Position and MotionInexpensive scientific probes (sensors) are now available from a number of sources. For a few hundred dollars, a classroom or lab can be equipped to electronically capture a variety of scientific data: position/motion, voltage and current, light intensity and wavelength, temperature, sound waves, even simple chemical composition of solutions. Typically the data is recorded tens, hundreds, or even thousands of times per second, and with much more accuracy than can be achieved by hand. The data can then be analyzed and mathematically modeled on a computer, using software such as Mathscribe. We believe that this approach has tremendous educational value, and indeed seems to be sweeping our nation's colleges and high schools.
This document presents just a few fascinating examples of sensor analysis, using data captured by some of AccuLab's SensorNet probes (209-522-8874). The experiments here are grouped into the following areas:
The data and mathematical models for these examples are stored in Mathscribe ".mm" files, and are available on-line as part of the standard Mathscribe download. To open one of these files, start Mathscribe, choose Open, open the "reference" folder, open the "more_mm_files" folder inside it, and then open the example file(s) you are interested in. You can then graphically explore and tune the file's mathematical model yourself, or use Cut and Paste commands to replace its data with your own data, or just use it as a starting point for analyzing an experiment of your own.
Position and Motionfree fall.mm - A "picket fence" object with markings every 4.9 centimeters is dropped, and the time of each marking's passage is recorded. We thus get the distance dropped d in centimeters as a function of the elapsed time t in seconds. Following Sir Isaac Newton, we then model (approximate) this data using the equation d=a+b*t+c*t^2, for suitable parameters a, b and c. By choosing good values for these parameters, we can get roughly 1-millimeter accuracy for this experiment's data - not bad for an object in free fall! What is the physical meaning of each parameter? The best value for c seems to be about 490 here; what value does this yield for the acceleration of gravity?
terminal velocity1.mm, terminal velocity10.mm, terminal
velocity25.mm - 
In
practical examples of falling bodies, we often must consider the
effect of air resistance. To study this, we dropped packets of
1, 10 and 25 coffee filters from a height h
= 3 meters at time t = 0.6 seconds (approximately), and
recorded the height every 0.05 seconds. A naive argument suggests
that the filters should obey the law 
, for
suitable a, b, c and r. Using Mathscribe
we see that this formula can be made accurate to approximately
4, 10 and 15 millimeters for 1, 10 and 25 coffee filters respectively,
if we choose the parameters carefully. (If you think this isn't
very accurate, *you* should try measuring falling coffee filters
sometime!) In these three cases, we find the (downward) terminal
velocity b is roughly 1.122, 3.31 and 7.62 m/sec respectively.
slinky.mm - A body with moving parts must exert varying
forces to keep the parts from scattering apart. A slinky (coiled
spring toy) is a simple example of such an object. In this experiment,
we suspended an oscillating slinky and recorded the height h of its bottom end, in meters, every
0.1 seconds. We discovered this height can indeed be fairly accurately
predicted by a decaying exponential times a simple sine wave:
, for suitable values of the constants. In
fact, we were able to get a root-mean-square error of about 17
millimeters, which as the accompanying graph shows is quite accurate.
light-distance.mm - 
Light
has many remarkable properties. It is also of fundamental importance,
for instance as a means for humans to sense information. Light
photons are also the quantum mechanical mechanism behind interactions
between electrons and protons, and hence ultimately determine
virtually all of chemistry and biology. In this experiment, we
start by simply determining how the intensity i
of light varies depending on the sensor's distance s from
a light source. Apparently the data can be fairly accurately modeled
using the formula 
, if we choose
c and a correctly. In fact, this formula allows
us to limit the error to about 27 lux. Note that this gives us
a much better approximation that a pure inverse-square law (the
case where a = 0), presumably because our light source
is bigger than just a point source. If we changed the experiment
by shining the light through a small hole, what formula would
you expect?
heat-cool.mm - Temperature plays a key role in many
important processes. This first experiment measures the flow of
heat into our probes themselves, for instance to see how long
it takes for our sensors (thermometers) to reach an accurate reading.
At time t = 2 seconds, we switched probes T2
and T3 between a beaker of
hot water and one of cold water, and then recorded the temperature
in degrees Farenheit every 0.5 seconds. One might expect the probes
to reach equilibrium according to a simple exponential decay formula,
but modeling the data in Mathscribe shows this is not the case.
One can get a better fit by using a formula for an "S-shaped"
curve, such as the "logistic growth" formula from population
biology: 
, with suitable constants r,
H and K. Together with a similar ad hoc formula
for T3, we can model our data to within about 1 degree.
How does our data compare to the rated speed and accuracy of the
probes, published by their manufacturer?
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