9781118230725.pdf

toward zero. (2) The direction of (and thus of ) approaches the

direction of the line tangent to the particle’s path at position 1. (3) The average

velocity :vavgapproaches the instantaneous velocity at :v t 1.

:r/t v:avg

Figure 4-4The velocity of a

particle, along with the scalar

components of :v.

:v

Path

O

y

x

Tangent

vy

vx

v

The velocity vector is always

tangent to the path.

These are the x andy

components of the vector

at this instant.

Checkpoint 1

The figure shows a circular path taken by a particle.

If the instantaneous velocity of the particle is

, through which quadrant is the par-

ticle moving at that instant if it is traveling (a) clockwise

and (b) counterclockwise around the circle? For both

cases, draw on the figure.v:

(2 m /s)iˆ(2 m /s)jˆ

:v

y

x

66 CHAPTER 4 MOTION IN TWO AND THREE DIMENSIONS

The direction of the instantaneous velocity of a particle is always tangent to the

particle’s path at the particle’s position.

v:

The result is the same in three dimensions: is always tangent to the particle’s path.

To write Eq. 4-10 in unit-vector form, we substitute for from Eq. 4-1:

This equation can be simplified somewhat by writing it as

(4-11)

where the scalar components of are

(4-12)

For example,dx/dtis the scalar component of along the xaxis. Thus, we can find

the scalar components of by differentiating the scalar components of.

Figure 4-4 shows a velocity vector and its scalar xandycomponents. Note

that is tangent to the particle’s path at the particle’s position.Caution:When a

position vector is drawn, as in Figs. 4-1 through 4-3, it is an arrow that extends

from one point (a “here”) to another point (a “there”). However, when a velocity

vector is drawn, as in Fig. 4-4, it does notextend from one point to another.

Rather, it shows the instantaneous direction of travel of a particle at the tail, and

its length (representing the velocity magnitude) can be drawn to any scale.

:v

v:

:v :r

:v

vx

dx

dt

, vy

dy

dt

, and vz

dz

dt

.

:v

:vvxiˆvyjˆvzkˆ,

:v

d

dt

(xiˆyjˆzkˆ)

dx

dt

iˆ

dy

dt

jˆ

dz

dt

kˆ.

:r

:v

In the limit as , we have and, most important here, takes

on the direction of the tangent line. Thus,:vhas that direction as well:

t: 0 :vavg::v :vavg