Vernier Logger Pro Modeling, Fitting, and Linearization - Manual (Page 3)

Logger Pro Modeling, Fitting, and Linearization
©Vernier Software & Technology 3
An Example of Modeling and Fitting
The data shown in the graphs below come from the video analysis of a falling stack of ten coffee filters.
Since the nested filters have not reached terminal velocity, it is reasonable to ask if the motion can be
described by the kinematic equation that works for small, dense falling objects.
Analytic Mathematical Modeling: The investigators are told to see if the physical model of a point mass
in a uniform gravitational field near the earth describes the motion using the modeling method. They use
Equation 3, choosing A = –4.9 m/s
2 because it is half the acceleration of gravity, B = 0.00 m/s because the
initial velocity is zero, and C = 2.06 m because the filters are released that high above the
x-axis of the video. After making an overlay graph of the model on the data (Figure 2a), it is clear that
this model does not fit the data well. Small changes in the coefficients do not improve the agreement
significantly.
Figure 2: Three graphs of the same falling coffee filter data with different models or fits. [a] analytic mathematical
modeling using Eq. 3. [b] a curve fit to Eq. 3. [c] a curve fit to a cubic polynomial.
Curve Fitting: The investigators are told to use curve fitting to see if the quadratic function for objects
falling near the earth fits the data. They select the quadratic function under Curve Fit in Logger Pro. The
result is in Figure 2b. It is a fairly good fit, with RSME = 0.03 m. Since the investigators followed the
instructions and got a good fit, it may be hard to convince them that it is not a good model. The value of
coefficient A gives an acceleration of 5.4 m/s
2
, which is not the acceleration of gravity. The non-zero
value of B implies that the coffee filters had a significant initial velocity in this model, even though they
were actually dropped from rest. Moreover, this model does not predict that the filters will ever reach
terminal velocity. The pitfall for investigators is that if they place too much value in getting a good fit,
they are likely to overlook the reasonableness of the fitted coefficients or of the function itself.
Another pitfall occurs when the investigators are given the more general instruction to “find a simple
function that fits the data well.” Without the guidance of a physical model, they will probably try the next
higher order polynomial, a cubic function. The result is in Figure 2c. It is a much better fit than in
Figure 2a, having RSME = 0.004 m. The fit line goes beautifully through all the dots representing data
points. However, it is a terrible model – it has a local minimum because it is a cubic function, and it
predicts that after another second the filters will begin falling upward! With an over-reliance on the
mathematical goodness-of-fit, investigators may easily overlook the importance of understanding how the
analytic function and its coefficients must relate to a physical model.
The previous example shows some pitfalls with using curve fitting. We feel that Analytic Mathematical
Modeling is the most versatile and educational method for analyzing data that can be described by an
[a] [c] [b]