for all the things you do...
Sitting before you is a model rocket of your very own design. Everything from the rocket fuselage to the parachute was designed by you and you couldn't be prouder. The custom paint job turned out better than expected and the shining exterior brings a tear to the eye. To quote Keats, "a thing of beauty is a joy forever." The finished rocket is a wonderful testament to your prowess as a model rocket enthusiast.
But will it fly?
That's an important question, since a beautiful model rocket that doesn't fly can be a huge letdown! You want your rocket to soar to the heavens! You want to touch the clouds and challenge the gods! Well, maybe that's a little melodramatic, but you want your rocket to perform just as well as it looks.
To measure how well your rocket flies, you'll need to know how to measure the apogee of its flight. In rocketry, the apogee is defined as the highest point of the rocket's flight.
So, how do you go about measuring this point? Model rockets can travel for hundreds or even thousands of feet up in the air, so how on earth can we measure their maximum altitude? Fortunately, there are several methods for doing this and I'll explain a few of my favorites here.
Okay, so using an electronic altimeter isn't really "cheating," but it takes some of the fun out of calculating altitude with hand tools and mathematics. Nevertheless, electronic altimeters can save rocket enthusiasts a lot of time and hassle. At the time of writing (2011), a good electronic unit will cost anywhere from $50 to $100 and will weigh less than an ounce. The weight and size shouldn't be an issue for a large rocket, but a small rocket like the Estes Quark probably wouldn't be a good candidate for an electronic altimeter. Also, if you're prone to losing rockets to trees, lakes, or shotgun-wielding neighbors (kidding), you might want to find a less expensive method of calculating altitude.
The streamer method for determining altitude is cheap, simple, and fairly accurate. The concept behind the streamer method is that a streamer of plastic with fixed dimensions attached to a small weight will fall at a reasonably constant velocity. The large surface area of the streamer creates a lot of wind resistance or drag and the weight will quickly reach its terminal velocity. Here is a diagram of the standard altitude marker, or SAM, developed by Bill Stine in 1974:
The weight has a mass of three grams and the dimensions of the polyethylene streamer are 12" x 1" x 0.0001". A streamer of this size and mass will fall with a terminal velocity of approximately 18 feet per second. Of course, it's possible to experiment with different weights and streamers. All you need is your custom streamer and a very tall place to test it! Once you know the terminal velocity of the streamer, you can calculate altitude simply by multiplying the velocity by the time it took the ejected streamer to fall to earth. For example, a streamer that falls at 15 ft/s for 60 seconds gives us:
The streamer itself would be placed in the model rocket above the parachute or recovery system. For tumble recovery rockets (like the aforementioned Quark), you might not have room for a streamer, which leads us to our next method.
So far, small rockets have proven troublesome in our desire to measure altitude. They're too small for an electronic system and they're too small to carry a streamer payload. To track something like the Quark, we'll need to use optical tracking. There are several methods of optical tracking and I'll discuss the simplest one here. Be forewarned, however. There's going to be some math up ahead. Remember when you were a kid and you wanted to know when you were ever going to use trigonometry? Well buddy, here it is.
The quickest and easiest way to determine your rocket's altitude with optical tracking is to use elevation-angle-only tracking. This is a very simple method and the amount of trigonometry isn't nearly as bad as I made it out to be. Let's begin with a diagram of the launch site. If we assume that the rocket travels reasonably straight up from the launchpad, then we can draw ourselves a nice right triangle.
If we know the distance from our observation point to the launch pad (d) and the angle at which we can see the top of the rocket's trajectory (θ), we can calculate the altitude with the following equation:
If you're using a calculator, make sure you're in degree mode and not radian mode. Of course, this is assuming you measured the angle in degrees, although I don't think I've ever met anyone who measured rocketry angles in radians. Measuring the angle can be done with an inclinometer. The Quest Skyscope and the Estes Altitrak are two types of inclinometers specifically designed for model rocketry. If you're in a pinch, you can make your own inclinometer with a protractor, a piece of string, a washer, and a straw.
Simply track the rocket to its highest point, tilt the homemade inclinometer so that you're looking at the apogee point through the straw, and pinch the string to the protractor once the string stops moving. Subtract the number on the protractor from 90 and you have the angle you need for your calculation! Here's an example of this type of calculation in action:
I find that it's easier to find the ejection point of the rocket if I put a little colored chalk in with the parachute. Not only does the chalk prevent the parachute from sticking, but it also makes a nice visible cloud when the ejection charge goes off.
There are other methods of optical tracking that are a little more involved, but I've skipped them in this tutorial since this is more of an introduction to the methods of altitude tracking. For a quick and relatively accurate estimation of altitude, the elevation-angle-only method works well. Other methods, such as the alt-azimuth tracking method, often require multiple tracking stations and spotters.
Source: Stine, G. Harry, Handbook of Model Rocketry, Sixth Edition, John Wiley & Sons, Inc. New York, 1994.
C.H. Seman is a chemical engineer and freelance writer living in Longview, Texas. You can learn more about him and read more math and science articles in his blog, chseman.blogspot.com.