Goal

Given Position vs. Time data for a particle moving in one dimension, investigate the particle's motion by examining derivatives of Position as a function of Time.

Reference

Chapter 2 of your text

Recording

Your report for this lab will consist of a Logger Pro (or Graphical Analysis) file that you will upload to the Moodle server. Sections of the instructions that require written responses are lettered and written in italics. When you are asked to discuss things with your partner, to record results from your analysis of the data, or to answer questions, enter your answers in a text box of your Logger Pro file, and identify your answers with the same letters as used in the instructions.

Download the first data set now:  Right click on pt1.ga3, select Save As.  Navigate to your personal directory on the network. Enter the complete filename, "pt1.ga3", in quotes and Enter.   Also download the file   pt1a.dat .

Opening the data set

The program that you'll be using to analyze the data is Logger Pro.

1. open Logger Pro
   
2. click on "File" and then "Open". Navigate to the directory where you saved pt1a.dat and open the file. If you're asked whether to save the data in memory, select "No". You're now ready to start working with the data.

Taking the first derivative

In order to see the complete data table, use the scroll bar to the right.  Examine the data. The position of a particle is given from 0.300 to 1.500 s in intervals of 0.050 s. The position-vs.-time graph looks like it could be quadratic. If this were the case, the motion would be uniformly-accelerated. The title of this lab, however, tells you to expect otherwise. In order to investigate the motion, you'll take a series of derivatives. The technique for taking the first derivative is described below.

On the main menu bar, click on "Analyze" and then "Tangent". Move (not drag) the cursor over the graph. As you do, you'll see the slope of the tangent line as calculated at each point. The slope is the Velocity, the first derivative of Position with respect to Time. Use the following procedure to create a new column in the data table for Velocity and plot a graph of Velocity vs. Time.

1. Move the cursor as far left on the graph as you can. Then drag all the way across. This process selects the data between and including the two vertical bars. If you need to reselect the data, just single click anywhere on the graph and drag again.  Make sure that you select all of the data.
2. On the main menu bar, click "Data, "New Calculated Column".
3. In the New Calculated Column dialogue box, enter Velocity as the both the column name and the short name and give the appropriate units. In order to create the formula, click on "Functions" and select "derivative()". Then click on "Variables" and select "Position". Note that "derivative("Position")" appears as the New Column Equation. This means take the derivative of Position with respect to the independent variable, Time in this case. Finally, click on "OK" and the new column will appear in the data table.
4. In order to plot Velocity vs. Time, click on the y-axis label (Position). In the Y-Axis Setup box, check "Velocity" and uncheck "Position". Select "Autoscale", if not already selected, and then "OK". Click once on the graph to remove the vertical selection bars.

A. Stop and think! How does the new graph compare to the previous one? Note that there's a kink at the next to the last data point. It's important for you to understand what creates this kink. It's an artifact of the method of finding derivatives. Discuss this with your partner and check with the instructor to make sure you understand how the kink arises, since this will guide you in making decisions as you continue working with the data. You may want to try taking a closer look at the region of the graph with the first few data points. In order to do this, drag the cursor over these points to select them. Then click on the y-axis label (Velocity), select "Manual Scaling" and an appropriate number for the "Top Limit". Click "OK" to view the select region enlarged. When you're finished, restore the original graph view.

Taking the second derivative: You're now ready to find Acceleration, the second derivative of Position as a function of Time. Logger Pro has no second derivative function, but you don't need it. Simply take the first derivative of Velocity as a function of Time. Repeat steps 1-4 above, substituting Velocity wherever you see Position and substituting Acceleration wherever you see Velocity. Then plot the graph of Acceleration vs. Time.

B. Stop and think! Note that the kinks at the ends are much more pronounced than for the Velocity vs. Time graph. Discuss the reasons for this with your partner before going on. Decide which region of the graph has valid data. Finally, give a complete description of the motion of the particle, using everyday words like forwards, backwards, speeding, slowing, constant, increasing, decreasing, etc..

A New Concept: While the Acceleration of the particle is not constant, the derivative of Acceleration with respect to Time is constant. We could say:

The quantity da/dt is defined as the "Jerk". The particle of this problem could therefore be said to be moving with uniform jerk. Use the "Analyze", "Tangent Line" command to find the value of the Jerk. What are its units?

Equation of Motion: Now you're ready to obtain the equation of motion of the particle. Display the original Position vs. Time graph. Drag the cursor over the graph to select all the data. Then click on "Analyze", "Curve Fit". From the "General Equation" window, select "Polynomial". Enter the degree (which you should know based on the work you've done so far). After you click "Try Fit", click "OK"; you'll see the results of the fit.

C. Record the equation of fit, using proper physics symbols, writing the fit coefficients with the correct units, and rounding them to the proper number of significant figures.

D. Use the power law for derivatives to obtain Velocity as a function of Time. Repeat to find Acceleration as a function of Time. Finally, state the initial values of the Velocity and Acceleration.

E. Describe the characteristics of the force required to produce the motion of this particle. Can you think of a way to produce such a force ?

Save Your Work now!

On the Logger Pro main menu, select "File", "Save As". The directory should be the same as the one into which you downloaded the original data file. Select a filename Elab1_xxxxx_yyyyy.ga3, where xxxxx and yyyyy are your and your partner's pid numbers. Click "OK" to save.

Download the second data set now:  Right click on pt2.ga3, select Save As.  Enter the complete filename, "pt2.ga3", in quotes and Enter.

A More Interesting Motion: Open Data Set 2 in Logger Pro. You'll see that's it's a bit more interesting than the last one. The Position of the particle peaks, returns to 0, and then rises rapidly. The method of analyzing this motion will be different than the last. There will also be much fewer instructions, since you should be familiar with the use of the program by now.

G. Begin by doing a polynomial fit of the Position vs. Time data. When you have the best fit that you think you can get, record the equation of fit in the same form as you did previously (proper symbols, units, significant figures).

H. Use the power law for derivatives to find the first derivative of the equation of fit. Using your resulting equation, have Logger Pro calculate a new column for the first derivative and plot a graph of these results vs. time. Label axes appropriately. Describe in words how the velocity changes as a function of time. Finally, give a complete description of the motion of the particle, using everyday words like forwards, backwards, speeding, slowing, constant, increasing, decreasing, etc..

I. Use the power law once more to find the second derivative of the equation of fit.  Using your resulting equation, have Logger Pro calculate a new column for the second derivative and plot a graph of these results vs. time. Again, select appropriate labels. Describe in words how the acceleration changes as a function of time.

J. Create a scenario to explain how an object could move according to the equations that you obtained above. You'll have to invent a force with the necessary characteristics. Be sure to consider the initial velocity and acceleration and how the velocity and acceleration change with time.

Save Your Work now!

Select a filename elab1_xxxxx_yyyyy.ga3, where xxxxx and yyyyy are your & your partner's pid numbers. Save the file in your working directory.

K. When you are ready, send your file to the instructor using the Moodle course server.  Be sure to save a copy for yourself, too.