36 Graphic solution of algebraic equations containing two unknowns_ 32. Purpose.-a. Graphs are used to represent pictorially the relationship between two quantities, that is, how one quantity varies with another quantity. The following example shows how the pressure of a gas varies with the volume at a temperature of 0° C: b. Graphs are used extensively to portray certain physical relations. Once the graph has been drawn, it can be used to make rapid approximate computations, thus saving valuable time and work. For this reason, the graph is of great value in aeronautics. c. To interpret fully the graphic process, it is necessary to understand the construction of graphs. In illustrating their construction, the procedure used by mathematicians will be presented briefly and applied to aeronautical information. 33. Axes and points.-a. Begin by taking two known straight lines intersecting at right angles in the same plane. The point of intersection is called the origin, O. The straight lines OX and OY are reference lines and are known as rectangular axes. b. The horizontal line is called the X-axis. The vertical line OY is called the Y-axis. c. The position of any point is fixed by reference to these two axes. Thus, the point P (fig. 27) is located by describing it as so many units horizontally from OY (in the X-direction) and so many units vertically from OX (in the Y-direction). In any graph, certain units are marked off on the X-axis, and the same or different units are marked off on the Y-axis. (See figs. 28 and 29.) d. It is to be noted that in going from OY to the right in the Xdirection, the numbers are positive, signifying the positive direction; while to the left the numbers are negative, signifying the negative direction. On the other hand, going from OX upward in the Y-direction, the numbers are positive, signifying the positive direction; while downward the numbers are negative, signifying the negative direction. Hence it takes two numbers, the coordinates, to locate a single point: one of them the Y-value, denoting the vertical distance, and the other, the X-value, denoting the horizontal distance. The word ordinate is sometimes used to designate the Y-value, and the word abscissa is sometimes used to designate the X-value. In figure 28, the point R is located as being 4 units horizontally from OY (4) units in the X-direction or X=4) and 5 units vertically from OX (5 units in the Y-direction or Y=5). Symbolically, the point is described by its coordinates (4, 5) meaning X-4, and Y=5. The coordinates of the point S are (-3, 1) meaning X=-3, and Y=1: that is, 3 units to the left of OY in the X-direction and 1 unit above OX in the Y-direction. e. This practice of plotting the position of a point by coordinate axes is used in locating the latitude and longitude of a point in a Mercator Chart. 34. Reading graphs.-To illustrate the procedure, suppose it is desired to know the calibrated air speed corresponding to an indicated air speed of 150 mph on the meter the calibration curve of which is shown in figure 30. From "150" on the horizontal axis, move up air speed in miles per hour. All graphs are read by following this general procedure, although occasionally some graphs may be complicated by several curves and multiple scales. a. Air speed meter calibration graph.-This graph (fig. 30) is based upon an air speed meter calibration of an airplane. (1) If the indicated air speed is 110 mph, find the calibrated air speed. (2) If the indicated air speed is 165 mph, find the calibrated air speed. 173 mph. Answer. (3) If the calibrated air speed is 198 mph, find the indicated air speed. b. Pressure-temperature graph.-This graph (fig. 31) shows the relationship between the pressure and the temperature of a gas at constant volume. (1) At a temperature of 0° C., the pressure is.. (2) If the temperature increases, the pressure. (3) If the pressure is decreased, the temperature of the gas(4) At a temperature of 100° C., the pressure is (5) If the pressure is 30 pounds/square inch, the temperature is c. 24-hour system.-The 24-hour system for stating time eliminates the use of the abbreviations AM and PM. The values for AM time are unchanged except that four figures are always used. For example, 9:15 AM becomes 0915 hour; 4:45 AM becomes 0445 hour, and 11:48. AM becomes 1148 hour. The values for PM time are increased by 1200, hence 1:30 PM becomes 1330 hour and 5:55 PM becomes 1755 hour. The use of this system decreases the chances for making errors and for this reason it has been adopted for use in the U. S. Army Air Forces. d. Sunset graphs.-These graphs enable the pilot to determine the time of sunset for any position on the earth. The 24-hour system of keeping time is used in sunset graphs. (1) Instructions for use. (a) Enter the top or bottom scale with proper date. (b) Move vertically down or up to the curve for observer's latitude (observer's position). (c) Move horizontally to the right or left and read local civil time of sunset on vertical scales at the side. (2) Find the sunset time for November 1 at latitude 30° N. (Follow instructions for use.) 1721 hr. Answer. (3) Find the sunset time for May 15 at latitude 50° N. (4) Find the sunset time for May 20 at latitude 30° N. 1848 hr. Answer. (5) Find the sunset time for June 10 at latitude 10° N. 1736 hr. Answer. (7) Find the sunset time for October 20 at latitude 30° N. e. Fuel consumption graph.-This graph (fig. 33) shows the relationship between the air speed and the fuel consumption. (1) At an air speed of 180 mph., the fuel consumption is gallons/hour. (2) At an air speed of 168 mph., the fuel consumption is gallons/hour. (3) If the fuel consumption is 53 gallons/hour, the air speed is mph. |