Lab Activity: Determine Speed and Error Analysis of Data

Purpose: Given the essential equipment, you and your partners will have to determine methods of measuring speed as precisely and with as little uncertainty as possible.  The focus is on data analysis, particularly on methods to determine uncertainties and propagate them to results.

Materials:      Marble             Ramp               Meter Stick      Stopwatch        Graph paper

Procedure:

            You are given the task of trying to measure the speed (distance / time) of a marble that rolls off a ramp.  You want to try and maximize accuracy and precision while limiting uncertainties.  As far as procedure, write detailed steps you decide on for a method that involves extracting a speed from a graph.

            Do not expect much help from me as far as measurements.  I suggest that you take a few minutes before starting to write down possible uncertainties (experimental, which you may be able to minimize) and then give this a try.

Data Table: Develop a neat, organized, and appropriate data table for all measurements for your procedure.  Your group will need to take some number of trials for time measurements for each distance that you use.  In your data table(s), include the standard deviation for each set of time measurements.

                             s_time = æ [S(t_avg – t_i )²] ö ^½                                           

                                          è       N – 1         ø

Questions/Analysis:

1.      What potential experimental errors did you and your partner(s) decide were the most significant?  Explain why these caused the most worry.  Do not just say something like “human error” or “calculation error;” but be detailed and thorough.

2.      Is it possible to “minimize” errors in this experiment?  Why or why not?  (Think about your equipment)  What did you try in order to minimize uncertainty on time measurements?

3.      Why is it important to be able to accurately determine speeds?  Think in terms of everyday events or situations; try to come up with two examples.

4.      Why is it important to be able to precisely determine speeds (i.e. measure down to more decimal places)?  Think in terms of everyday events or situations; try to come up with two examples.

5.      Make a graph of distance (y) versus time (x) using Excel.  Find the best-fit line and R² value for your data.  Then, put on error bars in the time dimension (this will likely have to be done by hand after printing out your graph.  Don’t forget to label axes, include units on the axes, and so on, when making a graph).  The error bars for time will make use of the standard deviation for that particular set of time trials.  It is OK for different points on a graph to have different sized error bars; in fact, you should expect to have different sized error bars for each data point.

6.      If you started your marble at the same height on the ramp each time you can assume it is going the same speed when it rolls off on a flat surface.  When you made your graph of distance versus time, this is why you would expect a straight line.  Are all your points on the line?  Don’t feel bad if they are not; rarely will data lie on the same line.  Determine and report the “best” speed of your marble as well as estimates of uncertainties on your speed result from your graph, using max and min slope lines as determined by your error bars.  In class, you will see an example of this.

7.   Calculate the uncertainty for your speeds by using the quadrature method for

      independent measurements (it is safe to assume that distance and time measurements

      are independent of each other and random since you’re using different measuring  

      devices for each).  Do this for each individual point of distance and time.  How do

      these uncertainties compare to those from your graph, using the max and min lines?

                       

                        dv = v [(dx/x)² + (dt/t)² ] ^½   , where dt is the standard deviation for that set

           of times, and dx = 1 cm = 0.01 m.

Note that dx/x and dt/t are called the fractional uncertainties of the distance and time measurements.  

We are adding the fractional uncertainties in quadrature (looks a lot like Pythagorean theorem!).

Grading:

Title (always have this)              1 point

Purpose (always have this)        1 point

Materials and Methods (Procedure)      3 points

- procedures should be written such that someone who has not done this before can recreate your lab

Data and Graph            5 points

Analysis Questions        10 points

Above and Beyond: If interested in learning the theory behind least-squares fits to linear data, look here.