9781118230725.pdf

REVIEW & SUMMARY 81

velocity of plane velocity of plane velocity of wind

relative to ground



relative to wind



relative to ground.

(PG)(PW)(WG)

This relation is written in vector notation as

(4-46)

We need to resolve the vectors into components on the co-
ordinate system of Fig. 4-20band then solve Eq. 4-46 axis by
axis. For the ycomponents, we find

vPG,yvPW,yvWG,y

or 0 (215 km/h) sin u(65.0 km/h)(cos 20.0°).

Solving for ugives us

(Answer)

Similarly, for the xcomponents we find

vPG,xvPW,xvWG,x.

Here, because is parallel to the xaxis, the component
vPG,xis equal to the magnitude vPG. Substituting this nota-
tion and the value u16.5°, we find

vPG(215 km/h)(cos 16.5°)(65.0 km/h)(sin 20.0°)

228 km/h. (Answer)

:vPG

sin^1

(65.0 km/h)(cos 20.0)

215 km/h

16.5.

:vPG:vPW:vWG.

θ

θ

vPG

vPW vWG

vPG

vPW vWG

N

y

N

E

20 °

x

(a)

(b)

This is the plane's actual

direction of travel.

This is the wind

direction.

The actual direction

is the vector sum of

the other two vectors

(head-to-tail arrangement).

This is the plane's

orientation.

Figure 4-20A plane flying in a wind.

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Review & Summary

Position Vector The location of a particle relative to the ori-
gin of a coordinate system is given by a position vector , which in
unit-vector notation is

(4-1)

Herex,y, and z are the vector components of position vector ,
andx,y, and zare its scalar components (as well as the coordinates
of the particle). A position vector is described either by a magni-
tude and one or two angles for orientation, or by its vector or
scalar components.

Displacement If a particle moves so that its position vector
changes from to , the particle’s displacement is

(4-2)

The displacement can also be written as

(4-3)

xyz. (4-4)

Average Velocity and Instantaneous Velocity If a parti-
cle undergoes a displacement in time interval t, its average ve-
locity for that time interval is

:vavg (4-8)

:r

t

.

:vavg

:r

iˆ jˆ kˆ

:r(x 2 x 1 )iˆ(y 2 y 1 )jˆ(z 2 z 1 )kˆ

:r:r 2 :r 1.

:r 1 :r 2 :r

iˆ jˆ kˆ :r

:rxiˆyjˆzkˆ.

:r

Astin Eq. 4-8 is shrunk to 0, reaches a limit called either the

velocityor the instantaneous velocity :

(4-10)

which can be rewritten in unit-vector notation as

(4-11)

wherevxdx/dt, vydy/dt,andvzdz/dt.The instantaneous

velocity of a particle is always directed along the tangent to the

particle’s path at the particle’s position.

Average Acceleration and Instantaneous Acceleration

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

average accelerationduringtis

(4-15)

Astin Eq. 4-15 is shrunk to 0,a:avgreaches a limiting value called

a:avg

v: 2 :v 1

t



v:

t

.

:v 1 :v 2

v:

v:vxiˆvyjˆvzkˆ,

:v

dr:

dt

,

:v

:vavg

either the accelerationor theinstantaneous acceleration :

(4-16)

In unit-vector notation,

(4-17)

whereaxdvx/dt, aydvy/dt,andazdvz/dt.

:aaxiˆayjˆazkˆ,

:a

dv:

dt

.

a: