Five Seconds per Mile Is Right Only on a Cold Night

See the flash, count the seconds until the thunder, divide by five. That is how many miles away the lightning struck. Everyone learns it as a kid, and it is close enough to decide whether to head indoors.

It is also, strictly speaking, exact at one temperature only. Sound covers a mile in 5.00 s when the air is about 4.41 °F. On a 68 °F summer evening it needs 4.69 s, so a 30 second count means 6.40 mi, and the rule says 6. That is 6.2% short, every time, at that temperature. None of this makes the rule useless. It makes it a rule of thumb with a temperature attached, and it is not the only shortcut about sound that people carry around without the fine print.

Below are six of those shortcuts, checked with the Speed of Sound Calculator, the Echo Distance Calculator and the Ohm's Law Calculator, plus one that turns out to be true. Every number came out of those tools.

Myth: five seconds of thunder always means one mile

Fact: the rule quietly assumes cold air. The speed of sound in dry air depends on temperature through the square root of the absolute temperature, v = 331.3 × √(T / 273.15 K), and nothing else about the weather matters much. Warm air carries sound faster, so each mile takes less than five seconds and the count undersells the distance.

The table below keeps the evening at 68 °F and changes only the count. Each number of seconds opens its own ready-made page with the delay, dry air and 68 °F already filled in.

The gap grows with the count, because it is a fixed percentage. At 2 seconds it is about 140 ft and nobody cares. At 30 seconds it is 0.40 mi.

The screenshot also shows the metric version of the rule, divide by three for kilometers. At 68 °F it lands on 10.00 km against the real 10.30 km, 2.9% short, so it happens to be the more accurate of the two on a warm night.

How much does the temperature itself move things? Here is one mile of air from a very cold day to a very hot one. The temperatures link to pages that show the full result for each.

Only the 0 °F row needs more than five seconds, 5.02 s. Everything warmer than about 4 °F is quicker. So yes, five seconds per mile is right on a cold night, and a little generous on every other.

Myth: sound has one speed, about 1,125 feet per second

Fact: that figure is dry air at roughly room temperature, and textbooks round it. The calculator gives 1,126.03 ft/s at 68 °F, but 1,050.98 ft/s at 0 °F and 1,159.67 ft/s at 100 °F. Up at airliner cruising height, around 11 km where the air is near -57 °C, it drops to 294.71 m/s, which is why Mach 1 is a slower speed up there than on the runway.

Physics problems usually give the temperature in Celsius. The common shortcut there is a straight line, 331.3 + 0.606 × T, and the table puts it next to the square-root law. Each Celsius value opens its own page.

Up to about 30 °C the shortcut is off by less than half a meter per second. By 100 °C it overshoots by more than 4 m/s: 391.90 m/s against 387.22 m/s. Fine for homework near room temperature, not for an oven.

Myth: air pressure changes the speed of sound

Fact: for an ideal gas at a fixed temperature, it does not. Higher pressure means denser air, and the two effects cancel exactly, as the Wikipedia article on the speed of sound puts it. What does change high up is the temperature, which is why the altitude example above is really a temperature example.

Humidity is the other weather variable people suspect. It does matter, a little: moist air is faster by about 0.1% to 0.6%, because water molecules are lighter than the nitrogen and oxygen they replace. The calculator assumes dry air, so on a muggy night the true distance to the storm is a touch longer still.

Myth: heavier materials carry sound more slowly

Fact: weight alone does not decide it. Steel is far denser than air and carries sound at 5,940 m/s, 17.31 times as fast. Speed depends on stiffness against density, and stiffness usually wins by a wide margin.

Usually, not always. Lead is denser than steel and soft, and sound moves through it at 2,160 m/s, 2.75 times slower than through steel. Neoprene rubber is heavier than water, yet only about 8.0% faster: 1,600 m/s against 1,481 m/s. Knowing that something is heavy tells you almost nothing about how fast sound goes through it.

Myth: echo time times the speed of sound gives the distance

Fact: it gives twice the distance. An echo goes out and comes back, so the gap is covered twice and the result has to be halved. A clap against a cliff heard 1.5 s later on a 59 °F day means a path of 1,674.58 ft and a cliff 837.29 ft away.

Sonar makes a second mistake easy: using one speed for all seawater. The speed at the surface depends on temperature and salt, and the Echo Distance Calculator runs the Mackenzie equation for it. A 4 s echo in 40 °F water at salinity 35 puts the bottom 1.825 mi down, at 4,817.38 ft/s.

Warm the same water to 77 °F and the same 4 s echo means 1.907 mi. The difference is about 132 m, roughly 433 ft, from temperature alone. Both figures use the surface speed; a survey ship measures how the speed changes on the way down, which is more than a single echo can tell you.

Myth: pick the standard resistor closest to the value you calculated

Fact: for an LED, round up. The series resistor sets the current, R = (supply voltage − LED forward voltage) ÷ current. A red LED at 2 V and 20 mA on a 9 V battery needs exactly 350 Ω, a value nobody sells.

The closest value in the common E12 series is 330 Ω, only 20 Ω away, but it lets 21.212 mA through, above the target. Rounding up to 390 Ω gives 17.949 mA, and the E24 series has 360 Ω for 19.444 mA. A little under the target current, the LED looks practically the same.

The same arithmetic for the supplies people search for most. The 2 V and 20 mA are assumptions: 2 V is the top of the 1.7 V to 2.0 V range given for red LEDs, and 20 mA is a common rating for small ones. Each voltage opens a page with those values filled in, ready to change to your LED's datasheet.

On 5 V the exact value, 150 Ω, happens to be a standard one. On 12 V the resistor burns 196.08 mW with the E24 part, 510 Ω, so the power rating starts to matter as much as the resistance.

The one that is true: sound really is faster on a hot day

This one survives every check. Between 0 °F and 100 °F the speed of sound in air rises from 1,050.98 to 1,159.67 ft/s, about 10.3%. The same 30 second thunder count means 5.97 mi on the coldest of those days and 6.59 mi on the hottest. On the cold one the divide-by-five rule is 0.5% over; on the hot one it is 8.9% short.

The one rule that holds

Most of these shortcuts are a real calculation with one input frozen. The thunder rule freezes the temperature. The 1,125 ft/s figure freezes it too. The echo shortcut forgets the return trip, the sonar shortcut freezes the water, and the resistor shortcut freezes the rounding direction. When a rule of thumb gives you a number, ask which input it quietly fixed, and put that input back. Usually that is a temperature.

Tools discussed in this article

• Speed of Sound Calculator: the speed of sound in air at any temperature and in seventeen other materials, a distance from a thunder delay, or the time a sound needs to travel
• Echo Distance Calculator: the one-way distance from a round-trip echo in air, water, seawater by temperature and salinity, or a solid, with times down to microseconds
• Ohm's Law Calculator: volts, amps or ohms from the other two plus the power, and an LED series resistor rounded up to E12 and E24 values

More physics tools

The rest of the physics shelf, in the order it was built:

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