Velocity vs time graphs are an important part of kinematics. Here's a way to make velocity vs time graph using Pivot Interactives data table and graphing tool. In general they are two common ways that students learn to make velocity vs time graphs from position vs time data:

- Students hand-calculate slopes of many tangent lines on their position vs time graph. They use the slope values to plot a new velocity vs time graph. Teachers report that this process is so time-consuming that students often lose sight of the overall goal.
- Many data analysis tools can convert a position vs time graph to a velocity vs time graph automatically. While this is super convenient for students, it obscures the process.

Pivot Interactives method balances these. Students select ranges of data on their position vs time graph, and use the linear regression tool to calculate the slope, which is the average velocity over that range. Students plot those slopes to create a velocity vs time graph.

Here are some pointers students will need to make this process work:

- For most situations, you don't need very many data points to see the velocity vs time graph. Usually 10 data points is sufficient.
- Use equal time intervals when collecting position vs time data.

Here's a video of the process:

Here is a written description showing the same process described in the video:

- When collecting position vs time data, use consistent time intervals between samples, such as 0.1 s.
- Once you've collected all position vs time data, make a graph of position on the vertical axis and time on the horizontal axis.
- Make a new column called
**velocity**, with appropriate units. - Next, use the linear regression tool on your position vs time graph. It will look strange because the position vs time graph is not linear. That's ok for now.
- Un-select all the data points except the first three. You can select/unselect data points either using the checkbox on the rows of the data table, or by clicking directly on a data point on the graph. When you click on data points on the graph, they change from magenta dots to grey
**x**'s and are no longer included in the linear regression calculation. Here's what the graph would look like when the first three data points are selected:

*In this position vs time graph, all the data points except the first three are un-selected (by clicking on them). The equation shown at lower left is the slope at t=0.05s.*

- The slope shown in the linear regression is the average slope from t=0 to t=0.1s. If the acceleration is constant, then this is the slope of the line
t=0.05s, the mid-time of the interval.*at* - Write this slope (295cm/s in this example) in the data table column for the velocity at the first mid-time (the second row of the table). Notice that you won't have a velocity measurement for the first time (t=0 in this case) because we don't know the slope at that instant.
- Un-select the first data point and select the fourth one. Your graph should look something like this:

To get the next velocity, unselect the first data point, and select the fourth data point. Now the slope shown in the linear regression formula is the velocity at t=0.1s.

- Now the slope shown in the linear regression formula is the slope of the line at the third time, t=0.1s in this case. Write this velocity in the data column for velocity for the second mid-time interval (the third row).
- Continue this process of un-selecting a data point on the left, and selecting the next one on the right to move the slope calculation to the next time. Continue to enter these new velocities in the appropriate cell in the velocity column on the data table. You won't have a velocity measurement for your first or last time, because we can't determine the slope at the first or last position measurement.
- When you have completed this, you can change the vertical axis on the graph to display your new velocity data.
- If you use the linear regression tool to find the slope of this new graph, make sure to re-select all the data points in the velocity vs time graph to include them in the linear regression calculations.
- Please let us know if you have questions or comments about this approach to making a velocity vs time graph by clicking the chat bubble at the lower right of your screen.

You can make an **acceleration vs time** graph using this process. Starting with the velocity vs time data, using the linear regression to find the slope at intervals along the velocity vs time graph. Use these slopes as instantaneous accelerations, and plot a new graph of acceleration vs time.