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77
Motion in Two Dimensions
CHAPTER OUTLINE
4.1
The Position, Velocity, and
Acceleration Vectors
4.2
Two-Dimensional Motion with
Constant Acceleration
4.3
Projectile Motion
4.4
Uniform Circular Motion
4.5
Tangential and Radial
Acceleration
4.6
Relative Velocity and Relative
Acceleration
±
Lava spews from a volcanic eruption. Notice the parabolic paths of embers projected into
the air. We will ﬁnd in this chapter that all projectiles follow a parabolic path in the absence
of air resistance. (© Arndt/Premium Stock/PictureQuest)
Chapter 4

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I
n this chapter we explore the kinematics of a particle moving in two dimensions. Know-
ing the basics of two-dimensional motion will allow us to examine—in future chapters—a
wide variety of motions, ranging from the motion of satellites in orbit to the motion of
electrons in a uniform electric ﬁeld. We begin by studying in greater detail the vector
nature of position, velocity, and acceleration. As in the case of one-dimensional motion,
we derive the kinematic equations for two-dimensional motion from the fundamental
deﬁnitions of these three quantities. We then treat projectile motion and uniform circular
motion as special cases of motion in two dimensions. We also discuss the concept of
relative motion, which shows why observers in different frames of reference may measure
different positions, velocities, and accelerations for a given particle.
4.1
The Position, Velocity, and
Acceleration Vectors
In Chapter 2 we found that the motion of a particle moving along a straight line is
completely known if its position is known as a function of time. Now let us extend this
idea to motion in the
xy
plane. We begin by describing the position of a particle by its
position vector r
, drawn from the origin of some coordinate system to the particle lo-
cated in the
xy
plane, as in Figure 4.1. At time
t
i
the particle is at point
±
, described by
position vector
r
i
. At some later time
t
f
it is at point
²
, described by position vector
r
f
.
The path from
±
to
²
is not necessarily a straight line. As the particle moves from
±
to
²
in the time interval
±
t
²
t
f
³
t
i
, its position vector changes from
r
i
to
r
f
. As we
learned in Chapter 2, displacement is a vector, and the displacement of the particle is
the difference between its ﬁnal position and its initial position. We now deﬁne the
dis-
placement vector
±
r
for the particle of Figure 4.1 as being the difference between its
ﬁnal position vector and its initial position vector:
(4.1)
The direction of
±
r
is indicated in Figure 4.1. As we see from the ﬁgure, the magnitude of
±
r
is
less
than the distance traveled along the curved path followed by the particle.
As we saw in Chapter 2, it is often useful to quantify motion by looking at the ratio
of a displacement divided by the time interval during which that displacement occurs,
which gives the rate of change of position. In two-dimensional (or three-dimensional)
kinematics, everything is the same as in one-dimensional kinematics except that we

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