| 2-3 |  | Velocity and Speed |
Suppose a student stands still or speeds up and slows down along a straight line. How can we describe accurately and efficiently where she is and how fast she is moving? We will explore several ways to do this.
Representing Motion in Diagrams and Graphs
Motion Diagrams: Now that you have learned about position and displacement, it is quite easy to describe the motion of an object using pictures or sketches to chart how position changes over time. Such a representation is called a motion diagram. For example, Fig. 2-6 shows a student whom we treat as if she were concentrated into a particle located at the back of her belt. She is standing still at a position 
from a point on a sidewalk that we choose as our origin. Figure 2-7 shows a more complex diagram describing the student in motion. Suppose we see that just as we start timing her progress with a stopwatch (so t = 0.0 s), the back of her belt is 2.47 m to the left of our origin. The x-component of her position is then x = –2.47 m. The student then moves toward the origin, almost reaches the origin at t = 1.5 s, and then continues moving to the right so that her x-component of position has increasingly positive values. It is important to recognize that just as we chose an origin and direction for our coordinate axis, we also chose an origin in time. If we had chosen to start our timing 12 seconds earlier, then the new motion diagram would show the back of her belt as being at x = –2.47 m at t = 12 s.
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| Figure 2-6 A motion diagram of a student standing still with the back of her belt at a horizontal distance of 2.00 m to the left of a spot of the sidewalk designated as the origin. |
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| Figure 2-7 A motion diagram of a student starting to walk slowly.The horizontal position of the back of her belt starts at a horizontal distance of 2.47 m to the left of a spot designated as the origin. She is speeding up for a few seconds and then slowing down. |
Graphs: Another way to describe how the position of an object changes as time passes is with a graph. In such a graph, the x-component of the object's position, x, can be plotted as a function of time, t. This position-time graph has alternate names such as a graph of x as a function of t, x(t), or x vs. t. For example, Fig. 2-8 shows a graph of the student standing still with the back of her belt located at a horizontal position of –2.00 m from a spot on the sidewalk that is chosen as the origin.
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| Figure 2-8 The graph of the x-component of position for a student who is standing still at x = –2.0 m for at least 3 seconds. |
The graph of no motion shown in Fig. 2-8 is not more informative than the picture or a comment that the student is standing still for 3 seconds at a certain location. But it's another story when we consider the graph of a motion. Figure 2-9 is a graph of a student's x-component of position as a function of time. It represents the same information depicted in the motion diagram in Fig. 2-7. Data on the student's motion are first recorded at t = 0.0 s when the x-component of her position is x = –2.47 m. The student then moves toward x = 0.00 m, passes through that point at about t = 1.5 s, and then moves on to increasingly larger positive values of x while slowing down.
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| Figure 2-9 A graph that represents how the position component, x, of the walking student shown in Fig. 2-7 changes over time.The motion diagram, shown below the graph, is associated with the grapht three points in time as indicated by the arrows. |
Although the graph of the student's motion in Fig. 2-9 seems abstract and quite unlike a motion diagram, it is richer in information. For example, the graph allows us to estimate the motion of the student at times between those for which position measurements were made. Equally important, we can use the graph to tell us how fast the student moves at various times, and we deal with this aspect of motion graphs next.
What can motion diagrams and x vs. t graphs tell us about how fast and in what direction something moves along a line? It is clear from an examination of the motion diagram at the bottom of Fig. 2-9 that the student covers the most distance and so appears to be moving most rapidly between the two times t1 = 1.0 s and t2 = 1.5 s. But this time interval is also where the slope (or steepness) of the graph has the greatest magnitude. Recall from mathematics that the average slope of a curve between two points is defined as the ratio of the change in the variable plotted on the vertical axis (in this case the x-component of her position) to the change in the variable plotted on the horizontal axis (in this case the time). Hence, on position vs. time graphs (such as those shown in Fig. 2-8 and Fig. 2-9),
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Since time moves forward, t2 > t1, so Dt always has a positive value. Thus, a slope will be positive whenever x2 > x1, so Dx is positive. In this case a straight line connecting the two points on the graph slants upward toward the right when the student is moving along the positive x-direction. On the other hand, if the student were to move “backwards” in the direction along the x axis we chose to call negative, then x2 < x1. In this case, the slope between the two times would be negative and the line connecting the points would slant downward to the right.
Average Velocity
For motion along a straight line, the steepness of the slope in an x vs. t graph over a time interval from t1 to t2 tells us “how fast” a particle moves. The direction of motion is indicated by the sign of the slope (positive or negative). Thus, this slope or ratio 
is a special quantity that tells us how fast and in what direction something moves. We haven't given the ratio 
a name yet. We do this to emphasize the fact that the ideas associated with figuring out how fast and in what direction something moves are more important than the names we assign to them. However, it is inconvenient not to have a name.The common name for this ratio is average velocity, which is defined as the ratio of displacement vector 
for the motion of interest to the time interval Dt in which it occurs. This vector can be expressed in equation form as
 | (2-4) | |
where x2 and x1 are components of the position vectors at the final and initial times. Here we use angle brackets 
to denote the average of a quantity. Also, we use the special symbol “≡” for equality to emphasize that the term on the left is equal to the term on the right by definition. The time change is a positive scalar quantity because we never need to specify its direction explicitly. In defining 
we are basically multiplying the displacement vector, 
by the scalar (
). This action gives us a new vector that points in the same direction as the displacement vector.Figure 2-10 shows how to find the average velocity for the student motion represented by the graph shown in Fig. 2-9 between the times t1 = 1.0 s and t2 = 1.5 s. The average velocity during that time interval is
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| Figure 2-10 Calculation of the slope of the line that connects the points on the curve at t1 = 1.0 s and t2 = 1.5 s. The xcomponent of the average velocity is given by this slope. |
The x-component of the average velocity along the line of motion, 
, is simply the slope of the straight line that connects the point on the curve at the beginning of our chosen interval and the point on the curve at the end of the interval. Since our student is speeding up and slowing down, the values of 
and 
will in general be different when calculated using other time intervals.
Average Speed
Sometimes we don't care about the direction of an object's motion but simply want to keep track of the distance covered. For instance, we might want to know the total distance a student walks (number of steps times distance covered in each step). Our student could be pacing back and forth wearing out her shoes without having a vector displacement. Similarly, average speed, 
, is a different way of describing “how fast” an object moves. Whereas the average velocity involves the particle's displacement 
, which is a vector quantity, the average speed involves the total distance covered (for example, the product of the length of a step and the number of steps the student took), which is independent of direction. So average speed is defined as
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Since neither the total distance traveled nor the time interval over which the travel occurred has an associated direction, average speed does not include direction information. Both the total distance and the time period are always positive, so average speed is always positive too. Thus, an object that moves back and forth along a line can have no vector displacement, so it has zero velocity but a rather high average speed. At other times, while the object is moving in only one direction, the average speed 
is the same as the magnitude of the average velocity 
. However, as you can demonstrate in Reading Exercise 2-4, when an object doubles back on its path, the average speed is not simply the magnitude of the average velocity 
.
Instantaneous Velocity and Speed
You have now seen two ways to describe how fast something moves: average velocity and average speed, both of which are measured over a time interval Dt. Clearly, however, something might speed up and slow down during that time interval. For example, in Fig. 2-9 we see that the student is moving more slowly at t = 0.0 s than she is at t = 1.5 s, so her velocity seems to be changing during the time interval between 0.0 s and 1.5 s. The average slope of the line seems to be increasing during this time interval. Can we refine our definition of velocity in such a way that we can determine the student's true velocity at any one “instant” in time? We envision something like the almost instantaneous speedometer readings we get as a car speeds up and slows down.
Defining an instant and instantaneous velocity is not a trivial task. As we noted in Chapter 1, the time interval of 1 second is defined by counting oscillations of radiation absorbed by a cesium atom. In general, even our everyday clocks work by counting oscillations in an electronic crystal, pendulum, and so on. We associate “instants in time” with positions on the hands of a clock, and “time intervals” with changes in the position of the hands.
For the purpose of finding a velocity at an instant, we can attempt to make the time interval we use in our calculation so small that it has almost zero duration. Of course the displacement we calculate also becomes very small. So instantaneous velocity along a line—like average velocity—is still defined in terms of the ratio of 
. But we have this ratio passing to a limit where Dt gets closer and closer to zero. Using standard calculus notation for this limit gives us the following definition:
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In passing to the limit the ratio 
is not necessarily small, since both the numerator and denominator are getting small together. The first part of this expression,
tells us that we can find the (instantaneous) velocity of an object by taking the slope of a graph of the position component vs. time at the point associated with that moment in time. If the graph is a curve rather than a straight line, the slope at a point is actually the tangent to the line at that point. Alternatively, the second part of the expression, shown in Eq. 2-6,indicates that, if we can approximate the relationship between 
and t as a continuous mathematical function such as 
, we can also find the object's instantaneous velocity by taking a derivative with respect to time of the object's position 
. When 
varies continuously as time marches on, we often denote 
as a position function 
to remind us that it varies with time.
Instantaneous speed, which is typically called simply speed, is just the magnitude of the instantaneous velocity vector, 
. Speed is a scalar quantity consisting of the velocity value that has been stripped of any indication of the direction the object is moving, either in words or via an algebraic sign. A velocity of (+5 m/s)
and one of (–5 m/s)
both have an associated speed of 5 m/s.
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| READING EXERCISE 2-4: | |
Suppose that you drive 10 mi due east to a store. You suddenly realize that you forgot your money. You turn around and drive the 10 mi due west back to your home and then return to the store. The total trip took 30 min. (a) What is your average velocity for the entire trip? (Set up a coordinate system and express your result in vector notation.) (b) What was your average speed for the entire trip? (c) Discuss why you obtained different values for average velocity and average speed.
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| READING EXERCISE 2-5: | |
Suppose that you are driving and look down at your speedometer.What does the speedometer tell you—average speed, instantaneous speed, average velocity, instantaneous velocity—or something else? Explain.
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| READING EXERCISE 2-7: | |
In Touchstone Example 2-2, suppose that right after refueling the truck you drive back to x1 at 35 km/h. What is the average velocity for your entire trip?
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| TOUCHSTONE EXAMPLE 2-2: Out of Gas |
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You drive a beat-up pickup truck along a straight road for 8.4 km at 70 km/h, at which point the truck runs out of gasoline and stops. Over the next 30 min, you walk another 2.0 km farther along the road to a gasoline station.
(a) What is your overall displacement from the beginning of your drive to your arrival at the station?
SOLUTION: Assume, for convenience, that you move in the positive direction along an x axis, from a first position of  to a second position of x2 at the station. That second position must be at  . Then the Key Idea here is that your displacement Dx along the x axis is the second position minus the first position. From Eq. 2-1, we have  | (Answer) | |
Thus, your overall displacement is 10.4 km in the positive direction of the x axis.
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, is changing with 
of the particle constant? (b) In which is the 
pointing in the negative x direction?
for the walk, but we lack the time interval 
for the drive. However, we know that for the drive the 
is 8.4 km and the average 
is 70 km/h. A Key Idea to use here comes from Eq. 
from the beginning of your drive to your arrival at the station? Find it both numerically and graphically.
for the entire trip is the ratio of the 
graphically, first we graph x(t) as shown in Fig. 
to the run 
, which gives us 
.
. The total time interval is 
. Thus, Eq. 