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Unformatted text preview: 53 3 CHAPTER Vectors and Two-Dimensional Motion O U T L I N E 3.1 Vectors and Their Properties 3.2 Components of a Vector 3.3 Displacement, Velocity, and Acceleration in Two Dimensions 3.4 Motion in Two Dimensions 3.5 Relative Velocity In our discussion of one-dimensional motion in Chapter 2, we used the concept of vectors only to a limited extent. In our further study of motion, manipulating vector quantities will be- come increasingly important, so much of this chapter is devoted to vector techniques. Well then apply these mathematical tools to two-dimensional motion, especially that of projectiles, and to the understanding of relative motion. 3.1 VECTORS AND THEIR PROPERTIES Each of the physical quantities we will encounter in this book can be categorized as either a vector quantity or a scalar quantity . As noted in Chapter 2, a vector has both direction and magnitude (size). A scalar can be completely specified by its magni- tude with appropriate units; it has no direction. An example of each kind of quan- tity is shown in Figure 3.1 (page 54). As described in Chapter 2, displacement, velocity, and acceleration are vector quantities. Temperature is an example of a scalar quantity. If the temperature of an object is 5 8 C, that information completely specifies the temperature of the ob- ject; no direction is required. Masses, time intervals, and volumes are scalars as well. Scalar quantities can be manipulated with the rules of ordinary arithmetic. Vectors can also be added and subtracted from each other, and multiplied, but there are a number of important differences, as will be seen in the following sections. When a vector quantity is handwritten, it is often represented with an arrow over the letter ( ). As mentioned in Section 2.1, a vector quantity in this book A : Legendary motorcycle stuntman Evel Knievel blasts off in his custom rocket-powered Harley-Davidson Skycycle in an attempt to jump the Snake River Canyon in 1974. A parachute prematurely deployed and caused the craft to fall into the canyon, just short of the other side. Knievel survived. B e t t m a n n / C o r b i s 54 Chapter 3 Vectors and Two-Dimensional Motion will be represented by boldface type with an arrow on top (for example, ). The magnitude of the vector will be represented by italic type, as A . Italic type will also be used to represent scalars. Equality of Two Vectors. Two vectors and are equal if they have the same magnitude and the same direction. This property allows us to translate a vector parallel to itself in a diagram without affecting the vector. In fact, for most purposes, any vector can be moved parallel to itself without being affected. (See Fig. 3.2.) Adding Vectors. When two or more vectors are added, they must all have the same units. For example, it doesnt make sense to add a velocity vector, carrying units of meters per second, to a displacement vector, carrying units of meters....
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