will it fly combined 1 by oozEHOK

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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

								
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