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Teacher Lesson Plan Page 1 of 9 Will it fly? The mathematics behind invention A lesson plan by Richard Messick For Algebra 1 students Overview: This lesson will use data from the Wright Brother’s wind tunnel experiments of 1901 to determine mathematically whether the 1903 Wright Flyer is capable of sustained, controlled flight. The lesson will then require students to choose an inventor or invention and discover what role mathematics played it the inventive process. Guiding Question: What is the role of mathematics in invention? Objectives: L1.2.4 Organize and summarize a data set in a table, plot, chart, or spreadsheet; find patterns in a display of data; understand and critique data displays in the media. A1.2.1 Write equations and inequalities with one or two variables to represent mathematical or applied situations, and solve. A1.2.3 Solve (and justify steps in the solutions) linear and quadratic equations and inequalities, including systems of up to three linear equations with three unknowns; apply the quadratic formula appropriately. A1.2.8 Solve an equation involving several variables (with numerical or letter coefficients) for a designated variable, and justify steps in the solution. A2.4.1 Write the symbolic forms of linear functions (standard [i.e., Ax + By = C, where B ≠ 0], point-slope, and slope-intercept) given appropriate information, and convert between forms. S3.1.4 Design simple experiments or investigations to collect data to answer questions of interest; interpret and present results. NEH Workshop Lesson Plan R. Messick Teacher Lesson Plan Page 2 of 9 Materials/Resources: Student Handout Graphing Calculator Internet and other resources Assessment: Students will be assessed on the completion of problems in the student handout as well as a required research paper or project. Instructional Sequence: For Activities 1-3 on the Student Handout, divide the class into small groups of 2 to 3 students each. Each student should have a copy of the student handout, each group needs a graphing calculator capable of calculating a linear regression of bivariate data. Begin by reading, silently or aloud, page 1 – 4 of the student handout. This explains who the Wright Brothers were and what they accomplished as well as explaining the very basics of aeronautics. Page 5 contains the data for the activities to follow. The students will be calculated lift and drag based on the data in the table. For Activity 1, Part 1, they will calculate lift using the given formula for each of the 24 sets of data and place their answers on the table. For Activity 1, Part 2, they will calculate drag using the given formula for each set of data and record their answers on the table. Their answers should be similar to those on the chart on the next page. This concludes Activity 1. Acitivity 1 could be modified to using a computer spreadsheet program like Excel. The students could enter the data or it could be provided to them and then, they could use a formula in the spreadsheet to calculate lift and drag. That was the method used to develop the table on the next page. The Wind Tunnel data is based on wind tunnel tests conducted by NASA on a replica of the 1903 Wright Flyer that was built, tested, and flown to commemorate the centennial of the Wright Brothers’ First Flight in 2003. NEH Workshop Lesson Plan R. Messick Teacher Lesson Plan Page 3 of 9 Wind Tunnel Data with Calculated Lift and Calculated Drag Angle of Attack Velocity CL, w CD, w Calculated Lift Calculated Drag -4.0013 25.18 0.3342 0.0421 356.54 44.92 -4.0013 24.98 0.3158 0.0458 331.60 48.13 -3.9997 24.93 0.3900 0.0437 407.96 45.69 -2.0006 25.08 0.4767 0.0472 504.45 49.92 -1.9998 25.07 0.5429 0.0491 574.21 51.94 -1.9990 24.98 0.5089 0.0469 534.43 49.21 -0.0048 24.83 0.7021 0.0619 728.66 64.21 -0.0009 25.08 0.6313 0.0703 668.16 74.36 -0.0009 24.93 0.6460 0.0554 675.74 57.98 0.0006 24.98 0.6999 0.0626 734.98 65.78 0.0006 25.19 0.6848 0.0613 731.05 65.47 0.0006 24.93 0.6773 0.0569 708.53 59.55 0.0006 25.07 0.6672 0.0573 705.47 60.62 0.0006 25.03 0.6635 0.0576 699.57 60.73 0.0006 24.99 0.6430 0.0558 675.74 58.69 0.0006 25.09 0.6352 0.0574 672.92 60.78 1.9983 25.18 0.8267 0.0807 882.11 86.05 1.9983 25.08 0.8183 0.0752 865.95 79.62 1.9983 25.14 0.7894 0.0705 839.50 75.03 3.9984 25.03 0.9332 0.0958 983.80 101.04 3.9991 25.14 0.9595 0.1052 1020.38 111.92 3.9991 25.17 0.9411 0.0989 1003.57 105.47 5.9985 24.82 1.0910 0.1292 1130.91 133.89 5.9992 24.94 1.0931 0.1316 1144.36 137.77 Activity 2, Part 1, involves inputting data from the table above into lists in a graphing calculator (TI83/84 or similar). List L1 should contain Angle of Attack data List L2 should contain the Calculated Lift data List L3 should contain the Calculated Drag data List L4 should contain the Coefficient of Lift data List L5 should contain the Coefficient of Drag data As a timesaver, this data could already by loaded into the calculators before the beginning of the lesson or it could be loaded on one calculator and then transmitted to the others by link cable. Alternatively, if Activity 1 was created on Excel, you could use the trendline feature of a scatterplot and find the line of best fit equation NEH Workshop Lesson Plan R. Messick Teacher Lesson Plan Page 4 of 9 directly from Excel. The Student Handout is designed to be used with a graphing calculator but could easily be modified to work with a spreadsheet. In Activity 2, Part 2 – A, Using lists L1 and L2, the students will use the Linear Regression function of the calculator to find the equation of the line of best fit for the Calculated Lift. The answers appear below. Screen Capture from TI-84+ Alternatively, if using Excel, select Angle of Attack and Calculated Lift data. Use chart wizard to construct a scatterplot. Right-click the data and add a trendline. Under options, click the show equation box. Calculated Lift by Angle of Attack 1400.00 1200.00 y = 77.735x + 693.71 1000.00 800.00 Lift 600.00 400.00 200.00 0.00 -6.0000 -4.0000 -2.0000 0.0000 2.0000 4.0000 6.0000 8.0000 Angle of Attack So, the equation of the line of best fit is y = 77.73 x + 693.71 For Activity 2, Part 2 - B, the inequality is 750 < 77.73 x + 693.71 For Activity 2, Part 2 - C, the solved inequality is 0.72 < x or x > 0.72 This means that the angle of attack must be greater than 0.72. NEH Workshop Lesson Plan R. Messick Teacher Lesson Plan Page 5 of 9 In Activity 2, Part 3 – A, we will be repeating the steps from Part 2, using Calculated Drag instead of Lift. So, using Lists L1 and L3 this time: Or in Excel, using Calculated Drag by Angle of Attack: Calculated Drag by Angle of Attack 160.00 140.00 y = 8.8914x + 68.425 120.00 100.00 Drag 80.00 60.00 40.00 20.00 0.00 -6.0000 -4.0000 -2.0000 0.0000 2.0000 4.0000 6.0000 8.0000 Angle of Attack Activity 2, Part 3 – A, the answer is y = 8.89 x + 68.43. Activity 2, Part 3 – B, the answer is 8.89 x + 68.43 < 132 Activity 2, Part 3 – C, the answer is x < 7.15 This means the angle of attack must be less than 7.15. For Activity 2, Part 4 – A, all the students need to do is pick an angle of attack. Based on Parts 2 and 3, they must choose a value somewhere between 0.72 and 7.15. 0.72 < angle of attack < 7.15 NEH Workshop Lesson Plan R. Messick Teacher Lesson Plan Page 6 of 9 In Activity 2, Part 4, the students need to calculate the lift and drag for the angle of attack they chose in part A. In order to calculate lift for this angle of attack, they need to determine the coefficient of lift. Again, they will find a line of best fit, this time using angle of attack (L1) and coefficient of lift ( L4 ). Or if using Excel, make a scatterplot of coefficient of lift by angle of attack. Coefficient of Lift by Angle of Attack 1.2000 y = 0.074x + 0.6577 1.0000 Coefficient of Lift 0.8000 0.6000 0.4000 0.2000 0.0000 -6.0000 -4.0000 -2.0000 0.0000 2.0000 4.0000 6.0000 8.0000 Angle of Attack So the equation of the line of best fit is y = 0.07 x + 0.66 For Activity 2, Part 4 – C, they must substitute the value they chose for angle of attack in place of x and solve for y to determine the coefficient of lift for their chosen angle of attack. A chart after the next section shows the coefficients of lift and drag for various angles of attack. NEH Workshop Lesson Plan R. Messick Teacher Lesson Plan Page 7 of 9 For Activity 2, Part 4 – D, Again, the students must find the equation for a line of best fit. This time they use angle of attack (L1) and coefficient of drag (L5) Or if using Excel, make a scatterplot of coefficient of drag by angle of attack. Coefficient of Drag by Angle of Attack 0.1400 Coefficient of Drag 0.1200 y = 0.0085x + 0.0649 0.1000 0.0800 0.0600 0.0400 0.0200 0.0000 -6.0000 -4.0000 -2.0000 0.0000 2.0000 4.0000 6.0000 8.0000 Angle of Attack So, the equation for the line of best fit is y = 0.01 x + 0.06 For Part 4 – E, the student needs to substitute their value for angle of attack in place of x and solve for y. This will give them the coefficient of drag for their angle of attack. The chart on the next page shows the coefficients of lift and drag for various angles of attack. NEH Workshop Lesson Plan R. Messick Teacher Lesson Plan Page 8 of 9 Angle of Attack Coefficient of Lift Coefficient of Drag Calculated Lift Calculated Drag 0.75 0.71 0.07 725.67 68.75 1.00 0.73 0.07 743.49 71.29 1.25 0.75 0.07 761.32 73.84 1.50 0.77 0.08 779.14 76.39 1.75 0.78 0.08 796.96 78.93 2.00 0.80 0.08 814.79 81.48 2.25 0.82 0.08 832.61 84.02 2.50 0.84 0.09 850.43 86.57 2.75 0.85 0.09 868.26 89.12 3.00 0.87 0.09 886.08 91.66 3.25 0.89 0.09 903.90 94.21 3.50 0.91 0.10 921.73 96.76 3.75 0.92 0.10 939.55 99.30 4.00 0.94 0.10 957.38 101.85 4.25 0.96 0.10 975.20 104.39 4.50 0.98 0.11 993.02 106.94 4.75 0.99 0.11 1010.85 109.49 5.00 1.01 0.11 1028.67 112.03 5.25 1.03 0.11 1046.49 114.58 5.50 1.05 0.12 1064.32 117.13 5.75 1.06 0.12 1082.14 119.67 6.00 1.08 0.12 1099.96 122.22 6.25 1.10 0.12 1117.79 124.76 6.50 1.12 0.13 1135.61 127.31 6.75 1.13 0.13 1153.43 129.86 7.00 1.15 0.13 1171.26 132.40 The above chart not only provides the coefficients of lift and drag but then calculates the values of Lift and Drag for various angles of attack. These are useful figures for checking the work in Activity 3. Activity 4 gives the students a chance to follow their own interests. They will be given limited time in class to work on the project during the project period. This will be an individual assignment, although I would allow some teaming on a worthwhile large project. Students will be free to work on it after their daily work is completed and I will give them two or three days during the project period along with the day in the media center. Note: While this lesson is designed for Algebra 1 students, it could easily be modified for Algebra 2 and Pre-Calculus students by altering the linear regression to either quadratic regression or exponential regression. NEH Workshop Lesson Plan R. Messick Teacher Lesson Plan Page 9 of 9 Drawing from original Wright patent on the Wright Flyer NEH Workshop Lesson Plan R. Messick Student Handout Page 1 of 13 Will it fly? The mathematics behind invention STUDENT HANDOUT Kitty Hawk, North Carolina December 17, 1903 The temperature was 34 with winds between 20 and 30 miles per hour. The wind chill was about 8 as Wilber and Orville Wright hauled their “aeroplane” from the shed. There was ice and puddles of water on the dusty dunes but the brothers were determined to fly. They had already run into delays waiting on parts and they were running out of time. They had promised to be home in Dayton, Ohio in time for Christmas. Despite the less than favorable conditions, they were confident they would succeed. Part of the reason for this confidence was that they had done the math. They knew their plane would fly. At about 10:30am on December 17, 1903, success! With Orville at the controls, the plane left the ramp and flew, 125 feet in 12 seconds. Three other flights were made that day. After the fourth flight, a gust of wind blew the Wright Flyer over and a wing was damaged. The Wright Brothers returned to Dayton, Ohio and continued development of airplanes. For more information about the Wright Brothers and early aviation, check the sources at the end of this handout or visit your library. “I would hardly think today of making my first flight on a strange machine in a 27-mile wind…I look with amazement upon our audacity in attempting flights with a new and untried machine under such circumstances. Yet faith in our calculations and the design of the first machine, based upon our tables of air pressures, secured by months of careful laboratory work, and confidence in our system of control…had convinced us that the machine was capable of lifting and maintaining itself in the air…” -Orville Wright NEH Workshop Lesson Plan R. Messick Student Handout Page 2 of 13 The Basics of Flight The four basic forces of flight: LIFT – A force acting perpendicular to the flight path that must overcome weight. WEIGHT- The force to due earth’s gravity. (The actual weight of the plane, crew and equipment) THRUST- A force acting in the direction of the flight path. DRAG- A force acting opposite to the direction of the flight path, caused by air friction and pressure distribution. To achieve flight, Lift must be greater than Weight and Thrust must be greater than Drag. Lift > Weight Thrust > Drag Even at the time the Wright Brothers were studying aeronautics, the formulas for Lift and Drag were well established L = kSV2cL D = kSV2cD Where L = Lift generated in pounds D = Drag generated in pounds k = Smeaton coefficient of air pressure (a constant) S = Surface area of the airfoil V = Velocity relative to the wind in miles per hour cL = Coefficient of lift cD = Coefficient of drag NEH Workshop Lesson Plan R. Messick Student Handout Page 3 of 13 Smeaton Coefficient John Smeaton was a British engineer who studied water and wind mills during the mid 18th Century. In 1759, he published a paper on his findings. In an appendix to that paper he included a value called the Smeaton coefficient which when multiplied by the square of the velocity gives the pressure in pounds per square foot on any flat surface presented at right angles to the wind. The number was determined by Smeaton to be a constant with a value of approximately 0.00492, which was rounded to 0.005 for ease of calculation. This was the value of the Smeaton coefficient that the Wright Brothers used in their calculations in 1900 and 1901. Airfoil and Angle of Attack An airfoil is any part of an aircraft that produces lift. The wing is the primary airfoil but propellers and tail surfaces can be airfoils as well. The gray area in the cutaway above shows an airfoil and its parts. The leading edge is the front of the airfoil that meets the air first. The trailing edge is the back of the airfoil where the airflow over the top of the airfoil meets the airflow from the bottom of the airfoil. The chord is the imaginary line that joins the leading edge to the trailing edge. The camber of an airfoil is the curve of its upper and lower surfaces. The upper camber refers to the upper curve of the airfoil while the lower camber refers to the lower curve. Cambers are measured by their distance from the chord. The direction of the air flowing past an airfoil’s leading edge is called the relative wind. The relative wind is always parallel and in the opposite direction of the flight path. The angle of attack is the angle at which relative wind meets an airfoil’s leading edge. An increase in the angle of attack increases lift. However, if the angle of attack is too large, it can decrease lift and at a critical point can cause an aircraft to stall. A stall is a condition where air no longer flows smoothly over the upper surface and lift is reduced. The angle of attack also causes a force backward (drag). NEH Workshop Lesson Plan R. Messick Student Handout Page 4 of 13 Coefficient of Lift and Drag The Wright Brothers’ interest in flying began from a small helicopter toy that their father had given them when they were children. Their interest intensified when they read of the death of the German hang glider and aeronautics authority Otto Lilienthal. Lilienthal was killed in a glider accident in 1896. He had pioneered the study of aeronautics by using piloted gliders to develop tables of the coefficients of lift and drag. It was Lilienthal’s data that the Wrights used to calculate the lift and drag of their first gliders in 1900 and 1901. The brothers also corresponded with Octave Chanute, a civil engineer, who was in his sixties when he began to study aviation. He designed and built gliders and provided the Wrights with encouragement and practical advice. In 1900, the Wrights built a large glider to test their theories about flying. They chose Kitty Hawk, North Carolina along the coast because of its strong sustained winds, soft sand for landing, few curious onlookers and even fewer trees. They built a second glider in 1901 and again went to Kitty Hawk to test the new glider. The 1901 glider did not do well. It did not produce near the lift that the Wrights had expected from their calculations. Their final test flight of 1901 ended in a crash with Wilber sustaining some bruises and a black eye. On the train ride back to Dayton, Ohio, Wilber, frustrated from the results of their tests, remarked that man would not learn to fly within their lifetimes. The Wrights considered abandoning their aviation investigations. Had it not been for an invitation from Octave Chanute for Wilber to speak at a conference for the Western Society of Engineers, the Wright Brothers may have given up on flying. The invitation was waiting for Wilber when the brothers arrived home in Dayton following the disappointments at Kitty Hawk in 1901. Wilber accepted the invitation and used his presentation to call into question the data on coefficients of lift that Lilienthal had published. The conference was the spark that the brothers needed to continue their work. Convinced that Lilienthal’s data was flawed, the brothers built a wind tunnel, one of the first in the United States, and began an impressive series of experiments and calculations. They developed their own tables of coefficients of lift and drag. Based on these experiments they build a new glider and returned to Kitty Hawk for flight tests in 1902. While much is made of their first successful flight, it is their wind tunnel experiments which transform the brothers from aviation enthusiasts into aeronautical engineers. The success of the 1903 Wright Flyer is a direct result of their research. The Wrights discover that Lilienthal’s work was essentially correct. The flaw was in the Smeaton constant. The accepted value of the constant was 0.005. The Wrights determined from their research that the value was actually closer to 0.0033. Today, aeronautical engineers use the value of 0.00339. Since Lilienthal had used the accepted value of 0.005, his data appeared to be in error. With their new value for Smeaton’s constant, the Wrights were ready to fly. NEH Workshop Lesson Plan R. Messick Student Handout Page 5 of 13 Activity 1: Calculating Lift and Drag The following data is from wind tunnel experiments conducted by NASA’s Ames Research Laboratory on a replica of the 1903 Wright Flyer. The replica was built and flown to celebrate the centennial of the Wright Brothers’ First Flight. Angle of Coefficient Coefficient Attack Velocity of Lift of Drag Calculated Lift Calculated Drag -4.0013 25.18 0.3342 0.0421 356.54 44.92 -4.0013 24.98 0.3158 0.0458 331.60 48.13 -3.9997 24.93 0.3900 0.0437 407.96 45.69 -2.0006 25.08 0.4767 0.0472 504.45 49.92 -1.9998 25.07 0.5429 0.0491 574.21 51.94 -1.9990 24.98 0.5089 0.0469 534.43 49.21 -0.0048 24.83 0.7021 0.0619 728.66 64.21 -0.0009 25.08 0.6313 0.0703 668.16 74.36 -0.0009 24.93 0.6460 0.0554 675.74 57.98 0.0006 24.98 0.6999 0.0626 734.98 65.78 0.0006 25.19 0.6848 0.0613 731.05 65.47 0.0006 24.93 0.6773 0.0569 708.53 59.55 0.0006 25.07 0.6672 0.0573 705.47 60.62 0.0006 25.03 0.6635 0.0576 699.57 60.73 0.0006 24.99 0.6430 0.0558 675.74 58.69 0.0006 25.09 0.6352 0.0574 672.92 60.78 1.9983 25.18 0.8267 0.0807 882.11 86.05 1.9983 25.08 0.8183 0.0752 865.95 79.62 1.9983 25.14 0.7894 0.0705 839.50 75.03 3.9984 25.03 0.9332 0.0958 983.80 101.04 3.9991 25.14 0.9595 0.1052 1020.38 111.92 3.9991 25.17 0.9411 0.0989 1003.57 105.47 5.9985 24.82 1.0910 0.1292 1130.91 133.89 5.9992 24.94 1.0931 0.1316 1144.36 137.77 Here’s something to think about as you’re doing these calculations: The Wright Brothers had no computers or calculators to help them. NEH Workshop Lesson Plan R. Messick Student Handout Page 6 of 13 Part 1: Calculating Lift Use the formula for Lift: L = kSV2cL to calculate the Lift for each of the angle of attacks in the table on page #5. Round your answer to the nearest hundredth and write it on the table in the fifth column. Part 2: Calculating Drag Use the formula for Drag: D = kSV2cD to calculate the Drag for each of the angle of attacks in the table on page #5. Round your answer to the nearest hundredth and write it on the table in the sixth column. k = Smeaton’s constant = 0.0033 S = Surface Area of the airfoil = 510 (for the 1903 Wright Flyer) V = Velocity (See table for value) cL = Coefficient of lift (See table for value) cD = Coefficient of drag (See table for value) Wright Brothers 1902 Glider NEH Workshop Lesson Plan R. Messick Student Handout Page 7 of 13 Activity 2: Determining Angle of Attack for the First Flight Part 1: Inputting Lists Using your graphing calculator input five lists. For L1, input the Angles of Attack from the table on page 5. For L2, input the Calculated Lifts from the table on page 5. For L3, input the Calculated Drags from the table on page 5. For L4, input the Coefficient of Lift from the table. For L5, input the coefficient of Drag from the table. Part 2: Finding the Linear Regression of Lift by Angle of Attack Once your lists have completed inputting your three lists, use the STAT CALC menu to find the slope and y-intercept of the line of best fit for the data in lists L1 and L2[LinReg (y=ax+b)]. Key press sequence on a TI-83/4 to find Linear Regression A) Write the equation of the line of best fit in the space below. Round slope and y-intercept to the nearest hundredth. y = ______ x + ________ SLOPE (A) Y-INTECEPT (B) Note: y is the dependent variable (Lift) and x is the independent variable (angle of attack) Recall that in order to fly, Lift > Weight. The weight of the 1903 Wright Flyer was approximately 605 pounds. Add an additional 145 pounds for Orville and the total weight of the Wright Flyer was about 750 pounds. Therefore, Lift > 750 pounds. B) From the equation in A) above, we know that Lift = (a) x + b, so to find the minimum angle of attack for a successful flight, we need to solve the inequality: 750 < ______ x + __________ SLOPE (A) Y-INTERCEPT (B) C) Solve this inequality to find the minimum angle of attack for a successful flight of the 1903 Flyer. Round your answer to the nearest hundredth. NEH Workshop Lesson Plan R. Messick Student Handout Page 8 of 13 Part 3: Finding the Linear Regression of Drag by Angle of Attack Next we will again use the STAT CALC menu to find the slope and y-intercept of the line of best fit for the data in lists L1 and L3 [LinReg (y=ax+b)]. Key press sequence on a TI-83/4 to find Linear Regression A) Write the equation of the line of best fit in the space below. Round slope and y-intercept to the nearest hundredth. y = ______ x + ________ SLOPE (A) Y-INTECEPT (B) Note: y is the dependent variable (Drag) and x is the independent variable (angle of attack) Recall that in order to fly, Thrust > Drag. The engine on the 1903 Wright Flyer was designed and built by the Wright Brothers and machinist Charlie Taylor. The engine weighed approximately 125 pounds and produced around 8 horsepower. The Wrights calculated the thrust of the engine to be 132 pounds in November 1903. So, Drag < 132 pounds. B) From the equation in A) above, we know that Drag = (a) x + b, so to find the maximum angle of attack for a successful flight, we need to solve the inequality: ______ x + __________ < 132 SLOPE (A) Y-INTERCEPT (B) C) Solve this inequality to find the maximum angle of attack for a successful flight of the 1903 Flyer. Round your answer to the nearest hundredth. Replica of the Wright Wind Tunnel NEH Workshop Lesson Plan R. Messick Student Handout Page 9 of 13 Part Four: Selecting an Angle of Attack and Finding the Coefficients of Lift and Drag A) In Part 2 C), you determined the minimum angle of attack to maintain lift. In Part 3 C), you determined the maximum angle of attack to maintain thrust. The angle of attack for the First Flight must be in the interval between these minimum and maximum values. Choose a value for an angle of attack between the minimum and maximum values and write it in the space below. B) In order to determine the lift for the angle of attack which you chose in A), we must first determine the coefficient of lift for this angle of attack. We will again use the STAT CALC menu to find the slope and y-intercept of the line of best fit for the data in lists L1 and L4 [LinReg (y=ax+b)]. Key press sequence on a TI-83/4 to find Linear Regression Write the equation of the line of best fit in the space below. Round slope and y-intercept to the nearest hundredth. y = ______ x + ________ SLOPE (A) Y-INTECEPT (B) Note: y is the dependent variable (coefficient of Lift) and x is the independent variable (angle of attack) C) Using the equation in B) above, substitute your angle of attack from A) in place of x and solve for y. The solution for y is the coefficient of lift for your angle of attack. Original Wright Brothers’ Notebook with calculations. NEH Workshop Lesson Plan R. Messick Student Handout Page 10 of 13 D) In order to determine the drag for the angle of attack which you chose in A), we must first determine the coefficient of drag for this angle of attack. Use the STAT CALC menu to find the slope and y-intercept of the line of best fit for the data in lists L1 and L5 [LinReg (y=ax+b)]. Key press sequence on a TI-83/4 to find Linear Regression Write the equation of the line of best fit in the space below. Round slope and y-intercept to the nearest hundredth. y = ______ x + ________ SLOPE (A) Y-INTECEPT (B) Note: y is the dependent variable (coefficient of Drag) and x is the independent variable (angle of attack) E) Using the equation in B) above, substitute your angle of attack from A) in place of x and solve for y. The solution for y is the coefficient of drag for your angle of attack. Orville and Wilber Wright NEH Workshop Lesson Plan R. Messick Student Handout Page 11 of 13 Activity 3: Will it fly? (Putting it all together) Here’s your chance to soar! In Activity 2, you chose an angle of attack and determined the coefficients of both lift and drag for that angle of attack. Write those figures in the spaces below. Angle of Attack: Coefficient of Lift: ( cL) Coefficient of Drag: ( cD) The other information you’ll need: k (Smeaton’s constant) = .0033 S (Surface area of the airfoil) = 510 V (Velocity) = 24.6 Weight of the aircraft and Orville = 750 Thrust of the Wrights’ engine = 132 1) Using the information above, calculate Lift. Remember: L = kSV2cL 2) In order to fly, Lift > Weight. Write an inequality that compares the calculated lift and the weight given above. Is Lift > Weight? 3) Using the information above, calculate Drag. Remember: D = kSV2cD 4) In order to fly, Thrust > Drag. Write an inequality that compares the calculated drag and the thrust given above. Is Thrust > Drag? 5) Based on your calculations, will it fly? NEH Workshop Lesson Plan R. Messick Student Handout Page 12 of 13 “Although a general invitation had been extended to the people living within five or six miles, not many were willing to face the rigors of a cold December wind in order to see, as they no doubt thought, another flying- machine not fly. The first flight lasted only twelve seconds, a flight very modest compared with that of birds, but it was, nevertheless, the first in the history of the world in which a machine carrying a man had raised itself by its own power into the air in free flight, and sailed forward on a level course without reduction of speed, and had finally landed without being wrecked.” -Orville & Wilber Wright from “The Wright Brothers Aeroplane” as published in Century Magazine, September 1908. “To invent an airplane is nothing. To build one is something. But to fly is everything.” ~Otto Lilienthal Sources and Suggested Readings: Crouch, Tom D. First Flight: The Wright Brothers and the Invention of the Airplane. National Park Service. Division of Publications. 2002. Engler, Nick. Lift and Drill: The Story of The Wright Brothers Wind Tunnel. Wright Brothers Aeroplane Company. www.first-to-fly.com. Kelly, Fred C. The Wright Brothers: A Biography. Dover. 1989. Kelly, Fred C. Orville Wright: How We Invented the Airplane. Dover. 1953. McCredie, Patty. The Wright Brothers Adventure. NASA. McCullough, Robert N. The Wright Stuff: The Mathematics of the Wright Brothers. Wright State University. www.libraries.wright.edu. Verma, Shilpi. Can the Wright Flyer Handle It? NASA. www.quest.arc.nasa.gov. Wright Brothers Wind Tunnel Test Data. NASA. www.quest.arc.nasa.gov. 1999. NEH Workshop Lesson Plan R. Messick Student Handout Page 13 of 13 Activity 4: Your Turn Choose an inventor or an invention in which you have an interest. Research the inventor or invention to see what role mathematics played in the inventive process. Then, take your research and either 1) Write a 3 to 5 page essay 2) Prepare and present a 3 to 5 minute speech 3) Design a PowerPoint presentation (5 slide minimum) 4) Design a poster or diorama 5) Find some other creative way to convey the information You should include material from three different sources, only two of which can be internet sources. Every attempt should be made to make at least one a primary source. Remember the focus of the project is the role of mathematics in the inventive process. Don’t just describe the inventor and invention. Explain how mathematics was used to create the invention. We will spend one class period in the Media Center to get you started and you will have three weeks to complete the project. If you need help or have questions, ask. This assignment is worth 100 points and is equivalent to a test score. NEH Workshop Lesson Plan R. Messick