Project Title

Ready, Aim, Fire!

 

Introduction

Mathematical models are used to help make real-life decisions.  Models are developed by analyzing data patterns and finding the equation that best fits the data.  Many models are based on a linear pattern where the dependent variable increases or decreases at a fixed rate based on the independent variable.  Models of motion are often more complex quadratic models based on a variety of factors including the pull of gravity.

 

Task

Working together with your group, you will collect and analyze experimental data.  Once your data is collected and recorded, you will use your graphing calculator to develop models based on the data.  Using your equations, you will then get to test the accuracy of the models.  Have you ever wanted to shoot rubber bands at school?  Now is your chance!

 

Instructions

Materials Needed:  rubber bands, ruler, protractor, measuring tape, and lab safety goggles; TI-83+ or 84+ graphing calculator to calculate the regression equations; digital camera to capture the experiment; and, a small action figure for the conclusion of the experiment. 

 

Complete each problem in order keeping careful notes of the results.  Also, be sure to take lots of pictures of the experiment.  You will create a multimedia presentation of your results at the conclusion of the project.

 

1                     First problem:  Horizontal Flight Distance vs. Stretch of Rubber Band

·                     This experiment is best performed outdoors on an open, level surface.  Gather all of the necessary materials, including paper and pencil to record the results, and head outside.  Within your group, assign the following tasks:  spotter, recorder, holder, and launcher.  In order to maintain safety, all group members should wear lab safety goggles to protect themselves from flying rubber bands.  

 

·                     The holder will hold the ruler level at about waist height.  The launcher will place one end of the rubber band on the end of the ruler and pull back the elastic to measure the starting length at rest.  (At this point the rubber band should not be stretched.  This measurement is just the starting point.)  For the first trial, the launcher will stretch the rubber band 1 cm beyond the starting point and release.  The spotter will measure the horizontal flight distance and the recorder will record the results in the table below.

 

Hint:  For more accurate data, the spotter needs to take note of where the rubber band first hits the ground, and measure to that point, rather than where the rubber band finally comes to rest. 

 

·                     Repeat the 1 cm stretch for a total of three trials, then, find the average of the three trials and record.  Continue launching rubber bands at 2 cm, 3 cm, 4 cm, etc. until the table is completed or the rubber band breaks.

 

 

2                     Second problem: Use your TI-83+ or 84+ graphing calculator to create a scatter plot of your data.

 

NOTE: Clear the STAT area of your calculator by pushing the STAT key, select 4:ClrList, 2^nd ,L1, enter a comma from the keyboard, 2^nd, L2, and then ENTER.  The line should look like this prior to pressing ENTER:

 

 

·                     Enter your data into a table by pushing the STAT key and then ENTER.  Enter the stretch beyond rest value into L1 and the average distance value into L2.  Be sure that you have 10 entries in the L1 column and 10 entries in the L2 column before going on to the next step. 

 

Hint:  If you have made an error, use the arrow keys to highlight the error and then simply type over the error.

 

·                     Once your data is in the table, create a scatter plot by pushing 2^nd and then Y=.  Hit ENTER and then move the cursor, using the arrows, to ON and hit ENTER.  Now to see the points on the graph, push ZOOM 9.  Each data point will now appear on your screen.

 

·                     Now we will use the calculator to find the line of best fit of the data.  Push the STAT key then use the right arrow to highlight CALC.  Push the number 4, and then ENTER.  The calculator will give the line of best fit in the form ax+b, where a is the slope and b is the y-intercept.  Record the given equation on your paper with each coefficient to the nearest thousandth. 

 

·                     To see how closely the line fits with the data, push Y= and enter the equation into the calculator.  Then push the GRAPH key.  You should now see the data points with the line of best fit.

 

            Record various observations about the data points and the line of best fit.  How close are the points to the line?  Are there any points that are very far away?  What might have caused the discrepancy?  

 

            You will need to include a graph of your data in your final project.  You can either neatly graph the data and line of best fit by hand and take a picture or use a computer generated graphing program, such as GeoGebra.  The GeoGebra program can be downloaded for free at http://www.geogebra.org/cms/en/download.

  

3                     Third problem:  Horizontal Flight Distance vs. Angle of elevation

·                     This experiment will require the holder and the launcher to work together carefully.  Again, the holder will keep the ruler level at about waist height.  The launcher will place one end of the rubber band on the end of the ruler and pull back the elastic to measure the starting length at rest. For all the trials, the launcher will stretch the rubber band 5 cm beyond the starting point and release.  Just like in the previous experiment, the spotter will measure the horizontal flight distance and the recorder will record the results in the table below.

 

·                     The difference is that now the ruler will be positioned with varying angles of elevation.  The first trials, where the ruler is level, is an angle of 0 degrees elevation.  To achieve an angle of 10 degrees, the ruler will need to be positioned on a protractor and angled up to the 10-degree mark.  Remember, the rubber band should be stretched 5 cm and three trials should be done at each angle of elevation.   

 

 

4                     Fourth problem: Use your TI-83+ or 84+ graphing calculator to create a scatter plot of your data.

 

NOTE: Clear the STAT area of your calculator by pushing the STAT key, select 4:ClrList, 2^nd ,L1, enter a comma from the keyboard, 2^nd,.L2, and then ENTER.

 

·                     Enter your data into a table by pushing the STAT key and then ENTER.  Enter the angle of elevation into L1 and the average distance into L2.  Be sure that you have 10 entries in the L1 column and 10 entries in the L2 column before going on to the next step. 

 

            Hint:  If you have made an error, use the arrow keys to highlight the error and then simply type over the error.

 

·                     Once your data is in the table, create a scatter plot by pushing 2^nd and then Y=.  Hit ENTER and then move the cursor, using the arrows, to ON and hit ENTER.  Now to see the points on the graph, push ZOOM 9.  Each of the data points will now appear on your screen.

 

            Hint:  How does this graph differ from the previous graph?  Does the data appear to be linear or parabolic?  

 

·                     Now we will use the calculator to find the function that best fits the data.  In order to determine the best fit for our data, we will need to go to the Catalog by pushing 2^nd and then 0.  Scroll down using the arrow keys until you see Diagnostic On.  Position the cursor beside it and hit ENTER.  Then hit ENTER again and your calculator should say Done. 

 

·         Let’s assume that the function is linear:  Push the STAT key then use the right arrow to highlight CALC.  Push the number 4 then Enter.  The calculator will give the line of best fit in the form ax+b, where a is the slope and b is the y-intercept.  Record the given equation on your paper with each coefficient written to the nearest thousandth.  Also, record the r value given.

 

·         Let’s now assume that the function is quadratic: Push the STAT key then use the right arrow to highlight CALC.  Push the number 5 then ENTER.  The calculator will give the quadratic regression in the form .  Record the given equation on your paper with each coefficient written to the nearest thousandth.  Also, record the  value given.

 

Hint:  The  value or  value gives us a numerical picture of how closely the function fits the data.  The closer the value is to 1 (or negative 1 in the case of a negative slope), the better fit of the function to the data set.  It is generally accepted that a value of .9 or greater is a good fit.         

                                   

·                     Choose which model best fits your data and enter the function into Y=.  Then push the GRAPH key.  You should now see the data points and the function graphed.

 

            Record various observations about the data points and the function.  How close are the points to the regression?  Are there any points that are very far away?  What might have caused the discrepancy?  

 

            You will need to include a graph of your data in your final project.  You can either neatly graph the data and regression by hand and take a picture or use a computer generated graphing program, such as GeoGebra.  The GeoGebra program can be downloaded for free at http://www.geogebra.org/cms/en/download.

 

5         Fifth Problem:  Analyze the data

 

·                     Find the vertex of the quadratic model.  Is the vertex a maximum or minimum?  What does the vertex represent in this situation?

 

Hint:  The TI graphing calculator can quickly calculate the vertex.  With the quadratic function entered into Y=, push 2^nd and then TRACE.  Select Maximum, number 4.  The calculator now says Left Bound on the bottom of the screen.  Use the arrows to position the cursor to the left of the maximum.  Then hit ENTER.  The calculator now says Right Bound.  Use the arrows to position the cursor to the right of the maximum.  Hit ENTER.  The calculator will now say Guess.  Hit ENTER.

 

·                     Use your linear model from part 2 to determine how far the rubber band must be stretched to travel the maximum horizontal distance found in part 4 above. 

 

Hint:  Substitute the horizontal distance, y, into the linear model and solve for x.  

 

·                     Now test your calculation.  How close did your rubber band come to the maximum horizontal distance?

 

Collaboration:

Trade rubber bands and equations with a neighboring group.  Each group should now have their neighbor’s linear equation to model flight distance vs. stretch and a quadratic equation to model flight distance vs. angle of elevation.  The neighboring group will position a small action figure on the ground within range of the rubber band shooter.  The goal is use the two equations to hit the action figure in the least number of tries. 

 

Begin with the linear model.  Measure the distance to the action figure in cm.  Substitute the measurement in to the linear equation to determine the stretch distance necessary to hit the figure.  You many need to adjust slightly.  How many tries did it take?

 

Now try the quadratic model.  Substitute the measurement in to the quadratic equation to determine the angle of elevation necessary to hit the figure.  Remember that the rubber band should be stretched 5 cm.  How many tries did it take?

 

Which model allowed you to hit the action figure in the least number of tries?  With which model was the neighboring group more successful?  What factors contribute to the inaccuracy of the model?  What could be done to minimize the error?

 

Conclusions

Your final product will be a multimedia presentation of your experiment photos.  You can use the free program Animoto to create a slide show of your photos.  The program is available for free at http://animoto.com/create.  Another option is Windows Movie Maker or iMovie.  Additionally, you will need to present your data tables, graphs, and regression models.  You can incorporate the data into your slide show, make a separate poster, or include a typed lab report to go along with your slideshow. 

 

Grade

Your project will be given a score of 1 to 4, with 4 being the highest score possible. You will be evaluated based on the following criteria:

Score	Content	Presentation
4	Your project appropriately answers each of the problems.  Completed data tables, graphs, and equations are included.    Evidence of careful data collection is apparent.  Data points are tightly grouped around the functions.	Your project contains information presented in a logical and interesting sequence that is easy to follow.   Your project is professional looking with graphics and attractive use of color.
3	Your project appropriately answers each of the problems.  Completed data tables, graphs, and equations are included.    Evidence of fairly careful data collection is apparent.  Data points are for the most part grouped around the functions.   Minor errors may be noted.	Your project contains information presented in a logical sequence that is easy to follow.   Your project is neat with graphics and attractive use of color.
2	Your project attempts to answer each of the problems.  Partially completed data tables, graphs, and equations are included.    Evidence of inaccurate data collection is apparent.  Data points are not tightly grouped around the functions.   Errors and inaccuracies may be noted.	Your project is hard to follow because the material is presented in a manner that jumps around between unconnected topics.   Your project contains low quality graphics and colors that do not add interest to the project.
1	Your project attempts to answer some of the problems.  Data tables, graphs, and equations are not complete.    Evidence of inaccurate data collection is apparent.  Data points are not tightly grouped around the functions.   Major errors are noted.	Your project is difficult to understand because there is no sequence of information.   Your project is missing graphics and uses little to no color.

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The NROC Project 2019