To determine the acceleration and its uncertainty using the data provided, we can perform a linear regression analysis. In this context, acceleration will be the slope of the line obtained by fitting the data points (time vs. velocity) to a linear equation:

$\text{Velocity} = (\text{Acceleration}) \times (\text{Time}) + \text{Initial Velocity}$

Steps to Perform Linear Regression in Excel:

1. Input the Data:

   • Enter the time data into one column and the corresponding velocity data into another column.

2. Use the Data Analysis ToolPak:

   • Go to the Data tab and click on Data Analysis.
   • Select Regression from the list and click OK.
   • For Input Y Range, select the velocity data.
   • For Input X Range, select the time data.
   • Ensure Labels is checked if you included column headers.
   • Choose an output range or new worksheet.
   • Click OK to run the regression analysis.

3. Interpret the Results:

   • In the regression output, look for the Coefficients table.
   • The coefficient corresponding to Time is the slope, which represents the acceleration.
   • The Standard Error associated with this coefficient represents the uncertainty in the acceleration.

Manual Calculation:

If you prefer to perform the calculation manually or confirm the Excel results, here's a brief outline:

1. Calculate the Slope (Acceleration): The slope $m$ is given by: $m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$ where:

   • $n$ is the number of data points.
   • $\sum xy$ is the sum of the products of corresponding x and y values.
   • $\sum x$ and $\sum y$ are the sums of x (time) and y (velocity) values, respectively.
   • $\sum x^2$ is the sum of the squares of x values.

2. Calculate the Standard Error of the Slope: The standard error $SE_m$ of the slope is given by: $SE_m = \sqrt{\frac{\frac{1}{n-2} \sum (y_i - \hat{y}_i)^2}{\sum (x_i - \bar{x})^2}}$ where:

   • $y_i$ are the observed velocities.
   • $\hat{y}_i$ are the velocities predicted by the regression line.
   • $\bar{x}$ is the mean of the time values.

Performing the Analysis

Since I cannot directly use Excel, I recommend following the above steps in your Excel software. The output will provide you with the slope (acceleration) and the standard error.

If you prefer, I can also do the calculations manually with the given data. Would you like me to do that?

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Follow-Up Questions

1. What is the physical significance of the slope in a velocity-time graph?
2. How would the initial velocity affect the linear regression result?
3. How can you verify the linearity of the data before performing regression?
4. What are potential sources of error in a real-world experiment involving falling objects?
5. How would air resistance affect the velocity measurements and the calculated acceleration?

Tip

Always check the R-squared value in your regression output to assess how well your data fits the linear model. A value close to 1 indicates a good fit.