What do fireworks and thunderstorms have in common? Actually, a lot of things. First of all, with the 4th of July holiday just around the corner and the summer thunderstorm season in full swing, they’re both things that a lot of us here in the United States can expect to see in the next week. But most importantly for us today, they’re also both things that make noise…a lot of it. While loud noises are usually just annoying, in the case of fireworks and thunderstorms they’re actually useful. How? You can use them to figure out the distance to those spectacular fireworks explosions and lightning strikes. In the case of lightning, that’s not just useful—it could actually save your life! So let’s figure out how it works.
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Fun Facts About Fireworks
Before we start calculating distances to fireworks and thunderstorms, let’s look at some fun math facts about these two visual and audible wonders. Given that 4th of July celebrations are coming up in the U.S., let’s start with fireworks. Perhaps the most amazing fact about fireworks is that although they seem high-tech, they’re really quite ancient. Well, at least the idea behind them is ancient. In truth, modern fireworks use lots of sophisticated technology in their construction, but at their heart is a design that was developed over 1,000 years ago in China.
The fireworks used for big modern displays are shot out of launch tubes (think of little cannons pointed up) at almost 200 miles per hour to a height of nearly 1,000 feet. The clever folks who build these fireworks use lots of precise math and physics (which you can read about here) to make sure the timing works out perfectly so that each firework reaches its peak just as it bursts into a spectacular display of brilliant light and thunderous sound. And speaking of thunder…
Fun Facts About Thunderstorms
Have you ever wondered where thunder comes from? No, I don’t just mean knowing that it comes from thunderstorms, I mean actually knowing why thunderstorms produce thunder in the first place. I’ll leave it to Everyday Einstein to explain all the detailed science, but the basic idea behind the origin of thunder is pretty simple. In a nutshell, thunder comes from 100-million Volt bolts of lightning (that’s huge, by the way…your household electrical system runs at a measly 120 Volts). These lightning bolts quickly heat the air surrounding them up to tens of thousands of degrees Fahrenheit (which causes the air to “expand”), and the subsequent cooling of the air (which causes it to “contract”) results in the production of sound waves—aka, thunder. Voilà!
Speed of Sound vs. Speed of Light
Now that we understand a bit about why fireworks and thunderstorms produce their spectacular sights and sounds, let’s see what we can do with this knowledge. The first thing we need to understand is how light and sound travel through air. In particular, the fact that light and sound travel at very, very different speeds. More precisely, a sound wave (which is just air molecules progressively bumping into each other on their way to bumping into your eardrum) travels through air at about 770 miles per hour (although the precise value depends a bit on things like humidity, altitude, and temperature).
While that’s definitely faster than your car or even a passenger jet, it’s tortoise-slow compared to light waves that travel about 670,000,000 miles per hour! Just to make this remarkable difference clear, light travels about 870,000 times faster than sound…which means that in the time it takes light to travel from Los Angeles to New York, a sound wave will travel only about 15 feet.
In the time it takes light to travel from Los Angeles to New York, a sound wave will travel only 15 feet.
How to Estimate Distance Using Sound
It’s now time to use the difference between the speeds of light and sound to our advantage. In particular, let’s use the fact that we see fireworks and lightning before we hear them to estimate the distance to the fireworks show or thunderstorm. Since light travels really fast, we can assume that we see fireworks and lightning as soon as they happen (while that’s not precisely true, it’s a really good approximation). With this assumption, all we have to do to estimate the distance to fireworks and thunderstorms is time the number of seconds between the flash and the bang. You can use a stopwatch if you’d like, but counting “one thousand one, one thousand two, …” should be accurate enough to get a decent estimate.
Since we know that sound travels about 770 miles per hour, we also know that it travels about 770 / 60 = 12.8 miles per minute, and thus about 12.8 / 60 = 0.2 miles per second. Which means that for every 5 second delay between a flash and a bang, the distance to the fireworks or thunderstorm must be about 5 x 0.2 = 1 mile. So if you counted 10 seconds, the distance must be about 10 x 0.2 = 2 miles. If you counted 15 seconds, the distance must be about 15 x 0.2 = 3 miles. Just remember that you can multiply the number of seconds delay by 0.2 to get the distance in miles.
Use Math to Keep Safe!
If you live in a city where there are lots of big fireworks displays, you can use this technique to figure out how far away they all are from you. More importantly, if you live somewhere with lots of thunderstorms, you can use this technique to keep safe. The rule of thumb is that you should seek shelter whenever a thunderstorm is within 3 miles of you. Which, using our distance estimating technique, means that you should seek shelter whenever you hear a thunderclap within 3 x 5 = 15 seconds of seeing a lightning bolt flash. Thunderstorms move fast and can be unpredictable, so this technique won’t always be perfectly accurate—but it should at least give you a fair warning. So this 4th of July and summer thunderstorm season, remember to use math to keep safe!
Wrap Up
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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
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Fireworks image, bayasaa at Flickr, CC BY 2.0