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(Chris Devlin) #1

Acceleration


When a particle’s velocity changes, the particle is said to undergo acceleration(or
to accelerate). For motion along an axis, the average accelerationaavgover a time
intervaltis

(2-7)


where the particle has velocity v 1 at time t 1 and then velocity v 2 at time t 2 .The
instantaneous acceleration(or simply acceleration) is

(2-8)


In words, the acceleration of a particle at any instant is the rate at which its velocity
is changing at that instant. Graphically, the acceleration at any point is the slope of
the curve of v(t) at that point. We can combine Eq. 2-8 with Eq. 2-4 to write

(2-9)

In words, the acceleration of a particle at any instant is the second derivative of
its position x(t) with respect to time.
A common unit of acceleration is the meter per second per second: m/(s s)
or m/s^2. Other units are in the form of length/(time time) or length/time^2.
Acceleration has both magnitude and direction (it is yet another vector quan-
tity). Its algebraic sign represents its direction on an axis just as for displacement
and velocity; that is, acceleration with a positive value is in the positive direction
of an axis, and acceleration with a negative value is in the negative direction.
Figure 2-6 gives plots of the position, velocity, and acceleration of an ele-
vator moving up a shaft. Compare the a(t) curve with the v(t) curve — each
point on the a(t) curve shows the derivative (slope) of the v(t) curve at the
corresponding time. When vis constant (at either 0 or 4 m/s), the derivative is
zero and so also is the acceleration. When the cab first begins to move, the v(t)

a

dv
dt




d
dt

dx
dt




d^2 x
dt^2

a

dv
dt

aavg

v 2 v 1
t 2 t 1




v
t

,


20 CHAPTER 2 MOTION ALONG A STRAIGHT LINE


2-3ACCELERATION


Learning Objectives


2.12Given a graph of a particle’s velocity versus time, deter-
mine the instantaneous acceleration for any particular time
and the average acceleration between any two particular
times.

●Average acceleration is the ratio of a change in velocity v
to the time interval tin which the change occurs:


The algebraic sign indicates the direction of aavg.


aavg

v
t

.


●Instantaneous acceleration (or simply acceleration) ais the
first time derivative of velocity v(t)and the second time deriv-
ative of position x(t):

.

●On a graph of vversust, the acceleration aat any time tis
the slope of the curve at the point that represents t.

a

dv
dt




d^2 x
dt^2

After reading this module, you should be able to...
2.10Apply the relationship between a particle’s average
acceleration, its change in velocity, and the time interval
for that change.
2.11Given a particle’s velocity as a function of time, calcu-
late the instantaneous acceleration for any particular time.

Key Ideas

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