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CHAPTER 4

Motion in Two and Three Dimensions

4-1POSITION AND DISPLACEMENT

Learning Objectives

After reading this module, you should be able to...

4.01Draw two-dimensional and three-dimensional position

vectors for a particle, indicating the components along the

axes of a coordinate system.

4.02On a coordinate system, determine the direction and

magnitude of a particle’s position vector from its compo-

nents, and vice versa.

4.03Apply the relationship between a particle’s displace-

ment vector and its initial and final position vectors.

Key Ideas

●The location of a particle relative to the origin of a coordi-

nate system is given by a position vector , which in unit-

vector notation is

Herex,y, and z are the vector components of position

vector , and and zare its scalar components (as well

as the coordinates of the particle).

●A position vector is described either by a magnitude and

:r x,y,

iˆ jˆ kˆ

:rxiˆyjˆzkˆ.

:r

one or two angles for orientation, or by its vector or scalar

components.

●If a particle moves so that its position vector changes from

to , the particle’s displacement is

The displacement can also be written as

xiˆyjˆzkˆ.

:r(x 2 x 1 )iˆ(y 2 y 1 )jˆ(z 2 z 1 )kˆ

:r:r 2 :r 1.

:r 1 :r 2 :r

What Is Physics?

In this chapter we continue looking at the aspect of physics that analyzes

motion, but now the motion can be in two or three dimensions. For example,

medical researchers and aeronautical engineers might concentrate on the

physics of the two- and three-dimensional turns taken by fighter pilots in dog-

fights because a modern high-performance jet can take a tight turn so quickly

that the pilot immediately loses consciousness. A sports engineer might focus

on the physics of basketball. For example, in a free throw(where a player gets

an uncontested shot at the basket from about 4.3 m), a player might employ the

overhand push shot,in which the ball is pushed away from about shoulder

height and then released. Or the player might use an underhand loop shot,in

which the ball is brought upward from about the belt-line level and released.

The first technique is the overwhelming choice among professional players, but

the legendary Rick Barry set the record for free-throw shooting with the under-

hand technique.

Motion in three dimensions is not easy to understand. For example, you are

probably good at driving a car along a freeway (one-dimensional motion) but

would probably have a difficult time in landing an airplane on a runway (three-

dimensional motion) without a lot of training.

In our study of two- and three-dimensional motion, we start with position

and displacement.