Issue 008 - Volcanic hazards - Atmospheric transport
How far can Mayon volcanic ash travel before falling?
Reuters reported that the Philippines' Mayon volcano continued spewing lava, rocks, and ash, with PHIVOLCS saying the eruption had persisted for 190 consecutive days. Use a simple wind-and-settling model to estimate whether ashfall danger is local, regional, or farther downwind.
The problem
Estimate how far volcanic ash from Mayon can travel before falling to the ground.
If an ash plume rises hundreds of meters to a few kilometers above the volcano, is the danger zone for ashfall more like 1 km, 10 km, 100 km, or farther downwind?
Because Fermi problems target an order of magnitude, I normally use no more than two significant digits and write most calculations in scientific notation; the Fermi reference explains both conventions.
Before checking sources
Matt's first pass
I started with a few assumptions. Larger rock or ash pieces should fall near free-fall, without much effect from wind resistance. Smaller, less dense ash might fall at one-third to one-fourth of free-fall because air resistance keeps it aloft much longer.
I guessed that wind near the ground might move at about 5 to 15 mph, while higher-atmosphere winds might gust above 40 mph. For time in the air, I used:
time in air ~= sqrt(height / g)
g ~= 10 m/s^2Then, to account for air resistance for lighter ash, I tripled or quadrupled the time in the air.
I used a best-case and worst-case scenario to find a range.
Best case: ash is launched about 200 m high, experiences only low-speed near-surface winds, and has little wind resistance slowing its descent.
fall time ~= sqrt(200 m / 10 m/s^2)
~= 4.5 seconds
downwind distance ~= 7 m/s x 4.5 seconds
~= 32 metersThat gave me a very small ejecta radius.
Worst case: ash is launched about 2,000 m high, is extremely light and low density, gets a 4x multiplier to the time in the air, experiences wind gusts up to about 40 mph, and has extra height to fall because Mayon's summit is much higher than nearby lower-elevation areas.
fall height ~= 2,000 m plume + 2,500 m volcano height
~= 4,500 m
fall time ~= 4 x sqrt(4,500 m / 10 m/s^2)
~= 88 seconds
downwind distance ~= 20 m/s x 88 seconds
~= 1.8 x 10^3 m
~= 1.8 kmSo my first-pass range was roughly 30 meters to 2 km, depending on particle size, height, and wind.
Calibration Score
Matt's Calibration Score: 10 / 100
Higher is better: earn points for accurate pegs, sound models, correct math, and a result close to the sourced answer. The image shows percent full of it: 100 minus the Calibration Score.
Pegs: 0/30. The settling-speed and plume-transport pegs were not in the right regime.
Model: 0/30. Free-fall time was the wrong model for fine ash; terminal settling speed controls the estimate.
Math: 0/10. The fall-time equation also missed the factor of two.
Result: 10/30. The answer was off by several orders of magnitude, even though the reflection caught why.
Grounding facts
PHIVOLCS maintains a 6-km permanent danger zone around Mayon for direct volcanic hazards, but ashfall is a different kind of risk. It is not a neat circle around the cone; it follows wind direction, particle size, plume height, rain, and eruption duration. AP reported earlier in 2026 that nearly 200,000 residents across 124 villages were affected by Mayon ash, with more than 5,400 people fleeing to shelters.
That helps explain why a local-looking volcanic event can become a regional transportation, agriculture, water, aviation, and public-health problem.
After checking sources
Check and recalibrate
The free-fall calculation is the wrong main model for fine ash. The small correction is that free-fall time should use sqrt(2h / g), not sqrt(h / g), but that only changes the answer by about 1.4x. The much larger issue is that small ash particles quickly reach a terminal settling speed, so the useful first-pass model is:
downwind distance ~= wind speed x fall time
fall time ~= release height / settling speed
therefore:
downwind distance ~= wind speed x release height / settling speedA useful rounded wind peg is 10 m/s, which is 36 km/h or about 22 mph. For release height, a modest Mayon ash plume might put ash about 1 to 3 km above nearby downwind ground, especially if the ash starts above a 2.5-km summit and falls toward lower-elevation communities.
For coarse ash, use a settling speed around 1 m/s:
coarse fall time ~= (1 x 10^3 to 3 x 10^3 m) / (1 m/s)
~= 1 x 10^3 to 3 x 10^3 seconds
coarse distance ~= (1 x 10^1 m/s) x (1 x 10^3 to 3 x 10^3 seconds)
~= 1 x 10^4 to 3 x 10^4 m
~= 10 to 30 kmFor finer ash, use a settling speed around 0.1 m/s, and remember that still finer ash can settle even more slowly:
fine fall time ~= (1 x 10^3 to 3 x 10^3 m) / (1 x 10^-1 m/s)
~= 1 x 10^4 to 3 x 10^4 seconds
fine distance ~= (1 x 10^1 m/s) x (1 x 10^4 to 3 x 10^4 seconds)
~= 1 x 10^5 to 3 x 10^5 m
~= 100 to 300 kmIf the ash is very fine, reaches stronger winds aloft, or remains suspended through turbulent mixing, the range can extend farther. USGS notes that the smallest volcanic ash particles can travel tens to thousands of kilometers downwind depending on wind speed, eruption volume, and eruption-column height.
A good Fermi answer is therefore: coarse ash mostly falls within tens of kilometers, while fine ash can plausibly travel about 100 km or more downwind. For this problem's answer choices, the danger zone is much closer to 100 km or farther than to 1 km.
Post-check reflection
Matt's reflection
My biggest errors were using the wrong equation to calculate time to fall to the ground, because I left off the 2x height, and more importantly using that equation at all instead of a steady falling velocity after ash reaches terminal velocity almost immediately. My intuitions about how to handle this problem were way off, and that pushed my answer off by several orders of magnitude.
I'm familiar with the idea that ash from volcanic eruptions can be distributed hundreds to thousands of kilometers away, so I should have anticipated that my calculation was not reliable. I may have assumed that only uncommon conditions were necessary for ash to travel that far. I was startled by how incorrect I was, but after looking at the explanation, I should not have been.
For the news item, it is easy to read about ash and lava and take for granted what is being described. Digging into the estimate makes the regional footprint much more vivid.
Recommended memory peg
Remember 10 m/s wind ~= 36 km/h ~= 22 mph, and for settling problems use distance ~= wind speed x height / settling speed. Coarse ash may settle around 1 m/s, while fine ash can be closer to 0.1 m/s or slower.
Reader results
Bars show how submitted estimates sort into the answer choices from the gut-check prompt.