Modeling Projectile Range

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 range equation to model experimental projectile range data.
   2. Apply a maximum R² square fitting procedure to evaluate the success of the range equation model and extract the launch speed as a free parameter of the model.
   3. Recognize the assumptions built into the range equation and evaluate the validity of those assumptions for this experimental situation.

MOdeling

The range equation (below) allows us to predict the launch distance, or range, from the launch angle and launch speed.

\begin{equation}

R = \frac{v_0^2 sin(2\Theta)}{g}

\end{equation}

The range equation is derived from the kinematic equations assuming a constant downward acceleration equal to g and zero horizontal acceleration. Therefore, the range equation intrinsically neglects the effects of air resistance. We would like to test the range equation to verify the validity of those assumptions. We can test the equation examining how well it can model experimental range vs. angle data.

Experimental Methods

Range vs. launch angle data was collected according to the video below. The data is available in an online spreadsheet. You can copy and paste the data into your own spreadsheet for plotting and analysis.

Analysis Methods

Graphing

1) Copy and paste the range vs. angle data into the first two columns of your spreadsheet, leaving an empty row at the top for the column labels.

2) Label the two columns, including units.

3) Plot the data. Place the dependent variable (we don’t control) on the vertical axis and the independent variable (what we control) on the horizontal axis. Be sure to provide a title and axes labels with units.

Uncertainty Estimation

4) Below is an image of the plumb-bob and scale use to determine the launch angle. Based on the image, estimate the uncertainty in the angle measurements. Explain your reasoning.

5) Slow motion video was used to determine where the ball landed in order to measure the range. Several individual frames of one of the videos are shown below. The numbers on the meter stick are 1 cm apart. Based on this information, estimate the  uncertainty in the distance measurements. Explain your reasoning.

6) Add horizontal and vertical error bars to your graph based on the uncertainty estimates for angle and range. The following video demonstrates how to make the graph and add the error bars.

Data Fitting

Now we will attempt to use the range equation to model the data and extract the unknown launch velocity as a free parameter. This is our only free parameter because all of the other parameters (inputs) to the model are known or measured. This video walks you through the steps, which are also written out below (through step 23).

7) Label the third column “angle (rads)” and enter an equation to convert the angle data from degrees to radians.

8) Choose two cells several columns off to the right of your sheet where you can input the launch speed and the gravitational acceleration. Enter the gravitational constant equal to 9.8 m/s/s. This is a known parameter of the model. Set the launch speed to 1 m/s (for now). We don’t actually know the launch speed, so this is a free parameter that we will attempt to extract from the model. Use adjacent cells to label your two parameter cells, including units, but don’t put the units in the same cell as the values.

9) Label the fourth column “range model” and drop down to the second row of the column.

10) In the second row of the fourth column, enter the range equation to calculate the expected range by referencing the first cell containing the angle in radians and your two cells containing the parameter values. If you chose H2 to contain the gravitational acceleration and H3 to contain your launch speed, your equation would look like this in Excel:

=$H3$^2*SIN(2*C2)/$H2$

Notice that we have used dollar signs to make sure that this equation always references our parameter cells, even if we pull the equation to other cells.

11) Pull-down the cell containing the range equation to automatically calculate the expected range for all of your measured launch angles.

12) Add the model predictions to the existing plot as a new data series.

13) Add a legend to your plot.

14) Appropriately name the the two series “data” and “model” so that the names are reflected in the legend.

15) Make the model roughly agree with the data by making 2-3 attempts at manually adjusting the launch speed parameter.

We don’t want to rely on our eyes to determine how well the model fits the data. Instead, let’s do some basic statistics. We will use maximum R² estimation to find the launch speed because you have worked with R² values in previous labs. In other words, we will find the launch speed value that maximizes how well the range equation describes the observed variation in the range data. In previous labs we asked Excel to do the minimum R² estimation for us, using a function that we chose from the available options. However, none of those functions have a $sin(2\Theta)$ in them like the range equation does, so this time we we will do our own maximum R² estimation.

16) Create a new column to the right of your model predictions and enter a formula that calculates the difference the observed (measured) and expected (predicted) values and squares that difference.

17) Pull down the formula to apply it to all the pairs of observed and expected range values. Now you have a set of always positive values that are larger when the percent difference between prediction and observation is larger. These are known as the squares of the residuals.

18) In another cell beneath your parameter cells (off the the right), apply the SUM formula add up all values calculated in the previous step. This value is known as the residual sum of squares, or SS_res.

19) Below the cell you chose above, enter a formula to calculate the average of all the measured range values.

20) At the top of the column to the right of squares of residuals, enter a formula that calculates the square of the difference between the measured value and the average of all measured values.

21) Again use  dollar signs to ensure that when you pull this formula down all of the new formulas will continue to reference the same cell containing the average. Pull the formula down.

22) Now in another cell below your residual sum of squares, sum up all the values you just calculated. This is the total sum of squares, or SS_tot.

23) Finally, in a new cell calculate the R² as: 1 – SS_res/SS_tot

24) Now we can start changing the value of the launch speed to maximize the R². Don’t spend more than a minute doin so. What value did you arrive at for the launch speed?

You may have noticed that by continuing to add more digits to the speed you could continue to maximize the R² indefinitely. We need to factor in uncertainties  to decide how precise our estimate of the launch speed parameter should be.

25) There are sophisticated ways to estimate the uncertainty in the free parameter, but for the purpose of this lab we will take a very rudimentary approach. You may improve the R² by changing the launch speed, but stop once most of the predicted values are within the error bars on the data. Now check to see how much you need to change the speed in each direction to cause those predictions to move outside the error bars. That range of speeds will define our speed uncertainty.

26) Use the center of the range of speeds you found as the speed value. Use the range of speeds to write the speed uncertainty as + half of the size of that range. Adjust the significant figures in your estimated launch speed to align with your uncertainty and record the speed and uncertainty below.

27) Write your final R² value:

Conclusions

28) Do the results of your experiment support or refute your hypothesis? Explain.

29) Are the kinematic equations, and the range equation that results from them, a reasonable model for everyday projectiles like small plastic balls, despite the the fact that they neglect air resistance? Explain your reasoning based on the results of this experiment.

Further Questions

If this was a new, unverified model then you would have provided some evidence that your model is valid under the conditions in this experiment because it can describe the data well. You also have used your model to extract the launch speed as a free parameter.  If you then measured the velocity another way and found agreement with your model prediction, then you would even stronger evidence that your model is valid under the conditions in this experiment.

30) If you were to repeat the experiment, how might you experimentally measure the launch speed of the ball? Explain what equipment you would use and what procedure you would follow to determine the launch speed.

More on R² Fitting

You may have noticed that the data looks kind of like a parabola and maybe you thought about applying a quadratic fit. Doing so, you will see a good fit with very high R² and we might consider using that fit as an empirical model to make predictions of range from angle. However, the we would have the range equation model suggesting the range should depend on $sin(2\theta)$ and a quadratic model suggesting that range depended on $\Theta^2$!  Both our range equation and the quadratic equation had very high R², so how would we decide which is the better model?

1. The range equation model only required one free parameter to fit the data, but the quadratic equation had three free parameters that were adjust to give the best fit (the constant, the coefficient in front of the linear term and the coefficient in front of the quadratic term). From that difference alone the range equation is the better model.
2. Due to the issues above, the quadratic equation fails at the limits. If you input a zero or 90 degree launch angle into the quadratic fit equation you will get a prediction for the range that is not zero! (You launch the ball straight up, but it doesn’t come straight down?) Meanwhile, the range equation correctly predicts the limiting cases. From that difference alone the range equation is the better model.
3. Finally, the range equation is a physical model based on established theory, but the quadratic equation was simply an empirical model based on the data alone. Considering that both models performed equally well, the range equation is the better model.

Just because a model gives a high R², that doesn’t necessarily mean it has a real connection to the mechanisms being studied. Conversely, just because a model gives a low R², that doesn’t mean the model is fundamentally flawed.