - it's a back-of-the-envelope calculation that will give us a rough answer. This is a dramatic simplification of the hairy physics that's actually going on inside a volcano. Nonetheless, readers of this blog will know that I like these toy models, because they give you some insight in exchange for not a lot of work.

With that in mind, let's go on.

Picture the moment that the volcano explodes. Inside the volcano, hot gases have built up a huge pressure, whereas outside, everything is normal. At the top of the volcano sits a chunk of rock, like a cork on a champagne bottle, that's being pushed up from the inside by high pressure gas. We can use an equation called Bernoulli's equation to relate the pressure at the top and at the bottom of this chunk of rock.

$latex P_{inside}$ is the pressure inside the volcano at the moment before it explodes. This is what we want to find. $latex P_{outside}$ is the pressure outside the volcano at this moment, which is good 'ol atmospheric pressure. *ρ* is the density of mud sitting on top of the volcano. And $latex v$ is the speed of the rock at the breaking point. I'm assuming that this is somewhere near the speed we calculated above, of 63 meters/second.