Monday, February 17, 2014

Falling objects and the (largely) unsuccessful search for simple altitude measurements

Between the birth of model rocketry in 1958 and the appearance of small barometric altimeters in the 1990's, there was no small degree of thought and debate given to measuring the altitude of model rockets, especially for competition purposes. Since the beginning (and for the competition purists, even now), optical tracking using two or more theodolites was deemed best, as it could measure altitudes to better than 5%. However, it had/has a few issues:
  1. Equipment fairly difficult to construct and bulky (tripods, etc)
  2. Measured altitude is that at ejection, rather than apogee. This is because it is easy for a ground observer to spot the ejection at a distance, especially if colored tracking powder (chalk) is used to create a small "cloud" when the rocket separates.
  3. The personnel operating the trackers must be adequately trained, else there will be numerous "lost tracks", due to losing sight of the rocket or erroneous altitude/azimuth readings.
The above are not minor issues; in fact, it was proposed in the late 1970's that altitude events be dropped from international FAI rocket competitions because of them. Altitude events were "just too hard to do." After some debate, the proposal was voted down, but many competitions here in the U.S. and other countries featured only duration events (which required just a couple of stop watches) because of the difficulties associated with altitude tracking. There had to be a better way (and with modern altimeters there is, despite the misgivings of a few aging competition rocketeers). In the early 1970's, a group of rocketeers began to think that timing a small falling object ejected from the rocket might be the much sought after Holy Grail of altitude determination.
The Triple-Track Optical Tracker by Trip Barber, Model Rocketeer, February1979,  pp. 8-10
The first order theory behind a falling object in air is pretty simple, and can even be solved analytically without resorting to computers. Imagine for a minute, that you are like my friend Eddie, who likes to jump out of perfectly good airplanes. As you fall out of the sky, there are only two forces acting on your body - gravity, which pulls you downward to the very hard ground below, and drag induced by your motion through the air, which tends to push you up. We all know what the force of gravity is - it's your weight. Drag is a little more tricky, as it depends on your speed (if you ain't moving, there is no drag), your area (the more of you there is, the more drag), and your shape. We can express this drag force in mathematical terms as

where Cd is the dimensionless drag coefficient (related to shape), A is the area of the falling body in the direction of motion, and V is the speed of fall.

So this is what happens when you fall out of the plane. Initially, your vertical speed is zero, which means the drag is zero, and gravity accelerates you downward. As it does so, you pick up speed, which causes the drag to increase, which in turn causes your speed to increase more and more slowly. At some point, the drag force equals your weight, at which point the net force on you is zero. We know from good old Newton's laws of motion that an object with no net force will move at a constant speed, so your falling speed stops increasing. This speed, which we call terminal velocity, is the key to using falling objects for rocket altitude measurements. If I can find an object that reaches its terminal velocity quickly, say within a few feet, I can determine the distance it fell (which is roughly the same as the rocket altitude when the object was ejected) by simply measuring the time of fall and multiplying by the object's terminal velocity. Much simpler than the fancy trig involved in optical tracking, and I can use the same stop watch used for duration events.

But what object do I use?

That is the topic of a future post - stay tuned :)

Side note (because I know you are going to ask): Terminal velocity for humans in a random posture is between 117-125 mph. Some experienced skydivers can assume a bullet-shaped position, which can increase their terminal velocity to over 200 mph. Why they would want to do this is beyond me.

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