9781118230725.pdf

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