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4-3 AVERAGE ACCELERATION AND INSTANTANEOUS ACCELERATION 69

Sample Problem 4.03 Two-dimensional acceleration, rabbit run

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For the rabbit in the preceding two sample problems, find

the acceleration at time t15 s.

KEY IDEA

We can find by taking derivatives of the rabbit’s velocity

components.

Calculations:Applying the axpart of Eq. 4-18 to Eq. 4-13,

we find the xcomponent of to be

Similarly, applying the aypart of Eq. 4-18 to Eq. 4-14 yields

theycomponent as

We see that the acceleration does not vary with time (it is a

constant) because the time variable tdoes not appear in the

expression for either acceleration component. Equation 4-17

then yields

(Answer)

which is superimposed on the rabbit’s path in Fig. 4-7.

To get the magnitude and angle of , either we use a

vector-capable calculator or we follow Eq. 3-6. For the mag-

nitude we have

(Answer)

For the angle we have

However, this angle, which is the one displayed on a calcula-

tor, indicates that is directed to the right and downward in

Fig. 4-7. Yet, we know from the components that must be

directed to the left and upward. To find the other angle that

:a

a:

tan^1

ay

ax

tan^1 

0.44 m/s^2

0.62 m/s^2 

 35 .

0.76 m/s^2.

a 2 ax^2 ay^2  2 (0.62 m /s^2 )^2 (0.44 m /s^2 )^2

a:

:a(0.62 m /s (^2) )iˆ(0.44 m /s (^2) )jˆ,
ay
dvy
dt



d

dt

(0.44t9.1)0.44 m /s^2.

ax

dvx

dt



d

dt

(0.62t7.2)0.62 m /s^2.

:a

:a

:a

x (m)

0

20

40

–20

–40

–60

y (m)

20 40 60 80

x

a 145°

These are the x andy

components of the vector

at this instant.

Figure 4-7The acceleration of the rabbit at t15 s. The rabbit

happens to have this same acceleration at all points on its path.

a:

has the same tangent as 35° but is not displayed on a cal-

culator, we add 180°:

35°180°145°. (Answer)

This isconsistent with the components of because it gives

a vector that is to the left and upward.Note that has the

same magnitude and direction throughout the rabbit’s run

because the acceleration is constant. That means that

we could draw the very same vector at any other point

along the rabbit’s path (just shift the vector to put its tail at

some other point on the path without changing the length

or orientation).

This has been the second sample problem in which we

needed to take the derivative of a vector that is written in

unit-vector notation. One common error is to neglect the unit

vectors themselves, with a result of only a set of numbers and

symbols. Keep in mind that a derivative of a vector is always

another vector.

:a

:a

Checkpoint 2

Here are four descriptions of the position (in meters) of a puck as it moves in an xyplane:

(1)x 3 t^2  4 t 2 and y 6 t^2  4 t (3)

(2)x 3 t^3  4 t and y 5 t^2  6 (4)

Are the xandyacceleration components constant? Is acceleration constant?:a

:r(4t (^3)  2 t)iˆ3jˆ
:r 2 t (^2) iˆ(4t3)jˆ