# Constant Acceleration with Carts

**Course Outcome 5**

### Materials:

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

### Observation

Wheeled carts running level tracks are sometimes used to learn about various types of motion. Sometimes fans are attached to the carts to cause them to accelerate. The video below demonstrates shows just such a setup being used to produce different types of motion. The kinematic equations are used to model motion with constant acceleration, but we don’t know if the fans will produce constant accelerations, so we don’t know if the kinematic equations can be used to model the cart motion.

### Question Generation

Our observations generated a question that we will answer by the end of this lab:

- Do the kinematic equations provide a reasonable model for analyzing the motion of the carts produced by the fans?

### Existing Knowledge Search

1) Spend a few minutes looking for an answer to the previous question.

- Did you find any useful information from a reliable source? If so, provide the source(s) and summarize the information below.

### Hypotheses Generation

2) Provide a of hypothesis on the use of kinematics for analyzing the motion of the carts.

- If the kinematic equations can analyze the cart motion, then using the kinematics equations to fit the acceleration, velocity, and position data will produce R
^{2}values of ___________ or better.

3) Explain your reasoning in choosing the minimum R^{2} value you provided above. Cite any sources that you used to help you decide on a value.

### Experimental Hypothesis Testing

#### Data Collection:

The data was collected during the demonstrations seen in the video at the top of the page. The data is available in an online spreadsheet. You can copy and paste the data into your own spreadsheet for plotting and analysis.

#### Data Analysis

The video below demonstrates how to make plot the data, label the graph, and add trendlines (using data from another lab as an example.) The video also shows you how to plot error bars, but you will not need to do that in this lab.

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

5) What kinematic equation should describe the position vs. time data for this accelerating object? Write it here:

6) Based on your answer above, apply the appropriate type of fit equation to the data, and record the resulting equation and the R^{2} here:

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

8) What kinematic equation should describe the velocity vs. time data for this accelerating object? Write it here:

9) Based on your answer above, apply the appropriate type of fit equation to the data, and record the resulting equation and the R^{2}.

10) Plot the acceleration vs. time data. Be sure to provide a title and axes labels with units.

11) For the kinematic equations to be applicable, the acceleration must be constant. Fitting a line to constant data should produce a slope of zero and a y-intercept equal to the constant value. Fit a line to the data and record the resulting equation and the R^{2}.

### Conclusions

(Outcome 2-2)

#### A note on R^{2}

We should note that the R^{2} value for your acceleration fit may be very low, even though the line you fit is able to reproduce the overall trend in the data. This is an opportunity for us to discuss the meaning of the R^{2 }value, it’s limitations, and why scientists often use other statistics to determine if a model does well at describing data (such as the statistic). The R^{2} value is a measure of how well the model reproduces the variation in the data away from the mean value. If the majority of the variation is causes by effects included in the model, then will be R^{2} high. For example, your velocity vs. time data shows a wide range of values, many are higher than what you would get for an average velocity and many are lower. This variation is caused by the acceleration, so the kinematics model (linear velocity function for a constant acceleration) is able to reproduce that variation very well. On the other hand, the acceleration data contains variations caused factors not included in the kinematics model, such as small changes to accelerations caused by the cart wheels and guide bar rubbing against the track, tiny debris on the track and the cart axles, variations in the fan speed, air currents, and measurement uncertainty. The kinematic model is able to reproduce the long-term trend in the data, which is constant, so fitting a line to the data produces a slope of nearly zero, but the flat line is very bad at reproducing the short term fluctuations in the data! That is where the low R^{2} comes from. The main point: *don’t evaluate the success of a model using R ^{2} alone! * The next lab in the series will give you more experience with evaluating a model from

*R*

^{2}*along with other important factors.*

You may be asking why we didn’t see similar obvious variations on the other graphs? That is because the sensor calculates the acceleration data from the velocity data. Small variations away from the expected velocity would not be obvious on the velocity graph, or contribute significantly to the R^{2}, but small variation in velocity that occurs over very short time will produce a large acceleration value that could be very different from the average.

12) Do the results of your three fits and their R^{2} values support or refute your hypothesis? Explain.

13) Our original question was: *Do the kinematic equations provide a reasonable model for analyzing the motion of the carts?* Based on the results of this experiment, how would you answer that question? Explain.

### Modeling

Let’s use the kinematic equations as a model to predict unknown quantities in a new situation. That is one of the main purposes for models, after all.

14) 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. We don’t know the initial and final values of position and velocity or the acceleration.

15) Fit the data with the appropriate function and record the fit equation and R^{2} value here:

16) 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 kinematic equation. Record the values below (don’t forget units).

You now know the initial position and velocity, even though you had no data on that. You have applied a kinematic model to the cart motion and *extracted* the values for three unknown *parameters* of the model. We can use those parameters to make predictions.

17) 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**.

18) 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.

describing what happens, but not how much happens