9781118230725.pdf

4-2 AVERAGE VELOCITY AND INSTANTANEOUS VELOCITY 65

Average Velocity and Instantaneous Velocity

If a particle moves from one point to another, we might need to know how fast it
moves. Just as in Chapter 2, we can define two quantities that deal with “how
fast”:average velocityandinstantaneous velocity.However, here we must con-
sider these quantities as vectors and use vector notation.
If a particle moves through a displacement in a time interval t, then its
average velocity is

or (4-8)

This tells us that the direction of (the vector on the left side of Eq. 4-8) must
be the same as that of the displacement (the vector on the right side). Using
Eq. 4-4, we can write Eq. 4-8 in vector components as

(4-9)

For example, if a particle moves through displacement in
2.0 s, then its average velocity during that move is

That is, the average velocity (a vector quantity) has a component of 6.0 m/s along
thexaxis and a component of 1.5 m /s along the zaxis.
When we speak of the velocityof a particle, we usually mean the particle’s
instantaneous velocity at some instant. This is the value that approaches
in the limit as we shrink the time interval tto 0 about that instant. Using the lan-
guage of calculus, we may write as the derivative

(4-10)

Figure 4-3 shows the path of a particle that is restricted to the xyplane. As
the particle travels to the right along the curve, its position vector sweeps to the
right. During time interval t, the position vector changes from to and the
particle’s displacement is.
To find the instantaneous velocity of the particle at, say, instant t 1 (when the
particle is at position 1), we shrink interval tto 0 about t 1. Three things happen
as we do so. (1) Position vector :r 2 in Fig. 4-3 moves toward :r 1 so that :rshrinks



:r

 :r 1 :r 2

:v

dr:

dt

.

:v



:v :v :vavg

:vavg

:r

t



(12 m)iˆ(3.0 m)kˆ

2.0 s

(6.0 m /s)iˆ(1.5 m /s)kˆ.

(12 m)iˆ(3.0 m)kˆ

:vavg

xiˆyjˆzkˆ

t



x

t

iˆ

y

t

jˆ

z

t

kˆ.

:r

:vavg

:vavg

:r

t

.

average velocity

displacement

time interval

,

:vavg

:r 

Figure 4-3The displacement of a particle

during a time interval , from position 1 with

position vector at time t 1 to position 2

with position vector at time t 2. The tangent

to the particle’s path at position 1 is shown.

:r 2

:r 1

t

:r

r 1

r 2

Path

Tangent

O

y

x

1

Δr^2

As the particle moves,

the position vector

must change.

This is the

displacement.