During a recent fireworks show, a friend had asked me if I knew how high the fireworks fly before it explodes. I didn’t know the answer but  I began to wonder how I could figure it out.

Judging the height of an object in the sky is not easy without a reference. Maybe if I knew the height of a nearby building or tree? With a height reference I could probably estimate the height of a firework with some  accuracy.

Low clouds can be a helpful indicator of height. I can simply gauge the altitude of the firework relative to the cloud. The altitude of a cloud can be found on weather websites like www.wunderground.com.

But wait, what if there are no clouds or buildings nearby?

I gave this a thought and came up with a way to estimate the height using a grade-school trick and math.

Thunderstorms, the Speed of Light and the Speed of Sound

Lightening and thunder occur during a thunderstorm. Lightening is the discharge of electricity between clouds, from a cloud to the ground, or vice versa. Most people don’t see the actual lightening bolt, but they’ll often see a flash. Thunder is the sound a lightening bolt makes and is often heard after a flash is seen.

In grade school,  I learned to tell how far away a lightening strike is by counting the number of seconds between seeing a lightening flash and hearing the sound of thunder.  Each second is about 1000 feet (1114 feet to be exact), so if I counted 5 seconds between lightening flash and hearing thunder, then the strike was about 5000 feet away (or about 1 mile).

This trick works because light travels incredibly fast when compared to sound.

What does this have to do with fireworks?

When watching the fireworks show with my friends, I could see the explosion before could hear it. So, just like I do for a thunderstorm, I began counting the number of seconds between the time I saw a firework explode and when I heard the explosion.

Most of fireworks I saw that night had a delay of 2  seconds or less. This corresponds to about 2000 feet. Keep in mind that this does not mean that the fireworks fly 2000 feet high. Take a look at the diagram below:

(Please forgive the quality of the diagrams in this post as I made them while eating lunch at work.)

As you can see from the diagram, I drew a triangle to help me solve the height problem.

Notice that the 2000 feet I estimated using the timing trick corresponds to the distance from me to the fireworks. In geometry speak, this is called the hypotenuse, the longest side of the triangle.  You can also see that the other two sides of the triangle corresponds the height and my distance to the launcher, which we don’t know. Now, let’s figure out the height using trigonometry.

Wait…trigonometry!?! That’s a scary word!

Actually, trigonometry is not a scary word once you know what it means. “Trigon” refers to a triangle and “ometry” refers to measuring. Are you afraid of measuring triangles? I didn’t think so.

Several thousand years ago, some clever people figured out how to solve problems like ours using triangles. Even to this day, triangles are used to solve problems in engineering, medicine, architecture and aviation.

How to solve our problem?

When dealing with a triangle, we can use some easy rules of trigonometry to figure out our problem. For instance, there are two useful right triangles (triangles with a 90° angle) that can make problem solving really easy. These two triangles are the 30°-60°-90° right triangle and the 45°-45°-90° right triangle shown below:

In these two types of triangles, the sides exist in a fixed ratio. Always!  This means, if you know the length of one side, and an angle,  you can figure out the length of the other sides by using the ratios.

Back to the fireworks problem

To figure out the height of the firework, I need to figure out how much I tilted my head to see the fireworks. This will help me figure out which triangle to use. Since this is an estimate, it’s okay to guess. I know I wasn’t craning my neck up or looking straight ahead. I’d guess that I was looking about 30° to 45° up.

Take a look at the diagram below:

If I was looking about 30° up, I would use the 30°-60°-90° right triangle. Using the ratios, a hypotenuse of 2000 feet would correspond to a height of 1000 feet.

If I was looking about 45° up, I would use the 45°-45°-90° right triangle. Using the ratios, a hypotenuse of 2000 feet would correspond to a height of 1414 feet.

Therefore, using the thunderstorm timing trick and trigonometry, I would estimate that fireworks I saw that night flew up to about 1000 to 1500 feet.

Keep in mind that approach is not perfect, that’s why it’s an estimate. An estimate gets us close to the exact answer. There were issues about timing between seeing and hearing the explosion, the angle the firework was launched,  as well as effects of the wind.

Even if we had the exact data, we would still use trigonometry to solve the problem.

Enjoy!

Dr. Dave