4-2AVERAGE VELOCITY AND INSTANTANEOUS VELOCITY
64 CHAPTER 4 MOTION IN TWO AND THREE DIMENSIONS
0 x (m)
20
40
–20
–40
–60
y (m)
20 40 60 80
(b) 25 s 20 s
15 s
10 s
5 s
t = 0 s
This is the path with
various times indicated.
x (m)
0
20
40
–20
–40
–60
y (m)
20 40 60 80
(a)
–41°
r
This is the ycomponent.
To locate the
rabbit, this is the
xcomponent.
Figure 4-2(a) A rabbit’s position vector
at time t15 s. The scalar compo-
nents of are shown along the axes.
(b) The rabbit’s path and its position at
six values of t.
:r
:r
Additional examples, video, and practice available at WileyPLUS
Check:Althoughu139° has the same tangent as 41°,
the components of position vector indicate that the de-
sired angle is 139°180°41°.
(b) Graph the rabbit’s path for t0 to t25 s.
Graphing: We have located the rabbit at one instant, but to
see its path we need a graph. So we repeat part (a) for sev-
eral values of tand then plot the results. Figure 4-2bshows
the plots for six values of tand the path connecting them.
:r
which is drawn in Fig. 4-2a. To get the magnitude and angle
of , notice that the components form the legs of a right tri-
angle and ris the hypotenuse. So, we use Eq. 3-6:
(Answer)
and tan^1. (Answer)
y
x
tan^1
57 m
66 m
41
87 m,
r 2 x^2 y^2 2 (66 m)^2 (57 m)^2
:r
4.06In magnitude-angle and unit-vector notations, relate a parti-
cle’s initial and final position vectors, the time interval between
those positions, and the particle’s average velocity vector.
4.07Given a particle’s position vector as a function of time,
determine its (instantaneous) velocity vector.
Learning Objectives
After reading this module, you should be able to...
4.04Identify that velocity is a vector quantity and thus has
both magnitude and direction and also has components.
4.05Draw two-dimensional and three-dimensional velocity
vectors for a particle, indicating the components along the
axes of the coordinate system.
which can be rewritten in unit-vector notation as
where and
●The instantaneous velocity of a particle is always directed
along the tangent to the particle’s path at the particle’s
position.
:v
vxdx/dt, vydy/dt, vzdz/dt.
v:vxiˆvyjˆvzkˆ,
Key Ideas
●If a particle undergoes a displacement in time interval t,
its average velocity for that time interval is
●As tis shrunk to 0, reaches a limit called either the
velocity or the instantaneous velocity :
:v
dr:
dt
,
:v
:vavg
v:avg
:r
t
.
:vavg
:r
