9781118230725.pdf

Constant Acceleration: A Special Case

In many types of motion, the acceleration is either constant or approximately so.
For example, you might accelerate a car at an approximately constant rate when
a traffic light turns from red to green. Then graphs of your position, velocity,
and acceleration would resemble those in Fig. 2-9. (Note that a(t) in Fig. 2-9cis
constant, which requires that v(t) in Fig. 2-9bhave a constant slope.) Later when
you brake the car to a stop, the acceleration (or deceleration in common
language) might also be approximately constant.
Such cases are so common that a special set of equations has been derived
for dealing with them. One approach to the derivation of these equations is given
in this section. A second approach is given in the next section. Throughout both
sections and later when you work on the homework problems, keep in mind that
these equations are valid only for constant acceleration (or situations in which you
can approximate the acceleration as being constant).
First Basic Equation. When the acceleration is constant, the average accel-
eration and instantaneous acceleration are equal and we can write Eq. 2-7, with
some changes in notation, as

Herev 0 is the velocity at time t0 and vis the velocity at any later time t. We can
recast this equation as

vv 0 at. (2-11)

As a check, note that this equation reduces to vv 0 fort0, as it must. As a fur-
ther check, take the derivative of Eq. 2-11. Doing so yields dv/dta, which is the
definition of a. Figure 2-9bshows a plot of Eq. 2-11, the v(t) function; the function
is linear and thus the plot is a straight line.
Second Basic Equation.In a similar manner, we can rewrite Eq. 2-2 (with a
few changes in notation) as

vavg

xx 0

t 0

aaavg

vv 0

t 0

2-4 CONSTANT ACCELERATION 23

2-4CONSTANT ACCELERATION

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

2.13For constant acceleration, apply the relationships be-
tween position, displacement, velocity, acceleration, and
elapsed time (Table 2-1).

2.14Calculate a particle’s change in velocity by integrating

its acceleration function with respect to time.

2.15Calculate a particle’s change in position by integrating

its velocity function with respect to time.

●The following five equations describe the motion of a particle with constant acceleration:

These are notvalid when the acceleration is not constant.

xx 0 vt

1

2

xx 0  at^2.

1

2

v^2 v 02  2 a(xx 0 ), (v 0 v)t,

xx 0 v 0 t

1

2

vv 0 at, at^2 ,

Learning Objectives

Key Ideas

Figure 2-9(a) The position x(t) of a particle

moving with constant acceleration. (b) Its

velocityv(t), given at each point by the

slope of the curve of x(t). (c) Its (constant)

acceleration, equal to the (constant) slope

of the curve of v(t).

0

a

Slope = 0

a(t)

(a)

(b)

(c) t

Position

0

x 0

x

t

Slope varies

x(t)

Velocity

0

Slope = a

v(t)

v 0

v

t

Acceleration

Slopes of the position graph

are plotted on the velocity graph.

Slope of the velocity graph is

plotted on the acceleration graph.