PHYS111Q : Labs

1 Average Speed

Introduction

In addition to reading this assignment, also read Appendices A and B.

We will be using a motion detector to track the positions of objects with time in several lab exercises this semester. We will use a program called Logger Pro to control a LabPro interface connected to the USB port of your laptop to collect data from the motion detector and from other sensors. Your technical task in this first lab meeting is to install Logger Pro on your laptop, and collect some data. Your physics tasks are to

• measure your average walking speed and the average walking speed of your lab section,
• calculat the statistical uncertainties in these results, and
• consider the following question: Is your walking speed above or below average?

We will start with a theoretical model based on the assumption that your natural walking speed is constant. The position of an object moving in one dimension with constant speed $v$ is described as a function of time $t$ by

$$x_2 = x_1 + v \Delta t$$ (1)

where $\Delta t = t_2 - t_1$, and $x_1$ is the position of the object at $t_1$. This is a linear model. That is, it describes a position vs. time graph that is a straight line with slope $v$ and intercept $x_1$.

Experiments

Logger Pro

Place the installation CD in the CDROM drive of your Laptop. Run the .exe file on the CD. Use the password written on the CD. Choose the standard installation options by clicking Next several times and then Finish. There should now be a Vernier Software tab on the Start -> Programs menu containing one item called Logger Pro.

Average Walking Speed

The motion detector can measure positions within a range of approximately 0.5-6.0 m from its front face. The ultrasonic pulses it emits spread out into a cone with a pitch of approximately $20^\circ$. The detector tracks the ``loudest'' echo it receives, so it is important that the object of interest be the best reflector of sound within this cone. Chairs, lab tables, walls, and lab partners within the cone can lead to ambiguous results.

Although lab group members should help each other, each group member should complete the following steps with his/her own laptop to iron out any hardware or software problems. In future labs, one member of each lab group will collect data for the group.

1. Connect the LabPro interface to the USB port of your laptop with the cable provided.

2. Run Logger Pro. The software should automatically detect the LabPro interface and the motion detector. If it does, you will see a Collect button in the toolbar, a blank data table, distance vs. time, and velocity vs. time graphs.

3. Place your motion detector on the counter along the wall so that it faces across the room. If you clear the area of people and chairs, there should a clear ``walking lane'' in front of the detector over 5 meters in length.

4. Press the Collect button to start data collection, and walk either directly toward or away from the detector. Try to walk at a steady, natural pace. You should see a graph of your distance from the detector vs. time in the Graph Window and a table of time and distance data in the Table Window, generated in ``real time.'' Somewhere in the neighborhood of 5-6 m, the detector will lose you in reflections from objects on the far side of the room. This is a good place to turn around and walk back toward the detector.

5. By default, Logger Pro collects data for 5 seconds at a sampling rate of 30 points per second. You can adjust these parameters using the Data Collection ... button . The sampling rate is fine, but you may want to adjust the experiment length.

6. Use the motion detector and Logger Pro to measure your position vs. time as you walk toward or away from the detector at your normal walking speed. Gather at least 5 measurements covering as great a total displacement as possible. Click Experiment -> Store Latest Run to save the measurements you want to keep. A typical run may contain two or more intervals of walking separated by ``junk'' collected when the detector lost you or while you were turning around. This is not a problem, as the software allows you to select segments of a run for analysis.

Analysis

The slope of a graph of distance vs. time is velocity. You will use the linear regression function of Logger Pro to extract slopes and uncertainties from your measurements.

1. Drag with the cursor in the graph window to select a region on the graph that you would like to analyze.
   • This should be a time interval during which you were walking at a steady pace. (How should a steady pace look on the graph?)
   • If you have several stored runs, and would like to see only one displayed in the graph window, use Data -> Hide Data Set to hide the distractions. Use Data -> Show Data Set to bring back hidden runs.

2. Click the Curve Fit ... button . If you have several runs stored, select the one you'd like to fit. Choose the equation of a straight line (mx + b). Then click Try Fit and Ok.

3. The best fit line should be shown on the graph along with a box describing the fit. This includes the slope (m) and intercept (b) values with uncertainties (standard deviations), along with the root mean square error (RMSE) and the correlation coefficient (Correlation). If you are unfamiliar with these quantities, see Appendix B.

4. Fit all of your walking measurements, and record each walking speed and its uncertainty.

5. Use File -> Save to save your entire Logger Pro session as an experiment file. All of your stored runs, the configuration of the motion detector, and any fits displayed on your graphs will be saved. (You will need a representative graph and fit to hand in.)

6. Enter your walking speed measurements into a column in a spreadsheet, and use the average() function to calculate a ``best value'' of your walking speed. Calculate the standard deviation using the spreadsheet function stdev(). The uncertainty in your best value is the standard deviation of the mean (the standard deviation divided by $\sqrt{N}$). Write your result along with its uncertainty on the white board at the front of the lab.

7. Before you leave the lab, enter all of the individual average walking speeds from your lab section (on the white board) into another column of your spreadsheet. Calculate the average walking speed of your lab section and the associated standard deviation and standard deviation of the mean.

Before You Leave Lab

This is a chance to touch base with your instructor and identify details you may have missed before you leave lab.

• Go over the details of how you got the ``best values'' for your average walking speeds.

• Discuss preliminary answers to question 4 and 5 below.

• Are the various average walking speeds you found within your group significantly different from one another?

Group Assignment

Hand in a representative distance vs. time graph with fit, your spreadsheet, and answers to the following questions. Always give reasoning for your answers.

1. Explain why 30 points per second is an acceptable sampling rate by giving examples of sampling rates that are too fast and too slow, and explaining why in each case.

2. Explain the meaning$^*$ of the standard deviation of the slope reported by the fitting routine for each average walking speed measurement.

3. Explain the meanings$^*$ of both the standard deviation and the standard deviation of the mean of (a) the set of walking speed measurements of a single student and (b) the set of walking speed measurements collected by the entire lab section.

4. Based on your observations, how consistent is the walking speed of a typical Ursinus student (a) during a single short walk and (b) from walk to walk? Use quantitative results from your analysis in your responses.

5. On the basis of your analysis of the available data, compare the walking speed of each member of your lab group with that of a typical Ursinus student. Can you conclude that your walking speed is above average, or below average? (Assume here that your lab section is a representative sample of Ursinus students.)

$^*$Caution : ``Explain the meaning'' means relate the abstract mathematical quantity to things in the ``real world'' that are easier to understand. For example, if you were asked to explain the meaning of the average of your walking speed measurements, a technical response, such as the definition of the average

$$<v> = \frac{1}{N} \sum_i v_i$$

would not be helpful. Instead, we're looking for a response more like ``Assuming that the variations in my walking speed are random, the average of my walking speed measurements is approximately the speed with which I am most likely to walk. The more measurements I include in the average, the better the approximation.''

Copyright © 2003-2009, Lewis A. Riley

Updated Tue Oct 20 12:02:25 2009

This work is licensed under a Creative Commons License.