41. Purpose. The purpose of this section is to give the student airman an idea of what a vector is and of how vectors are used in triangle of velocity problems.

42. Scalars and vectors.-a. A scalar is a quantity which has magnitude only. Such quantities can be completely specified by number or letter. Whenever one number completely specifies a quantity, that quantity is a scalar: distance, time, and speed are scalars, as in 6 feet, 30°, 40 mph., or y miles.

b. A Vector is a quantity which has direction as well as magnitude. Thus two numbers must be given to determine a vector, one for magnitude and one for direction. An automobile traveling north at 10 feet/second needs a direction as well as a magnitude to describe its motion, so its velocity (speed and direction) is a vector quantity.

c. Representation of a vector.-Although a vector is determined by two quantities, it can be represented graphically by an arrow having a length which depends on the magnitude of the vector and a direction which is the given direction.

(1) To find the proper length for a vector, a scale must be chosen arbitrarily, and some appropriate length must be selected to represent the given magnitude. A scale should be chosen which will give the best sized diagram. Large diagrams give more accurate results. For example, if 1 inch represents 100 pounds of force, a line 2% inches long would stand for a force of 225 pounds.

(2) The direction is measured clockwise from the north.

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43. Addition of vectors.-a. The sum of two vectors, V1 and V2 is a third vector which is written V3-V1+V2. This is called the resultant of V1 and V2.

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b. The resultant, when applied alone, gives the same result obtained by applying the two original vectors separately.

(1) To add two vectors, V1 and V2, graphically, draw both from a point O, then complete the parallelogram which has V1 and V2 as

two of its sides. An arrow from 0 to the opposite vertex of the parallelogram is the resultant.

(2) Example: To add the vectors V1 and V2:

V3

FIGURE 50.

(V, is the sum or resultant of V1 and V2.)

c. An alternate method of adding vectors is drawing V1=OA from any point, O, and V2=AB from the arrow end of V1, with arrow away from arrow end of V1. Then join O and B by an arrow. V3=OB is the resultant. This method may be used to add any number of vectors. Merely add each vector to the preceding vector. The resultant is obtained by joining the point O to the arrow of the last vector.

(1) Example: To add the vectors V1 and V2:

The dotted lines are unnecessary but are added here to show the similarity to the previous method.

(2) Example: To add the vectors V1, V2, V3, and V1:

NOTE.

FIGURE 52.

The vectors may be added in any order without changing the resultant

as long as the direction and magnitude of each vector are preserved.

d. Exercises:

(1) Tell which of the following are vectors and which are scalars:

(a) 2 pounds.

(b) 20 mph. east.

(c) 10 feet/second.

(d) Acceleration of a falling body.

(e) A velocity.

(2) Graph the following:

(a) 30 mph. NW.

(b) 40 yards/minute north.

(c) 30 pounds force acting N 10° E.

(d) 17 pounds force acting S 50° W.

(e) 99 pounds at 182°.

(f) 14 mph. at 37°.

(g) 18 feet/second at 282°.

(3) Add the following vectors (V1+V2):

(a) V1: Length =340

Azimuth=323°

(b) V1: Length =10

Azimuth 200°

V2: Length =410

Azimuth 20°

V2: Length =15

Azimuth=140°
Length 22, azimuth=163°

Answer.

(4) A car moves at the rate of 40 mph. due north for 2 hours, then turns and goes due east for 1 hour. Find by vector diagram how far in a straight line the car is from its starting point. How much time would have been saved by traveling the straight route?

89.44; .764 hr. or 45.8 min. Answer.

(5) A body is subjected to a force of 50 pounds acting toward 50° and a force of 70 pounds acting toward 340° at the same time. Find the direction and magnitude of a third force that will counterbalance exactly the effect of the first two.

(6) A man heads the bow of his boat directly across a river which is flowing at the rate of 2 mph. If he rows r mph., how much off his course is he at the end of 1 hour? How far does he go in 1 hour?

2 miles; √2+4 Answer.

(7) A man pushes on the handle of a lawn mower with a force of 75 pounds in line with the handle. The handle makes an angle of 30° with the horizontal. Find how much force is

(a) Straight ahead.

(b) Into the ground.

44. Triangle of velocity.-a. The velocity of an object may be defined as a vector quantity the magnitude of which is the speed of the object and the direction of which is the direction of the object's motion. It can be represented by a directed line segment (vector). The velocity of a body which is acted upon by two forces can be found by considering the velocity resulting from each force individually. The actual, or total, velocity of the body is the resultant of the separate velocities. The velocity, relative to the ground, of an airplane in flight is the resultant of two velocity vectors: the air speed and heading vector, and the wind velocity vector. This resultant velocity vector is the course and ground speed vector.

b. The terms used are now defined as follows:

(1) Air speed. The true speed of an aircraft relative to the air. Speed is given in mph. or knots.

(2) True heading.-The angular direction of the longitudinal axis of the aircraft with respect to true north. The words "heading" and "true heading" will be used interchangeably.

(3) Wind direction and speed.-Wind is designated by the direction from which it blows. Its speed is expressed in mph. or knots.

(4) Ground speed. The actual speed of an aircraft relative to the earth's surface is mph. or knots.

(5) True course. The direction over the surface of the earth, expressed as an angle, with respect to true north, that an aircraft is intended to be flown. It is the course laid out on the chart or map.

Strictly, track is the actual path of an aircraft over the surface of the earth that has been flown. In practice, "track" and "course" are freely used interchangeably.

c. Some general rules for setting up vector diagrams follow:

(1) The reference system for direction of vectors is always the same. The true north line is the reference line and all angles are measured from it clockwise, through 360°.

(2) Choose a scale which will give lines large enough for accurate measurement.

(3) Label all vectors as they are drawn.

(4) "Heading" and "air speed" determine one vector.

(5) "Course" and "ground speed" determine one vector.

(6) Wind always blows the aircraft from the heading onto the course. 45. Type I.—The simplest triangle of velocity problem involves finding the true course and ground speed when wind speed, wind direction, air speed, and heading are known.

a. Example: The pilot has been told to fly a heading of 56° in an airplane the air speed of which is 100 mph. The wind is blowing 25 mph. from 110°. Find the true course and ground speed.

NOTE.-See back of manual for figure 54.