Modeling Accelerated Motion With the Kinematic Equations

This lab is designed to align with AAOT science outcome #1: Gather, comprehend, and communicate scientific and technical information in order to explore ideas, models, and solutions and generate further questions.

Materials:

• writing utensil
• calculator
• digital device with spreadsheet program
• digital device with internet access

Objectives

1. 1. Apply the kinematic equations to model experimental motion data.
   2. Recognize the meaning of model fitting parameters to extract motion quantities such as initial position, initial velocity, and acceleration.
   3. Evaluate the success of the kinematic equations at modeling this specific motion data.
   4. Apply the kinematic model to extrapolate from existing data and predict motion quantities outside the dataset.

Methods

Experimental Methods

Wheeled carts running level tracks are sometimes used to learn about various types of motion, including accelerated motion. For that purpose, fans can be attached to the carts to cause them to accelerate.  The video below demonstrates shows such a setup used for different accelerated motion experiments.

Analysis Methods

The data collected during the demonstrations seen in the video are available in an online spreadsheet. You can copy and paste the data into your own spreadsheet for plotting and analysis. Be sure to review the videos on plotting, adding trendlines, and interpreting R² values found in the introduction if you need a refresher on those skills.

1) If the acceleration is constant, what kinematic equation should describe the position of an object as a function of time? Write it below and state the type of function (e.g. constant, linear, quadratic).

2) Plot the position vs. time data for the cart that changed direction. Be sure to provide a title and axes labels with units.

3) Based on your previous answer about the function type, apply the appropriate type of fit equation to the data and record the resulting equation and the R² here:

4) Comparing your fit equation to the kinematic equation it represents, you will see that the fit equation provides you values for the acceleration and initial position and velocity. Write the values for initial position, velocity, and acceleration below. You have applied a kinematic model to the cart motion and extracted the values for those three parameters of the model. The acceleration was an unknown parameter because you did not have data on that value. (Keep in mind that your spreadsheet program doesn’t know you are doing kinematics so it might default to calling the vertical variable “y” and the horizontal variable “x” even though we plotted position, which we often call “x” on the vertical axis and time, which we usually call “t” on the horizontal axis. Replacing the fit  “x” with “t” and the “y” with “x” will help you make the comparison).

5) Calculate a % difference between the measured initial position (first position value in the data) and the initial position extracted from the model fit. Do the same for the initial velocity values. Show your work.

6) Does the position analysis suggest that the kinematic equations successfully model this data (and therefore the acceleration is constant)? Explain in terms of the R² value and the accuracy of the initial velocity and initial position parameters.

7) If the acceleration is constant, what kinematic equation should describe the velocity as a function of time? Write it below and state the type of function (e.g. constant, linear, quadratic).

8) Plot the velocity vs. time data for the cart that turned around. Be sure to provide a title and axes labels with units.

9) Based on your recent answer about the function type, apply the appropriate type of fit equation to the data, and record the resulting equation and the R² here:

10) Comparing your fit equation to the kinematic equation it represents, you will see that the fit equation provides you values for the acceleration and initial velocity. Write the values for initial velocity and acceleration below. You have applied a kinematic model to the cart motion and extracted the values for those three parameters of the model. The acceleration was an unknown parameter because you did not have data on that value.

11)  Calculate a % difference between the measured initial velocity (first velocity value in the data) and the initial velocity extracted from the model fit. Show your work.

12) Does the velocity analysis suggest that the kinematic equations successfully model this data (and therefore the acceleration is constant)? Explain in terms of the R² value and the accuracy of the initial velocity parameters.

13) You now have two acceleration values that were extracted from the kinematic models, one from the position model and one from the velocity model. Calculate a % difference between these two values to determine how well they agree. Also calculate the the average of these two acceleration values.

Conclusions

14) Overall, do the results of your analysis of position and velocity data indicate that the kinematic equations successfully model the motion of the cart and therefore the acceleration is constant? Explain.

15) Would you trust the average value of the acceleration that you found by applying the kinematic models? Explain.

Further Questions

The real advantage of building and testing models is that when a successful model is discovered then we can use it to make predictions of unknown quantities. Let’s use the kinematic equations as a model to predict unknown quantities in a new situation.

16) Plot the position vs. time data for the second trial, when the cart does not turn around. You will notice that the data for the beginning and end of the motion are missing and that no acceleration data is available. Therefore, we don’t yet know the acceleration or the initial and final values of position and velocity.

17) Fit the position vs. time data with the appropriate function and record the fit equation and R² value here:

18) Determine the initial position, initial velocity, and acceleration of the cart by comparing the coefficients (numbers) in your fit equation to the variables in the corresponding kinematic equation. Record the values below (don’t forget units).

You now know the initial position, initial velocity and acceleration even though you had no data on those values. You have applied a kinematic model to the cart motion and extracted the values for three unknown parameters of the model. We can now use those parameters to make predictions by simply applying the kinematic equation with those values in place (the fit equation).

19) Now that you know the values of the parameters in the kinematic equation, use it to predict what the final position of the cart would be after at 3 s.

20) Use the acceleration parameter and initial velocity parameter that you extracted to predict the final velocity of the cart at 3 s. (Use these in another kinematic equation).

Using the model to predict values outside the range of data you have is known as extrapolation, that is what we have just done. Using the model to predict values within the range of data you have is known as interpolation. We didn’t interpolate because our data points were very close together (we had high resolution) so we had no reason to interpolate in this case.