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