Type A Expmt Part 1 Exercise 2: Gaussian PDF

You are doing an at-home experiment on projectile motion to measure how far a ball which has been rolled down a ramp will travel horizontally. Your setup looks something like the schematic shown in the diagram. You start the ball from a marked position on the ramp and release it from rest. You then use a ruler to measure how far it lands from the edge of the ramp. You have access to paper that you can put on the floor that will mark when the ball hits it, which gives you a helpful visual aid for your measurements.

After your first couple of practice runs, you notice that there is significant dispersion on the measured distance regardless of how carefully you line up the ball with the mark on the ramp. You decide that you need to repeat the measurement many times to assess the feasibility of the experiment. These are your results:

Based on these results you think that this data may be modelled reasonably well by a Gaussian function. In the next few steps you will explore this.

a) The data is provided to you in an Excel file. In the “Data” Sheet you’ll find the measurements in column A. From the data create a frequency table (similar to what you did for the dice data earlier in the term). You will “bin” the data using the bins setup in the Excel file. See below for a bit of a guide for getting started.

To count the number of readings in each bin you may wish to use one of the following strategies.

1. 1. In Excel use the Data sort function to arrange the measurements from smallest to largest. Then count the number of readings that fall into each category.
   2. Another way to count the data is to use the “COUNTIFS” function to count occurrences that fall within a specified range. For the first bin, the function would look like this: =COUNTIFS(A3:A52,”>=420.0″,A3:A52,”<=424.9″)

Once you’ve completed the table, check that your number of readings sums to 50. What should the frequencies sum to?

Make a histogram (2D column plot/bar graph) of your binned data.

For this example we chose a bin “width” of 5 mm. Can you give a possible reason why this is a more appropriate choice than 1 mm or 20 mm for this data set?

b) Calculate the average and the standard uncertainty of your data.

c) Calculate p₀ and write down the Gaussian PDF expression.

d) On the “Gaussian” Sheet of the Excel file, you can compare your measured data to the Gaussian function. Copy your frequency data into column B. (Highlight the frequency data on sheet 1, right click for the menu and choose copy. Then move your cursor to cell B3 of sheet 2, right click for the menu, and paste special, values.) Fill in your calculated values of p₀, x₀, and σ to create the Gaussian function.