Science and Engineering

Fireworks, Half a Dozen ps, and the Fourth of July

Ever wonder how large fireworks are?
	Fox 4 Kansas City Weather

While watching fireworks on the Fourth of July, here’s how to estimate how big those bursts of fire are in the sky:

1. Count how many seconds of time elapse from the instant you see the light from a firework explosion until you hear the sound. Count out loud, "one thousand one," "one thousand two," ... to estimate the number of seconds. (This is the same "trick" to estimate how far away lightning is.)
   
2. Estimate the angle in degrees of the firework display as shown in Figure 1 below:
   

Figure 1. Estimating the Size of Fireworks in the Sky

Moving one arm up and down to trace out the angle of the firework might help you estimate the angle. (See additional information below for help in estimating angles.)

3. Calculate the approximate size in feet of the firework display using half a dozen ps in the following formula:

where time[seconds] is from Step 1 and angle[degrees] is from Step 2.

If you can’t remember that 6p » 18.85..., and if you want an easier formula so that you can do the math "in your head," use 20 as an approximation:

Example. You’re about a mile from the fireworks display. You count off 5 seconds between seeing the light and hearing the sound of the firework. You estimate a rather small angle traced out by the firework, only about 4 degrees.

Using the "easy" formula, you quickly estimate: size[feet] = 20 x 5 x 4 = 400 feet. To be a little more precise, you use a calculator: size[feet] = 6p x 5 x 4 = 377 feet.

How can you measure angles of something in the sky if you’re not very good at estimating angles?
Method 1. Courtesy of John Harper, School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand:
"And in my youth the NZ Army taught me a simple way to estimate angles.  Hold your fist out at arm's length and look at your knuckles.  You will see a ^-^-^-^ configuration: the gaps between knuckles are approx 3,2,3 degrees.  To estimate angles over 8 degrees, while keeping your arm stretched out put your thumb tip and little fingertip as far apart as you can.  The angle between them is near 19 degrees.   This works because most people with long arms also have big hands. (I think the army said 21 not 19, but that one's easy to check: right angles are easy to find, and 5 X 19 is much closer to 90 than 5 X 21 is.  Maybe one can't stretch one's thumb and little finger so far apart as one gets older.  The 19-year-old version of myself is no longer available.)"

Method 2. Since estimating angles is difficult, make a simple astrolabe to measure the angles. You will need the following:

1. six-inch plastic protractor (with hole)
2. plastic drinking straw
3. piece of string or heavy thread about two feet long
4. large paper clip or binder clip for a weight
5. some tape

To make the astrolabe, follow these simple steps (see Figure 2 below):

1. Tape the drinking straw to the straight-edge part of the protractor.
2. Make a loop with the string that passes through the hole in the protractor.
3. Attach the two ends of the string together and to the paper clip weight.

Figure 2. Simple Astrolabe

To use your astrolabe, hold it from the top with your thumb and index finger. Rotate the astrolabe up or down around the hole. Now locate something by looking through the straw. Read the angle on the protractor marked by the string. To measure the angle of a firework, measure the "top" and "bottom" angle with your astrolabe. Then take the difference of the two angles. A team of two works best: One team member observes through the straw while the other reads the angle.

Use your astrolabe to determine the size of fireworks — or to measure the position of the moon in the sky!

Why does the formula work? How accurate is the formula?

Figure 1 shows two right triangles that have the same dimensions. In a right triangle, the tangent of an angle in radians is defined to be the ratio of the length of the opposite side divided by the length of the adjacent side. See Figure 3 below:

Figure 3. Tangent of an angle in a right triangle

If we use the definition of a tangent with one of the right triangles from Figure 1, the following formula can be written:

We divide by 2 in two places since this formula only applies to one of the two right triangles. The full angle and full size are shared by both triangles.

For angles < 30 degrees, there is less than a 10% error in the following approximation:

For angles < 10 degrees, there is less than a 1% error in this approximation. Assuming we’re at a safe distance from the fireworks, this approximation is appropriate.

Given this approximation for small angles, our formula becomes:

Or, by rearranging and adding units to the equation:

size[feet] » angle[radians] ´ distance[feet]

This formula looks simple, but we don’t have an easy way to get the angle measured in radians, nor the distance from our eye to the fireworks in feet.

Converting an angle in radians to degrees is not difficult with the following formula:

But how do we estimate the distance to the fireworks? Have you ever measured how far away lightning is in miles by counting seconds and dividing by five? Since sound travels about 1089 feet/second, the distance to the fireworks can be estimated using this sound fact from physics:

distance[feet] = 1089 feet/sec ´ time[seconds]

Now we substitute the expressions for angle[radians] and distance[feet] back into our formula for size[feet]:

By rearranging, we have:

Since 1089/180 » 6, the final formula becomes:

In our example above, this formula gave an estimate of 377 feet for angle = 4 degrees and time = 5 seconds. The "exact" equation

gives a value of 380.3 feet. So even our "quick and dirty" equation was fairly accurate.

Related References

This article was the basis for Explosive educational opportunity on the Fourth 
in the Fort Leonard Wood Army Post "Guidon" newspaper, 4 July 2002

For a novel way to estimate the value of p, take a look at efg's Buffon's Needles Lab Report.

Updated 04 Jul 2002

since 1 Nov 1998