Learning to Fly Denise Russo September 17, 2010
Content Area: Trigonometry Grade Level: 11-12 Date: Sept. 17, 2010 Text Selection: Trigonometry Section 7.5: Vectors Author(s): Margaret L. Lial, Charles D. Miller Audience: The audience for this lesson is a class of 30 eleventh grade students in any metro Detroit high school trigonometry class. Vector analysis is typically introduced toward the end of a trigonometry course, so it is assumed that the students in this classroom are comfortable with the trig functions and solving triangles using the law of sines/cosines. Content: This lesson is an introduction to vector analysis with an application in the area of flight science. Students will learn the concepts of vectors, scalars, magnitude, direction, and vector addition. The concepts will be applied as the students learn to fly by plotting a flight course on the aeronautical sectional and constructing and solving a wind triangle. Michigan Merit Curriculum Content Expectations: Mathematics Strand 1: Quantitative Literacy and Logic L1.2.3: Use vectors to represent quantities that have magnitude and direction, interpret direction and magnitude of a vector numerically, and calculate the sum and difference of two vectors. Mathematics Strand 3: Geometry and Trigonometry G1.3.2: Know and use the Law of Sines and the Law of Cosines and use them to solve problems. Find the area of a triangle with sides a and b and included angle θ using the formula Area = (1/2) absin θ. National Mathematics Standards (set by NCTM): Use trigonometric relationships to determine lengths and angle measures. Understand vectors and matrices as systems that have some of the properties of the real-number system. Develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices. Goals: The goals of this lesson are: Improve student problem solving ability using applied mathematics. Prepare students to make logical well-reasoned life choices.
Objectives: At the end of this lesson, students should be able to: Understand that a vector quantity has magnitude and direction. Differentiate between vector and scalar quantities. Use the head-to-tail method to add two vectors. Apply the properties of vectors to solve problems in the context of velocity and force. Materials: Trigonometry by Margaret L. Lial, Charles D. Miller Aeronautical sectional map Protractor Flight planning worksheet Vector addition worksheet Introduction: How will I ever use this stuff? Poll the students to determinee if anyone knows anything about vectors. Lead a short discussion about what the students think vectors are and how they are used. Poll the students to determine if anyone has ever flown in a small airplane. If there are any private pilots in the audience (this is quite possible), ask them if they have ever done a flight plan. Explain to the class that we will be learning to fly. Development: What is a vector? The purpose of the lesson is to learn and understand vectors and applications of vectors. Why are vectors important? Vectors are used to solve problems involving force, velocity, and acceleration. Other applications include accident investigation/reconstruction, electronics, and all types of engineering. A brief discussion defining vectors precedes the class activity. A scalar quantity has magnitude. Distance, mass, and speed are examples of scalar quantities. A vector quantity has magnitude and direction. Velocity, force, and acceleration are examples of vector quantities. A vector quantity is represented by a directed line segment (an arrow). The length of a vector represents its magnitude. The direction of a vector is its position relative to a given coordinate system. Figures 1 and 2 show examples of vector magnitude and direction. Figure 1: Vector Magnitude
Figure 2: Vector Direction Two vectors are equal if and only if they have the same magnitude and direction. The vectors in Figure 3 have the same magnitude but different directions, therefore, they are not equal. Figure 3: Unequal Vectors Vector addition is commutative; that is, A+B = B+A. Two vectors are addedd using the head-to-tail method, as shown in Figure 4. Figure 4: Vector Addition Guided Practice: Learning to Fly Explain to the class that we will be plotting a course between two airports and calculating the aircraft heading based on current weather data. Distribute the flight planning worksheet and aeronautical sectional. Have the class look at the sectional map and try to point out some of its features; lines of longitude, airport designations, etc. Following the steps on the flight planning worksheet, demonstrate how to plot a course between Canton-Plymouth Mettetal and Marine City. Construct a wind triangle and use vector concepts along with law of sines/cosines to determine aircraft actual course.
Independent Practice: Becoming a Pilot Divide students into small groups. Have each group create a flight plan between the two cities of their choice on the sectional map. After the group has completed their flight plan and solved their wind triangles, have each group report their results to the class. Have students complete the vector addition worksheet for homework. Checking for Understanding: Are the students getting it? During the group exercise, observe the students and their interactions in the groups. Make sure all the students in each group are comfortable with the activity, and on track. Closure: How else can we use vectors? After the groups have reported their flight plans, lead a class discussion about the important concepts involved in the lesson. Solicit suggestions from the students for other applications of vectors. How else might they use these concepts in everyday life? Evaluation Collect vector addition worksheets and review them to assess the students understanding of the concepts.
Flight Plan 1. Locate Canton-Plymouth Mettetal and Marine City airports on the map. 2. Using a straightedge, draw a light pencil line between the two airports. 3. Locate a line of longitude along the course. Using the protractor, determine the angle between the vertical (north) line and the course. 4. Call 1-800-WXBRIEF to determine the current winds along the flight path. For this exercise, we will assume a wind speed of 17 knots at 10 o. 5. Determine airspeed; this will be in the flight manual for the specific aircraft that you will be flying. For this exercise, we will use 110 knots (knots = nautical miles per hour). 6. Use the data from above to construct a wind triangle and determine the aircraft heading. North (0 o ) c-w Course (C) c w Wind (W) h c h-c Heading (H) Using the law of sines, solve for h (aircraft directional heading): sin h = sin sin h = sin sin h 60 = 17 sin 60 10 110 h=66.8
Vector Addition Your boat is traveling in a river with a strong current. For each case in the table below, find the resultant velocity. Be sure to draw the velocity triangle for each case. Boat Velocity Water Velocity Resultant Velocity Velocity Triangle 15 kts 3 kts 0 o 90 o 5 kts 5 kts 90 o 270 o 10 kts 10 kts 135 o 225 o Challenge Question: As a jet plane takes off, its path makes a fairly steep angle to the ground. The plane itself makes an even steeper angle. Its velocity vector may be resolved into two components, as shown in the figure. The axial component (the one directed along the plane s axis) is the plane s velocity ignoring the action of gravity. The vertical component is the velocity at which the plane is falling under the influence of gravity. The velocity vector is 250 kph at an angle of 10 o to the ground, and the plane s axis makes an angle of 15 o with the ground. What is the axial velocity? At what speed is the plane falling?