Data Analysis

Extracting Physical Meaning from Experimental Data

Import the 15 data sets you collected in the experimental section to a computer. You can perform the analysis in any programming language of your choice. Make sure the code you wrote for the Theoretical Modelling portion of this experiment is available.

Determining the Coefficients of Drag

Start with the first data set collected using only one coffee filter. Using a function fit library (in Python this is available through SciPy curve fit) fit the position data to the position equation for linear drag and the position equation for quadratic drag. Use the initial conditions extracted from the data or that were a part of the experimental design. From the fit equations extract the coefficients of drag (b or k). Perform this analysis for the other two data sets for the one coffee filter. Average the results for b together to get an experimental b. Do the same for k. These are the b and k values we will use for the remainder of the analysis.

Repeat the above analysis for each set of stacked coffee filters. You should have 5 different b values and k values. 

Deliverable #7: Create a graph with the number of coffee filters on the x-axis and b/k on the y-axis. Plot the experimental values of b and k found above. Is the relationship linear, quadratic, exponential, or something else?

Determining the Type of Drag

We have found the coefficients of drag for both types of drag, but only one type of drag applies here. Let's determine if linear drag or quadratic drag is a better fit for our data. Let's start with the one coffee filter data set, just the first data run. For the times you have experimental data create two analytical data sets: one for linear drag, one for quadratic drag. Use the initial conditions from the data/experimental design and the coefficients of drag calculated above. On a single graph place time on the x-axis and position on the y-axis. Graph the experimental data, the analytical position for linear drag, and the analytical position for quadratic drag. Calculate the mean-squared error between the analytical position for linear drag and the experimental data. Do the same for the quadratic drag. Determine which analytical solution is a better fit.

Repeat this process for all of the stacks (you should have five graphs and 10 error scores by the end).

Deliverable #8: Submit the plots and your error scores. Justify rather linear drag or quadratic drag is at play here.

Determining the Terminal Velocity

Start with the first data set from the one coffee filter data set. Use a numerical differentiation algorithm to determine the derivative of your experimental position data (you can use a pre-written one from a library or create your own). This should give you the speed of the coffee filter as it fell. If you dropped your coffee filter from high enough you should notice that the speed levels off near the end of the data. This is the terminal velocity of the coffee filter. Average over the region that is relatively flat to determine the experimental terminal velocity. Repeat this for the other two coffee filter data sets and over all three results together to get the final experimental terminal velocity.

Repeat this for the stacks of coffee filters as well, resulting in five terminal velocities.

Deliverable #9: Create a visual that compares the experimentally determined terminal velocities to the terminal velocities derived from the analytical equation (Deliverable #2) for the five different stacks of coffee filters.

Comparing Analytical, Numerical, and Experimental Results

Deliverable #10: Create a graph that displays the analytical solution, numerical solution, and experimental results for the position of the coffee filter as it falls with respect to time. On the graph, add the mean-squared error scores for the analytical and numerical results plus the analytical and experimental results. Make sure your graph has axes labels and a legend. Create one graph per stack of coffee filters.