9781118230725.pdf

4-3 AVERAGE ACCELERATION AND INSTANTANEOUS ACCELERATION 67

(Answer)

and

(Answer)

Check: Is the angle130° or130°180°50°?

tan^1 1.19 130 .

tan^1

vy

vx

tan^1 

2.5 m /s

2.1 m /s

3.3 m /s

For the rabbit in the preceding sample problem, find the v 2 vx^2 vy^2  2 (2.1 m /s)^2 (2.5 m /s)^2
velocity at time t15 s.

KEY IDEA

We can find by taking derivatives of the components of
the rabbit’s position vector.

Calculations:Applying the vxpart of Eq. 4-12 to Eq. 4-5,
we find the xcomponent of to be

(4-13)

Att15 s, this gives vx2.1 m /s. Similarly, applying the
vypart of Eq. 4-12 to Eq. 4-6, we find

(4-14)

Att15 s, this gives vy2.5 m/s. Equation 4-11 then yields

(Answer)

which is shown in Fig. 4-5, tangent to the rabbit’s path and in
the direction the rabbit is running at t15 s.
To get the magnitude and angle of , either we use a
vector-capable calculator or we follow Eq. 3-6 to write

:v

v:(2.1 m /s)iˆ(2.5 m /s)jˆ,

0.44t9.1.

vy

dy

dt



d

dt

(0.22t^2 9.1t30)

0.62t7.2.

vx

dx

dt



d

dt

(0.31t^2 7.2t28)

v:

:v

:v

Sample Problem 4.02 Two-dimensional velocity, rabbit run

Additional examples, video, and practice available at WileyPLUS

Figure 4-5The rabbit’s velocity at :v t15 s.

–130°

x (m)

0

20

40

–20

–40

–60

y (m)

20 40 60 80

x

v

These are the x andy

components of the vector

at this instant.

4-3AVERAGE ACCELERATION AND INSTANTANEOUS ACCELERATION

the average acceleration vector in magnitude-angle and

unit-vector notations.

4.11Given a particle’s velocity vector as a function of time,

determine its (instantaneous) acceleration vector.

4.12For each dimension of motion, apply the constant-

acceleration equations (Chapter 2) to relate acceleration,

velocity, position, and time.

Learning Objectives

After reading this module, you should be able to...

4.08Identify that acceleration is a vector quantity and thus has
both magnitude and direction and also has components.

4.09Draw two-dimensional and three-dimensional accelera-
tion vectors for a particle, indicating the components.

4.10Given the initial and final velocity vectors of a particle
and the time interval between those velocities, determine

either the acceleration or the instantaneous acceleration :

●In unit-vector notation,

where and axdvx/dt, aydvy/dt, azdvz/dt.

:aaxiˆayjˆazkˆ,

:a

dv:

dt

.

:a

Key Ideas

●If a particle’s velocity changes from to in time interval
t, its average acceleration during tis

●Astis shrunk to 0, :aavgreaches a limiting value called

:aavg

:v 2 :v 1

t



:v

t

.

 

:v 1 v: 2