# The Range Equation

## Materials:

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

## Observation

We often model the motion of thrown balls and other everyday projectiles using the kinematic equations, but the kinematic equations do not account for air resistance and we know that projectiles launched in the open air will experience some level of air resistance. For example, the range equation (below) is derived from the kinematic equations and thus does not account for air resistance.

(1)

## Question Generation

Are the kinematic equations a reasonable model for everyday projectiles like small plastic balls, despite the effects of air resistance?

## 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 hypotheses on the use of kinematic equations for modeling projectile motion.

- If the kinematic equations are a valid model for small plastic balls launched at everyday speeds below _______
**m/s**, then the range equation will fit experimental data on range vs. launch angle with an R^{2}value of ___________ or better.

3) Explain your reasoning in choosing the maximum launch speed you provided above. Cite any sources that you used to help you decide on a value. [Hint: at what speeds would air resistance become a noticeable factor? For example, if you walk along and drop the ball you will find that it continues moving with you after you let go and thus hits the ground next to you. How fast would you need to run so that when you dropped the ball it would immediately slow down due to air resistance and fall way behind you?]

4) Explain your reasoning in choosing the R^{2 }value you provided above. Cite any sources that you used to help you decide on a value. [Hint: The R^{2} tells you how well the model can explain the observed differences for all of the measured values. For example, R^{2} = 0.5 means the range equation can only explain 50% of the differences among the range values and the other 50% comes from something the model can’t account for.]

## Experimental Hypothesis Testing

### Data Collection:

Range vs. launch angle data was collected according to the video at the top of this 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

5) 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. Label these two columns, including units.

6) 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.

7) 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.

8) 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.

9) 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, as well as the steps for modeling the data that are described in the rest of the lab.

### Modeling

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.

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

11) 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*.

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

13) 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:

=^2*SIN(2*C2)/

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.

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

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

16) Appropriately name the the two series “data” and “model” and add a legend to your plot.

17) Manually adjust the initial speed parameter to see if you can make the graph of the model roughly agree with the data. Don’t spend more than a couple of minutes doing this.

18) Can you make the model roughly agree with the data? If so, what speed value did you come up with?

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 R ^{2} estimation* to find the launch speed because you have worked with

*R*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

^{2}*minimum R*for us, using a function that we chose from the available options. However, none of those functions have a in them, so like the range equation does, so this time we we will do our own

^{2}estimation*maximum R*

^{2}estimation*.*

19) 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.

20) 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*.

20) 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}.*

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

22) 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.

23) 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.

24) 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}.*

25) Finally, in a new cell calculate the R^{2} as: 1 – SS_{res}/SS_{tot}

26) Now we can start changing the value of the launch speed to maximize the R^{2}. 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 R^{2} indefinitely. We need to factor in uncertainties to decide how precise our estimate of the launch speed parameter should be.

27) 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 R^{2 }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.

28) 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.

29) Write your final R^{2} value:

## Conclusions

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

31) Our original question was: *Are the kinematic equations a reasonable model for everyday projectiles like small plastic balls, despite the effects of air resistance?* Based on the results of this experiment, how would you answer that question? Explain.

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.

32) How might you experimentally determine the launch speed of the ball?

## More on Modeling

You may have noticed that the data looks kind of like a parabola and decided to apply a quadratic fit. Doing so, you will see a good fit with very high R^{2 }and we could use that fit as an empirical model to make predictions of range from angle, but our range equation model says the range should depend on not ! Both our range equation and the quadratic equation had very high R^{2}, how would we decide which is the better model?

- The range equation model only required one free parameter to fit the data, but the quadratic equation had three parameters available to adjust to whatever value gave the best fit. From that difference alone the range equation is the better model.
- The range equation is a physical model based on theory, but the quadratic equation was simply an empirical model based on the data alone. From that difference alone the range equation is the better model.
- 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?) The range equation correctly predicts the limiting cases.

Just because a model gives a high R^{2}, that doesn’t mean it is a physically meaningful model!

## Modeling with Air Resistance

33) The simulation at the bottom of this page allows you to investigate situations where air resistance is significant. The simulation is based on a model that accounts for air resistance. For example a pumpkin with 1 **kg** mass and 0.5 **m** diameter launched at 30 **m/s**. Use the simulation to determine what angle produces the maximum range for that situation by clicking the simulation, choosing the lab icon, and then changing the settings to match the example situation. Don’t forget to check the box that turns on air resistance. Record the angle you found here:___

34) According the the range equation, what angle produces the maximum range? (Hint: look back at your graph of the the predictions made by the range equation).

35) Does the range equation do well at describing the situation with the pumpkin?

describing what happens, but not how much happens