599 9 A Look Back, A Look Ahead This chapter is in two parts: Polar Coordinates (Sections 9.1—9.3) and Vectors (Sections 9.4—9.7). They are independent of each other and may be covered in either order. Sections 9.1—9.3: In Chapter 1, we introduced rectangular coordinates (the xy -plane) and discussed the graph of an equation in two variables involving x and y . In Sections 9.1 and 9.2, we introduce polar coordinates, an alternative to rectangular coordinates, and discuss graphing equations that involve polar coordinates. In Section 5.3, we discussed raising a real number to a real power. In Section 9.3, we extend this idea by raising a complex number to a real power which is easily accomplished using polar coordinates. Sections 9.4—9.7: In previous chapters we have developed skills for finding solutions to many different types of equations, which is essential in solving applied problems. In the last four sections of this chapter, we develop the notion of a vector and show how it can be used to model applied problems in physics and engineering. Outline 9. 1 Polar Coordinates 9. 2 Polar Equations and Graphs 9. 3 The Complex Plane; De Moivre’s Theorem 9. 4 Vectors 9. 5 The Dot Product 9. 6 Vectors in Space 9. 7 The Cross Product Chapter Review Chapter Test Cumulative Review Chapter Project Polar Coordinates; Vectors How Airplanes Fly Four aerodynamic forces act on an airplane in flight: lift , drag , thrust , and weight (gravity). Drag is the resistance of air molecules hitting the airplane (the backward force), thrust is the power of the airplane’s engine (the forward force), lift is the upward force, and weight is the downward force. So for airplanes to fly and stay airborne, the thrust must be greater than the drag, and the lift must be greater than the weight. This is certainly the case when an airplane takes off or climbs. However, when it is in straight and level flight, the opposing forces of lift, and weight are balanced. During a descent, weight exceeds lift, and to slow the airplane, drag has to overcome thrust. Thrust is generated by the airplane’s engine (propeller or jet), weight is created by the natural force of gravity acting on the airplane, and drag comes from friction as the plane moves through air molecules. Drag is also a reaction to lift, and this lift must be generated by the airplane in flight. This is done by the wings of the airplane. A cross section of a typical airplane wing shows the top surface to be more curved than the bottom surface. This shaped profile is called an airfoil (or aerofoil), and the shape is used because an airfoil generates significantly more lift than opposing drag. In other words, it is very efficient at generating lift. During flight, air naturally flows over and beneath the wing and is deflected upward over the top surface and downward beneath the lower surface. Any difference in deflection causes a difference in air pressure (pressure gradient), and because of the airfoil shape, the pressure of the deflected air is lower above the airfoil than below it. As a result the wing is “pushed” upward by the higher pressure beneath, or, you can argue, it is “sucked” upward by the lower pressure above. Source : Adapted from Pete Carpenter. How Airplanes Fly—The Basic Principles of Flight, http://www.rcairplane-world.com/how-airplanes-fly.html, accessed June 2019. © rc-airplane-world.com —See Chapter Project— Credit: kesu87/123RF
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