Object Tracker Part 3: Analyze Velocity vs. Time

STEMcoding Object Tracker

Part 3: Analyze Velocity vs. Time

Using the data from Part 1 (or Part 2), you will calculate velocities and analyze velocity vs. time plots. You will use these plots to make inferences aboutg and x-acceleration.

Step 0. Open the code

Click here to open the program

Here are some questions to consider

Step 1. Open the csv file and add velocity columns

Find the csv file you downloaded in Part 1 (or your spreadsheet from Part 2) and open it in Excel or Google Sheets.

Add two new columns to the spreadsheet and name them vx and vy. This is where you will calculate velocities.

Step 2. Calculate velocities v_x and v_y

Recall that the velocity of an object is given by the change in the object's position over the change in time. This could be written as v_x = (x_f - x_i)/(t_f - t_i) where x_i = initial x position, x_f = final x position, t_i = initial time, t_f = final time. However, in this case our 'initial' and 'final' points will not be the first and last data points, as this would give us the average velocity over the entire video. Instead, we want to estimate the velocity of the basketball in each frame of the video. Each row in the spreadsheet represents a single frame of the video.

To infer the velocity at a given frame, take the "initial point" to be the frame directly before it, and the "final point" to be the frame directly after it. This calculates the velocity over a very small interval of time (1/240th of a second for 240fps!) to give an approximation of the instantaneous velocity for that frame. This is sometimes called the central difference method.

In your spreadsheet ignore the rows that say "Object out of range?" and instead start with the first rows that do not have that message.

Estimate v_x and v_y using the method described above and making sure to reference the cells containing the values you want to use. For example, using the basketball data, to calculate v_x you can copy the equation in the screenshot above. The basic idea is that to estimate v_x at the time corresponding to row 39, you calculate Δ x = x_f - x_i where x_f is coming from row 40 and x_i is coming from row 38. Likewise use Δ t = t_f-t_i where t_f is from row 40 and t_i is from row 38. Take the ratio of these results to get v_x because v_x = Δ x / Δ t which is why the formula in the spreadsheet is = (B40 - B38) / (A40 - A38)

Now repeat this calculation for all the other frames! Excel and Google Sheets have an autofill feature that makes this easy, but it will only work if you have used cell references in your calculation. Click on the cell containing your first vx value and click the bottom right corner. Drag your cursor down to the cells below and the calculations will be automatically performed. You can click on the filled cells to check that the formula is referencing the correct cells. Do this for vy as well.

Note: Why did we not calculate velocity in row 38? What about the very last row (row 286 for the basketball data)?

What do you get for your initial v_x and v_y? What did you get for the final v_x and v_y?

Step 3. Create a scatter plot of v_y vs. t

First we will look at y-velocity vs time so that we can infer y-acceleration again. Before making the plot, think about what you expect it to look like based on your calculated velocities and your understanding of physics. How does y-velocity change over time, and what is causing it to change? Does it increase or decrease, linearly or nonlinearly?

If you aren't sure how to create the plot, refer to Step 1 in these instructions for Excel or Google Sheets. Does the plot look how you expected? What type of function does it resemble?

Step 4. Fit a trendline to your v_y vs. t plot

Fitting a trendline will give us a specific equation that models that relationship between v_y and t.

First, determine what type of trendline to use (linear, exponential, polynomial, logarithmic, or power). You can probably determine the appropriate trendline type just by looking at the plot, but you should also consider the physics behind it -- What type of function is v(t) when acceleration is constant?

Add the appropriate trendline and its equation to the plot. If you aren't sure how to do this, refer to Step 2 in these instructions for Excel or Google Sheets Record the trendline equation.

Ultimately your plot including the trendline should look something like this:

Step 5. Infer a_y using the trendline equation

Show the equation for your trendline. As mentioned in the last step, there are instructions for how to do this in Excel and Google Sheets

Recall that acceleration is given by the change in velocity over the change in time (a = Δv / Δt). The slope of a line is given by the change in y variable (velocity) over the change in x variable (time).

What do you get for y-acceleration? If you do not get -9.8 for the y acceleration, is your data still consistent with -9.8? Compare this to your result from Part 2.

What could we do to make the experiment more precise and/or accurate?

Step 6. Create a scatter plot of v_x vs. t

Now we will look at x-velocity vs time. Before making the plot, think about what you expect it to look like based on your calculated velocities and your understanding of physics. How does x-velocity change with time? What forces (if any) could be causing the basketball to accelerate in the x-direction?

Create the plot. What do you notice about it? Does it look how you expected? Are there points that look like outliers?

Step 7. Fit a trendline to your v_x vs. t plot

Which option makes the most sense -- linear, exponential, logarithmic, power, etc.?

Step 8. Infer a_x using the trendline equation from Step 7

Show the equation for your trendline and use it to infer the acceleration in the x direction (a_x). The acceleration should be small or zero, but what do you actually get?

Step 9. Turn in your questions

Turn in the answers to your questions wherever your teacher said you should turn it in!