85
0
10
20
30
40
50
02468101214
t (s)
(3.0 s, 28 m/s)
(5.0 s, 42 m/s)
(7.0 s, 47 m/s)(9.0 s, 52 m/s)
Section 3.3Positive and Negative Acceleration
x
t
large,
positive
a
x
t
a=0
x
t
a>0
a<0
x
t
small,
positive
a
x
t
large,
negative
a
x
t
a=0
line (not necessarily on the actual curve): (3.0 s, 28 m/s) and (5.0
s, 42 m/s). The slope of the tangent line is (42 m/s-28 m/s)/(5.0 s
- 3.0 s)=7.0 m/s
2
.
(b) Two points on this tangent line are (7.0 s, 47 m/s) and (9.0 s,
52 m/s). The slope of the tangent line is (52 m/s-47 m/s)/(9.0 s -
7.0 s)=2.5 m/s
2
.
Physically, what’s happening is that at t=3.0 s, the skydiver is not
yet going very fast, so air friction is not yet very strong. She
therefore has an acceleration almost as great as g. At t=7.0 s,
she is moving almost twice as fast (about 100 miles per hour),
and air friction is extremely strong, resulting in a significant
departure from the idealized case of no air friction.
In the above example, the x-t graph was not even used in the solution of
the problem, since the definition of acceleration refers to the slope of the v-t
graph. It is possible, however, to interpret an x-t graph to find out some-
thing about the acceleration. An object with zero acceleration, i.e. constant
velocity, has an x-t graph that is a straight line. A straight line has no
curvature. A change in velocity requires a change in the slope of the x-t
graph, which means that it is a curve rather than a line. Thus acceleration
relates to the curvature of the x-t graph. Figure (c) shows some examples.
In the skydiver example, the x-t graph was more strongly curved at the
beginning, and became nearly straight at the end. If the x-t graph is nearly
straight, then its slope, the velocity, is nearly constant, and the acceleration
is therefore small. We can thus interpret the acceleration as representing the
curvature of the x-t graph. If the “cup” of the curve points up, the accelera-
tion is positive, and if it points down, the acceleration is negative.
Since the relationship between a and v is analogous to the relationship