### Introduction to the tools

Open a Microsoft Word document to keep a log of your procedures and your results.  This log will form the basis of your lab report.  Address the points highlighted in blue.  Insert the requested tables and graphs and answer all questions.

Your main tool for analyzing data will be the Microsoft Excel spreadsheet program.  Let us go ahead and start using it.

Assume you have performed an experiment, measuring the position of an object moving along a straight line path as a function of time.  Your data are shown in the table on the right.  You suspect that the object moved with constant speed, covering equal distance in equal time intervals.  You want to verify this by producing a plot of position versus time and confirming that it is well fitted by a straight line.  If yes, then the slope of the straight line equals the speed of the object in units of m/s.

 Time (s) Position (m) 0 0 1 0.8 2 1.5 3 1.6 4 2.5 5 2.7 6 3.2 7 3.9 8 4.5 9 5 10 5.5

Basic instructions for producing the plot are given below.  Experiment with the various options Excel presents to you.

(a)  Open Excel and enter your data.

 Highlight the table in the browser, right click Copy, and paste the table into an Excel spreadsheet by putting your cursor into cell A1 and clicking the Paste button.

(b)  Produce a graph of position versus time.

 Highlight your data in Excel. On the Excel menu bar click Insert, Chart, XY (Scatter), and pick one of the subtypes.  Excel will plot the second column versus the first column. Highlight your graph, click the + sign next to it, and give the chart a title and label the axes. The label for the x-axis should be "Time (s)", and the label for the y-axis should be "Position (m)".

(c)  Study your graph.  The plot of position versus time should resemble a straight line.  The slope of the best fitting straight line should yield the average speed of the object.  You can find this slope by adding a trend line to your graph.

 Right-click your data and choose "Add Trendline".   Choose "Linear" , "Display equation on chart".  (You can also set the intercept at zero for this data set.)  An equation y = ax + b will appear on your graph, where the number a is the slope and the number b is the y-intercept.

(d)  The fit is not perfect.  The data you have collected contain experimental uncertainties.  To find the resulting uncertainty in the slope you must use the regression function.

 Position the cursor in an empty cell.  On the menu bar click Data, Data Analysis, Regression.  For the input y-range choose the entries in column B.  For the input x-range choose the entries in column A.  (Again, put your cursor in the appropriate textbox and highlight the chosen cells.)  Under output options check new worksheet, and under residuals choose residual plots and line fit plots.  Click OK. [If you do not find data analysis on the menu, then you have to add the Analysis ToolPak Add-In.  Click File, then click Excel Options and then click the Add-Ins.  Click the Go button while Excel Add-Ins is selected and check Analysis ToolPak.] The regression function finds the best fitting straight line for your data.  Under SUMMARY OUTPUT, X Variable, you will  find the slope of this line.  It should be the same value you got from the trendline.  The cell to the right of the slope contains the standard error in this slope from the fit.  The slope of the position versus time graph is the velocity.  The residual plot and line fit plot give you visual feedback on how well a straight line can be fitted to your position versus time data

Paste your labeled plot of position versus time (including the trendline) into your Word document and answer the questions below.

 What is the slope of the best fitting straight line?  What is the uncertainty (standard error) in this slope?  Do you think that a straight line produces a good fit to the data? What is the average speed and the uncertainty in the average speed of the object? Is the object moving with approximately constant speed, or is its speed increasing or decreasing?  Justify your answer. (How do you know?)

To practice entering and copying formulas, let us calculate the speed of the object for each small time interval from the raw data.

 Type Speed (m/s) into cell C1.

We want cell C2 to hold the speed of the object between t = 0 and t = 1 s.  Speed is distance covered divided by the time interval.  The distance covered is the difference between the entries in cells B3 and B2 and the time interval is the difference between the entries in cells A3 and A2.

 Type =(B3-B2)/(A3-A2) into cell C2.  This yields the average speed of the object in the small time interval between the first and the second measurement. (Formulas always start with an equal sign.  A formula can be a simple numeric expression such as =2*4, or it can include longer expressions involving other cells and statistical and mathematical functions.) Copy the formula into cells C3-C11.  To copy a formula, position your cursor in the cell that contains the formula, choose copy from the menu bar, highlight the cells that will receive the formula, and choose paste from the menu bar.

Construct a plot of speed versus time.  Let us use the a method that does not depend on the data occupying adjacent columns.

 Put your cursor in an empty cell. On the Excel menu bar click Insert, Chart, XY (Scatter), and pick one of the subtypes. Right-click the chart and choose Select Data, Add. Position your cursor in the X-Values text box and highlight entries 2 through 11 in the time column. Now position your cursor in the Y-Values text box, erase any entries in this box, and highlight entries 2 through 11 in the speed column. Type "speed versus time" into the Name text box. Give the chart a title, and label the axes. The label for the x-axis should be "Time (s)", and the label for the y-axis should be "Speed (m/s)".

Paste your labeled plot of speed versus time into your Word document.

There is a huge scatter in the values, because of experimental uncertainties in the measurements of small distances and time intervals.  But if we make many measurements we expect the average of these uncertainties to decrease with the number of measurements.  The fitting routine producing the trendline averages over all data points and therefore produces a speed value with a much smaller uncertainty.

Let us find the average value of all entries in column C.

 Into Cell D1, type Average Speed (m/s). Into cell D2 type the formula =average(C2:C11).

What is the value of the average speed.  How does it compare to the slope of the straight line fit?

### Understanding Motion – Distance and Time

Sometimes the best way to measure the position of a moving object as a function of time is to make a video recording and the analyze the video clip.  In this exercise you will analyze a clip showing a cart moving on an air track.  You will determine the position of the cart as a function of time by stepping through the video clip frame-by-frame and by reading the time and the position coordinates of the cart off each frame.  You will construct a spreadsheet with columns for time and position and a plot of position of the cart versus time.

To play the video clip or to step through it frame-by-frame click the "Begin" button.  The "Video Analysis" web page will open.  You can toggle between the current page and the "Video Analysis" page.

"Play" the video clip.  When finished, "Step up" to frame 0.  In some browsers you have to click "Pause" first..
In the setup window choose to track the x-coordinate of an object.
Click "Calibrate".  Then click "Calibrate X".
The video clip contains calibration markers.  Each white card is 10 cm = 0.1 m wide and the space between card is 10 cm = 0.1 m wide.  Position the cursor over some marked position in the left part of the frame, for example the left edge of the leftmost card, and click the left mouse button.  Then position the cursor over some marked position in the right part of the frame, for example the left edge of the sixth card from the left, and click the left mouse button again.  This will record the x-coordinates of the chosen positions.  Enter the distance between those positions into the text box in units of meter.  For the example positions, you would enter 1 into the text box.  Click "Done".
Click the button "Click when done calibrating".  A spreadsheet will open up.
Click "Start taking data".
Pick the point on the cart whose position you will track, for example the little post sticking out of the top of the cart.  Position the cursor over that point.  When you click the left mouse button, the time and the x-coordinate of your chosen point will be entered into the spreadsheet.  You will automatically step to the next frame of the video clip.
Repeat for each frame in the video clip.  Then click "Stop Taking Data".
Highlight and copy your table.  Open Microsoft Excel, and paste the table into an Excel spreadsheet.

Check if the cart moved with constant speed, covering equal distances in equal time intervals.  Produce a plot of position versus time as in exercise 1.  Add a trendline.

Paste your labeled plot of x(m) versus time (s) (including the trendline) into your Word document and answer the questions below.

 What is the slope of the best fitting straight line?  What is the uncertainty (standard error) in this slope?  Do you think that a straight line produces a good fit to the data? What is the average speed and the uncertainty in the average speed of the cart? Is the cart moving with approximately constant speed, or is its speed increasing or decreasing?  Justify your answer.