Observational Experiments:

Everybody has been on top of a tall object at some point and wondered how high up they were. (bridges, mountains, cliffs, buildings, etc.). There are many possible ways to find an answer to this question. Topological maps work well if we are out of doors, blueprints work well for buildings, or even very long tape measures. Physics provides us with tools that we can use to make a rough determination of height without a need for special materials.

Many of you have had a taste in lab of the difficulties that are inherent in doing any kind of scientific experiment. Even the simplest seeming problem can become very difficult if it is rife with variation. An important feature of this lab is the idea that by planning ahead and doing some exploration in controlled situations we can reduce the uncertainties in our data set. This also improves the believability of our results by suggesting to our readers that we understand (even if we can't control!) the variables in the problem at hand.

There is a common approach that scientists in all disciplines tend to use in order to get a handle on the potential for error and measurement difficulties in a new experiment. In this process the first step is to carefully plan out how the measurement will take place. This includes identifying the tools that will be used to collect the data. Along the way you will also be deciding what data you will need to complete the problem. In many experiments it is possible to immediately determine all of the data you will need to collect. In very complex experiments investigators often have to do other secondary experiments just to figure out how to do the original experiment. This can be a very frustrating experience akin to crawling through a dark room feeling for something important, but you're not sure what. On the other hand, if you love a good puzzle and the sense of success that comes with figuring it out, this type of experimental work can be very satisfying.

After you, as the investigator, have figured out how you are going to proceed it is now time to try your idea out against a similar situation where you know what the answer should be. This is critical to your eventual success! Particularly in a type of experiment where you may be using equipment in a new way, or even making an entirely new kind of measurement, you need to assure yourself and your readers that you understand what you are doing. Along the way you will also generate a strong sense of the variability and reliability of the data you are proposing to gather. More often than not you will discover that there are one or more problems with the process you are proposing. Here is the stage where you can fine tune your measurement techniques. You can explore alternative techniques to see of you can get closer to the known result. In extreme cases you may even decide to do the measurement by an entirely different method. This is a lot like the process of turning in a draft paper to your writing instructor. The main difference is that you have to be very alert to what your trial experiment is trying to tell you since it is unlikely to communicate as clearly as the writing instructor.

In this particular case you have been given a less than helpful tool for generating your data. This is being done so you can think about how you go about calibrating an unfamiliar piece of equipment. The most reasonable approach is to determine the behavior of the equipment in a variety of known situations. From this data you can decide how you wish to model the behavior of the device. You may choose a simple linear model or a much more complex one. Your decision will be based on how accurate you need to be as well as which model most nearly reflects the data. As in most things, it is wise to use the simplest model that is consistent with the data, Be aware that you can't actually calibrate your device in the range of motion you will use it in. This is an uncomfortable experimental situation. Under these circumstances experienced investigators look very carefully at the modeling data near the portion they must extrapolate past.

Physics Modeling:

Modeling is the process of constructing a description of how some system behaves. The most traditional form of a model is a mathematical function which describes a set of data points (like the motion plots). The function is particularly useful if it allows us to extrapolate where we expect future data points to be located. Be aware that two data points define a straight line at best and three data points isn't much better. Realistically it takes five data points to have any clear sense of what is going on and even then one should be careful not to overinterpret the curve. Most experimentalists would tell you that they usually start be seeing if the data makes sense as a straight line. This is a good starting point for a number of reasons. The fundamental reason to start with a line is that it is easy to visualize and understand intuitively. If a straight line doesn't work then there are a variety of more complicated options that usually begin with second order polynomials. However the data and curve fit turns out it is important to look at the plotted data and the curve fit to see if they are saying the same things. Your eye and your brain are magnificent tools for integrating data and putting together patterns.

Most mathematical models are built to have fitting constants built into them. The constants in a quadratic equation are one example of fitting parameters. If you believe an object is traveling at a constant velocity then the slope of the position plot is a fitting parameter that can be varied to provide the best fit to your data. While these are sometimes referred to as 'fudge factors' it is best to not be too judgemental. You have also seen these factors in MTH 252 where they are thought of as integration constants that are determined from boundary conditions and initial conditions. Just another version of fitting parameters.

At last it is time to go out and do the actual experiment. It is important to bear in mind that the real experiment may not be as similar to the trial experiment as you anticipated. Pay attention to the experimental process and be sensitive to perceived differences between your expectations and the actual experience. Does some of the data feel bad? Can you identify the reason you get this sense? If you can determine a concrete reason for your unease you can justifiably throw that data out (remember the spring release on the track carts?).

After the experiment there is still the question of how sure you are of your final calculated result. There are many ways to get at this and they tend to vary from instructor to instructor. In my case I would like to think that by the time the experiment is complete you will have a good sense of how much variation there is in the data you collected. Imagine a reasonable worst case and redo your calculation using this modified data. The difference between your “best” result and this worst case result is a good indicator of the potential error in your calculation. This potential error is most often expressed in terms of a percentage of the presumed `best' value.