9781118230725.pdf

68 CHAPTER 4 MOTION IN TWO AND THREE DIMENSIONS

Average Acceleration and Instantaneous Acceleration

When a particle’s velocity changes from to in a time interval t, its average

acceleration during tis

or (4-15)

If we shrink tto zero about some instant, then in the limit approaches the

instantaneous acceleration(oracceleration) at that instant; that is,

(4-16)

If the velocity changes in eithermagnitudeordirection (or both), the particle

must have an acceleration.

We can write Eq. 4-16 in unit-vector form by substituting Eq. 4-11 for to obtain

We can rewrite this as

(4-17)

where the scalar components of are

(4-18)

To find the scalar components of , we differentiate the scalar components of.

Figure 4-6 shows an acceleration vector and its scalar components for a

particle moving in two dimensions.Caution:When an acceleration vector is

drawn, as in Fig. 4-6, it does notextend from one position to another. Rather, it

shows the direction of acceleration for a particle located at its tail, and its length

(representing the acceleration magnitude) can be drawn to any scale.

a:

:a v:

ax

dvx

dt

, ay

dvy

dt

, and az

dvz

dt

.

a:

a:axiˆayjˆazkˆ,



dvx

dt

iˆ

dvy

dt

jˆ

dvz

dt

kˆ.

a:

d

dt

(vxiˆvyjˆvzkˆ)

:v

:a

dv:

dt

.

:a

:aavg

:aavg

:v 2 :v 1

t



:v

t

.

average

acceleration



change in velocity

time interval

,

:aavg 

:v 1 :v 2 

O

y

x

ay

ax

Path

a

These are the x andy

components of the vector

at this instant.

Figure 4-6The acceleration of a particle and the

scalar components of a:.

:a