Flight Theory and Aerodynamics. Joseph R. Badick

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SCALAR AND VECTOR QUANTITIES

      A quantity that has size or magnitude only is called a scalar quantity. The quantities of mass, time, and temperature are examples of scalar quantities. A quantity that has both magnitude and direction is called a vector quantity. Forces, accelerations, and velocities are examples of vector quantities. Speed is a scalar, but if we consider the direction of the speed, then it is a vector quantity called velocity. If we say an aircraft traveled 100 nm, the distance is a scalar, but if we say an aircraft traveled 100 nm on a heading of 360°, the distance is a vector quantity.

      Scalar Addition

      Scalar quantities can be added (or subtracted) by simple arithmetic. For example, if you have 5 gallons of gas in your car’s tank and you stop at a gas station and top off your tank with 9 gallons more, your tank now holds 14 gallons.

      Vector Addition

Vector addition is more complicated than scalar addition. Vector quantities are conveniently shown by arrows. The length of the arrow represents the magnitude of the quantity, and the orientation of the arrow represents the directional property of the quantity. For example, if we consider the top of this page as representing north and we want to show the velocity of an aircraft flying east at an airspeed of 300 kts., the velocity vector is as shown in Figure 1.3. If there is a 30‐kts. wind from the north, the wind vector is as shown in Figure 1.4.

To find the aircraft’s flight path, groundspeed, and drift angle, we add these two vectors as follows. Place the tail of the wind vector at the head of the arrow of the aircraft vector and draw a straight line from the tail of the aircraft vector to the head of the arrow of the wind vector. This resultant vector represents the path of the aircraft over the ground. The length of the resultant vector represents the groundspeed, and the angle between the aircraft vector and the resultant vector is the drift angle (Figure 1.5).

Figure 1.3 Vector of an eastbound aircraft.

Figure 1.4 Vector of a north wind.

Figure 1.5 Vector addition.

      The groundspeed is the hypotenuse of the right triangle and is found by use of the Pythagorean theorem

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      The drift angle is the angle whose tangent is V_w/V_a/c = 30/300 = 0.1, which is 5.7° to the right (south) of the aircraft heading.

      Vector Resolution

It is often desirable to replace a given vector by two or more other vectors. This is called vector resolution. The resulting vectors are called component vectors of the original vector and, if added vectorially, they will produce the original vector. For example, if an aircraft is in a steady climb, at an airspeed of 200 kts., and the flight path makes a 30° angle with the horizontal, the groundspeed and rate of climb can be found by vector resolution. The flight path and velocity are shown by vector V_a/c in Figure 1.6.

Figure 1.6 Vector of an aircraft in a climb.

Figure 1.7 Vectors of groundspeed and rate of climb.

In Figure 1.7, to resolve the vector V_a/c into a component V_h parallel to the horizontal, which will represent the groundspeed, and a vertical component, V_v, which will represent the rate of climb, we simply draw СКАЧАТЬ