Counting Between Flash and Thunder: A Thunderstorm over the Ridge
On a hot, humid August afternoon a student stands in a valley in Nagano and watches a thunderstorm build over the ridge to the west. She sees a flash and counts 5.0 s until she hears thunder. Ten minutes later a second flash gives a count of 2.0 s. The air near the ground is at about 30 °C; at 0 °C sound travels at 331 m/s, and the speed rises by about 0.6 m/s for each 1 °C increase. The cloud over the ridge is dark, with a flat spreading top, and the valley floor is still sunny and calm, although a cool gust arrives later. Her friend says the shorter count proves the storm is coming closer, so they can keep hiking and simply leave when a count falls below 3 s. The path they intend to take runs along an exposed ridge above the treeline.
1. Estimate the distance to each flash, state the uncertainty honestly, and say what precision the method deserves.
Read the worked answer
Speed of sound at 30 °C ≈ 331 + (0.6 × 30) = 331 + 18 = 349 m/s.
First flash: 5.0 s × 349 ≈ 1745 m ≈ 1.7 km. Second flash: 2.0 s × 349 ≈ 698 m ≈ 0.70 km.
Uncertainty, judged rather than ignored:
- Human reaction time is around ±0.2 to ±0.5 s at each end of the count, i.e. up to about ±175 m on the 5 s measurement. That is the dominant error.
- Temperature: ±5 °C changes the speed by ±3 m/s, only about 1%, i.e. ±20 m.
- The sound path is slanted through air at different temperatures and may be bent (refracted), so the value is an estimate of distance, not a survey measurement.
Correct reporting: about 1.7 km, then about 0.7 km, quoted to two significant figures, with the caveat of a few hundred metres of uncertainty. Quoting '1745 m' would overstate the precision.
2. Judge the friend's rule ('keep hiking until the count is under 3 s') and recommend what the group should do.
Read the worked answer
A 3 s count corresponds to 3 × 349 ≈ 1050 m, about 1 km. The rule confuses distance to the last flash with danger from the next strike, and it fails in three ways:
- Lightning is not confined to the rain core. Strikes routinely occur 10 km or more from the visible rain, so a flash 'far away' can still be followed by a strike nearby.
- A count gives no warning time. Light arrives almost instantly; from 1 km the thunder takes 3 s, and from 300 m only about 0.9 s. A rule that waits for a count of 3 s leaves about 3 s to react - not enough to move off an exposed ridge.
- The trend already observed is the warning. The count fell from 5.0 s to 2.0 s, so the storm is approaching at roughly 1 km per 10 minutes (about 6 km/h).
Recommendation: descend from the exposed ridge now, before the count gets short, and move to lower ground away from isolated trees and metal. Do not continue upward on the reasoning that there is time until the count drops below 3 s.
3. Explain why the storm forms over the ridge rather than over the valley floor, and why the valley is calm and sunny until the cool gust arrives.
Read the worked answer
Over the ridge: warm, humid afternoon air is forced up the mountain slope. As it rises the pressure around it falls, the air expands and cools (roughly 1 °C per 100 m of lift). Cooler air holds less water vapour, so the relative humidity rises until the dew point is reached and water condenses into cloud droplets.
Condensation releases latent heat, which warms the air slightly and keeps it buoyant, so the updraught strengthens and the cloud grows tall until it hits a stable layer and spreads into the flat anvil top that the student can see. The supply of humid air and the forced lifting both happen where the slope is, so the storm is anchored over the ridge.
Why the valley stays calm: the storm's rising column is balanced by a downdraught of cooled air, some of it chilled by evaporating rain, plus the downward drag of falling precipitation. That cold, dense air spreads out along the ground as a gust front and reaches the valley later. Before it arrives, the valley is in the quiet inflow region, still sunny - which is exactly the misleading calm that makes people keep walking.