60
2.3Graphs of Motion; Velocity.
Motion with constant velocity
In example (a), an object is moving at constant speed in one direction.
We can tell this because every two seconds, its position changes by five
meters.
In algebra notation, we’d say that the graph of x vs. t shows the same
change in position,
.
x=5.0 m, over each interval of
.
t=2.0 s. The object’s
velocity or speed is obtained by calculating v=
.
x/
.
t=(5.0 m)/(2.0 s)=2.5 m/
s. In graphical terms, the velocity can be interpreted as the slope of the line.
Since the graph is a straight line, it wouldn’t have mattered if we’d taken a
longer time interval and calculated v=
.
x/
.
t=(10.0 m)/(4.0 s). The answer
would still have been the same, 2.5 m/s.
Note that when we divide a number that has units of meters by another
number that has units of seconds, we get units of meters per second, which
can be written m/s. This is another case where we treat units as if they were
algebra symbols, even though they’re not.
In example (b), the object is moving in the opposite direction: as time
progresses, its x coordinate decreases. Recalling the definition of the
.
notation as “after minus before,” we find that
.
t is still positive, but
.
x
must be negative. The slope of the line is therefore negative, and we say
that the object has a negative velocity, v=
.
x/
.
t=(-5.0 m)/(2.0 s)=-2.5 m/s.
We’ve already seen that the plus and minus signs of
.
x values have the
interpretation of telling us which direction the object moved. Since
.
t is
always positive, dividing by
.
t doesn’t change the plus or minus sign, and
the plus and minus signs of velocities are to be interpreted in the same way.
In graphical terms, a positive slope characterizes a line that goes up as we go
to the right, and a negative slope tells us that the line went down as we went
to the right.
Motion with changing velocity
Now what about a graph like example (c). This might be a graph of a
car’s motion as the driver cruises down the freeway, then slows down to look
at a car crash by the side of the road, and then speeds up again, disap-
pointed that there is nothing dramatic going on such as flames or babies
trapped in their car seats. (Note that we are still talking about one-dimen-
sional motion. Just because the graph is curvy doesn’t mean that the car’s
path is curvy. The graph is not like a map, and the horizontal direction of
the graph represents the passing of time, not distance.)
Example (c) is similar to example (a) in that the object moves a total of
25.0 m in a period of 10.0 s, but it is no longer true that it makes the same
amount of progress every second. There is no way to characterize the entire
graph by a certain velocity or slope, because the velocity is different at every
moment. It would be incorrect to say that because the car covered 25.0 m
in 10.0 s, its velocity was 2.5 m/s . It moved faster than that at the begin-
ning and end, but slower in the middle. There may have been certain
instants at which the car was indeed going 2.5 m/s, but the speedometer
swept past that value without “sticking,” just as it swung through various
other values of speed. (I definitely want my next car to have a speedometer
calibrated in m/s and showing both negative and positive values.)
0
5
10
15
20
25
30
0246810
t (s)
x
(m)
(c) Motion with changing velocity.
0
5
10
15
20
25
30
0246810
t (s)
x
(m)
.
t
.
x
(b) Motion that decreases x is
represented with negative values of
.
x
and v.
0
5
10
15
20
25
30
0246810
t (s)
x
(m)
.
t
.
x
(a) Motion with constant velocity.
Chapter 2Velocity and Relative Motion
Next Page >>
<< Previous Page
Back to the Table of Contents