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(2) If an airplane must fly a 600-mile course, which is 350° in 4 hours, while a 40-mph. wind is blowing from 25°, what must its air speed be?

AS=184, H=357° Answer. (3) If an airplane with an air speed of 310 mph. must fly a 200° course in a 35-mph. wind blowing from 180°, what must its heading be, and what will its ground speed be?

(4) An airplane's air speed is 160 mph., its heading 10°. The wind velocity is 28 mph. from 116o. What are the track and ground speed?

GS=170, T=1° Answer. (5) An airplane with an air speed of 130 mph. must fly a true course of 127° in a 25-mph. wind from 221°. What must be the heading, and what will the ground speed be?

(6) An airplane with an air speed of 150 mph. and a heading of 65° flies a course of 80° with a ground speed of 140 mph. Find the wind velocity.

WS=40, WD=357° Answer. (7) An airplane's air speed indicator reads 205 mph, and its compass 67o. The navigator notes that the airplane passes over two landmarks which are given on his map. The distance between them is 310 miles and the direction of the line between them is 76°. If the airplane took 1 hour, 40 minutes to fly from one to another, what is the wind velocity?

(8) An airplane with an air speed of 245 mph. must fly a course of 310° and return to the same base along the same line. A 280° wind is blowing 28 mph. How long should the pilot fly out along the line if he has 3 hours' fuel? What heading must he take in and out?

T=1 hr. 39 min.; H in=133°; H out=307° Answer. (9) A pilot flying an airplane which has an air speed of 200 mph. takes off from field A to fly to field B which lies 800 miles and 315° from field A. A wind of 30 mph. is blowing from 0°. The pilot miscalculates his course and lands at field C to get his bearing. C is 565 miles from both A and B, and is due north of A, and due east from B. How much time did he lose by not flying a direct line?

49. Miscellaneous exercises.-(1) If an airplane with an air speed of 160 mph, and a heading of 312° flies in a 35-mph. wind from 20°, what are the track and ground speed?

(2) An airplane with an air speed of 135 mph must fly a course of 93° in a 30-mph. wind from 225°. What must be the heading, and what will the ground speed be?

H=102°; GS=153 Answer. (3) An airplane's heading is 35o. A 25-mph. wind is blowing 95°. If the airplane's air speed is 120 mph., what are its ground speed and track?

(4) An airplane's air speed is 210 mph. A 45-mph, wind is blowing 1950. If the airplane must fly a course of 248°, what must the heading be and what will the ground speed be? H=238°; GS=180 Answer.

(5) An airplane must fly a 75° course and maintain a ground speed of 165 mph. A 40-mph. wind is blowing from 160°. What must the air speed and heading be?

(6) In a 30-mph. wind from 350°, an airplane flies with a heading of 235° and an air speed of 170 mph. What are the track and ground speed?

T=226°; GS=185 Answer. (7) An airplane must fly a 23° course in a 25-mph. wind from 272o. If the airplane's air speed is 140 mph., what must the heading be? What will the ground speed be?

(8) An airplane must fly from A to B in 4 hours 30 minutes. These fields are 630 miles apart and the direction of the line between them is 210°. If the wind is 32 mph. from 105°, what must the heading and air speed be?

H=196°; AS=136 Answer. (9) An airplane with an air speed of 135 mph. and a heading of 355° passes over a landmark, and 1 hour 20 minutes later passes over another which is 200 miles and 10° from the other. What is the wind velocity?

(10) An airplane with an air speed of 170 mph. and a heading of 177° flies directly over a road the direction of which is 165o. The pilot sees smoke blowing from a chimney from 275o. What are the ground speed and wind speed?

GS=178; WS=37 Answer. (11) An airplane with an air speed of 190 mph. must fly along a course of 310° and return along the same line. Total flying time is to be 2 hours 30 minutes. If the wind is 40 mph. from 190°, how long should the pilot fly out along the line before turning back?

(12) A pilot flying an airplane with an air speed of 175 mph. must land at either field A or field B as soon as possible. Field A is 117 miles and 30° from the airplane, while field B is 141 miles and 195° from the airplane. If the wind velocity is 40 mph. from 335°, to which field should the pilot fly? Arrives at B 6 minutes sooner. Answer.

(13) An airplane with an air speed of 150 mph. takes off from a field and flies with a heading of 305o. A 35-mph. wind is blowing from 200°. Forty minutes later an airplane with an air speed of 190 mph. takes off from the same field to overtake the first airplane. What heading must the pilot fly, and how long will it take him to overtake the other airplane?

SECTION VIII

60

PLANE GEOMETRY

Paragraph Purpose and scope.

50 Basic definitions--

51 Elements of plane geometry --

52 Constructions with ruler and compass-

53 Special properties of triangles -

54 Relationships between two triangles-

55 Special properties of circles...

56 Special properties of quadrilaterals.

57 Special properties of miscellaneous figures

58 Solid geometry-definitions and properties of some geometric solids..

59 Intersection of a sphere and a plane---

50. Purpose and scope.—The purpose of this section and the one following is to provide a working knowledge of the properties of certain geometric figures. Certain useful ruler-compass constructions are also to be included.

51. Basic definitions.-a. Plane geometry.That branch of mathematics which deals with figures on flat or plane surfaces.

b. Axiom.-A mathematical statement whose truth is accepted without proof.

Examples: (1) If equals are added to equals, the sums are equal.

(2) If equals are multiplied by equals, the products are equal. These axioms are applicable to many branches of mathematics.

c. Postulate.-A geometric statement accepted to be true without proof.

Examples: (1) Two straight lines cannot intersect at more than one point.

(2) Two lines in the same plane must either be parallel or they must intersect. Postulates apply specifically to geometry.

d. Proposition.—A statement of a geometric truth. e. Theorem.-A proposition to be proved.

Example: The sum of the angles of a triangle is equal to 180°. A proof is the logical argument which shows a proposition is true.

f. Corollary.A theorem that is similar in content to a previously proved theorem. A corollary often follows as an added conclusion to a proved theorem with very little added proof. Example:

Theorem: If a triangle is isosceles, the angles opposite the equal

sides are equal.
Corollary: An equilateral triangle is equiangular.

52. Elements of plane geometry.a. Point and line.--The two undefined elements of plane geometry. A line has no definite length. A segment is a portion of a line with a definite length.

b. Straight line. A line in a certain direction.

c. Broken line. A line made up of straight line segments, not all of which are necessarily in the same direction.

d. Curved line.— A line no portion of which is a straight line segment.

e. Parallel lines (1/1).- Are lines that lie in the same plane and do not meet however far extended..

f. Angle.The opening formed by two straight lines drawn from the same point, called the vertex.

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(1) The angle is described as angle BAC (ZBAC), 2 CAB, or 21. The vertex of the angle is A. AC and BA are the sides. The size of an angle does not depend on the length of the sides, but on how much side AB must revolve about A to take the position of side AC.

(2) In labeling an angle the letter at the vertex must be in the middle. Thus, ZBAC is correct and Z ABC is incorrect.

g. Angles, classification of.-Angles may be classified as follows: (1) Acute angle.- An angle more than 0° and less than 90°.

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(2) Right angle.-An angle of 90°.
(3) Obtuse angle.-An angle more than 90° and less than 180°.
(4) Reflex angle.-An angle more than 180° and less than 360°.
(5) Straight angle.-An angle of 180°.

h. Perpendicular.-If two lines meet at right angles, they are said to be perpendicular.

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i. Polygon.— Any closed figure bounded by line segments. *

FIGURE 65.-Triangle-a polygon of three sides.

FIGURE 66.—Quadrilateral- A polygon of four sides.

j. Circle.-A closed curve, all points of which are equally distant from a point within, called the center.

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(1) Radius.-A line segment from the center to the circle.

(2) Diameter.--A line segment through the center terminated by the circle.

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