Velocity vs time graphs are an important part of kinematics. We offer two unique ways to find rate-of-change functions on the platform.
Methods:
Rate of Change Method
This method is one of the simplest methods for finding the rate of change, as our Rate of Change function will do it for you!
To do this:
Create a column to house your rate of change function. For this example, I'll use a velocity column.
Click on the column's options menu, then select Change Column Formula.
In the Formula Calculator, click the Rate of Change button on the bottom of the calculator. This will enter the formula RateOfChange("y","x"). Rate of change is defined as a "y" per unit "x".
Next, click on the variable you want as "y". Since I'm doing velocity, defined as position per unit time, I want position as "y". So, I'll select position first.
Then, click on the variable you want as "y". Since I'm doing velocity, defined as position per unit time, I want time as "x". So, I'll select time second.
Once you're done, click the Submit button to run your calculation. You will notice that your calculated values are greyed-out, denoting that they were calculated, not measured.
Remember: When collecting position vs time data, use consistent time intervals between samples, such as 0.1 s.
Want more tips? Check out our YouTube video on the subject, where we go into detail about using this function.
3-Point Tangent Method
Automation is great, but it doesn't explain the math behind the button. If you're curious to understand how rate of change functions work, then this is the option for you.
Here's how to do this:
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, click the cog in the upper right of the graph and select Curve Fit.
Select Linear from the list of functions, and press Done. 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 by clicking directly on a data point on the graph. When you click on data points on the graph, they change from colored dots to grey xs and are no longer included in the curve fit 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 curve fit parameter shows the slope, or velocity of the object at that time.
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 at t=0.05s, the mid-time of the interval.
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 in your table).
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 apply a linear fit 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.