2.1 Uniform Motion

The simplest possible motion is motion along a straight line at a constant, unvarying speed. We call this uniform motion. Because velocity is the combination of speed and direction, uniform motion is motion with constant velocity.

FIGURE 2.1 shows the motion diagram of an object in uniform motion. For example, this might be you riding your bicycle along a straight line at a perfectly steady $5 m/s (\approx 10 mph)$ . Notice how all the displacements are exactly the same; this is a characteristic of uniform motion.

The figure consist of two parts that depict the motion diagram and position-versus-time graph for uniform motion.

FIGURE 2.1 Full Alternative Text

If we make a position-versus-time graph—remember that position is graphed on the vertical axis—it’s a straight line. In fact, an alternative definition is that an object’s motion is uniform if and only if its position-versus-time graph is a straight line.

《 Section 1.4 defined an object’s average velocity as $Δ \vec{r} / Δ t$ . For one-dimensional motion, this is simply $Δ x / Δ t$ (for horizontal motion) or $Δ y / Δ t$ (for vertical motion). Recall that $Δ x$ is the object’s displacement during the time interval $Δ t$ . You can see in Figure 2.1 that $Δ x$ and $Δ t$ are, respectively, the “rise” and “run” of the position graph. Because rise over run is the slope of a line,

v_{avg} \equiv \frac{Δ x}{Δ t} or \frac{Δ y}{Δ t} = slope of the position-versus-time graph

(2.1)

That is, the average velocity is the slope of the position-versus-time graph. Velocity has units of “length per time,” such as “miles per hour.” The SI units of velocity are meters per second, abbreviated $m / s$ .

NOTE The symbol $\equiv$ in Equation 2.1 stands for “is defined as.” This is a stronger statement than the two sides simply being equal.

The constant slope of a straight-line graph is another way to see that the velocity is constant for uniform motion. There’s no real need to specify “average” for a velocity that doesn’t change, so we will drop the subscript and refer to the average velocity as $v_{x}$ or $v_{y}$ .

An object’s speed $v$ is how fast it’s going, independent of direction. This is simply $v = | v_{x} |$ or $v = | v_{y} |$ , the magnitude or absolute value of the object’s velocity. Although we will use speed from time to time, our mathematical analysis of motion is based on velocity, not speed. The subscript in $v_{x}$ or $v_{y}$ is an essential part of the notation, reminding us that, even in one dimension, the velocity is a vector.

EXAMPLE 2.1 Relating a velocity graph to a position graph

FIGURE 2.2 is the position-versus-time graph of a car.

Draw the car’s velocity-versus-time graph.
Describe the car’s motion.

The figure illustrates a position-versus-time graph of a car and its corresponding velocity-versus-time graph.

FIGURE 2.2 Full Alternative Text

MODEL Model the car as a particle, with a well-defined position at each instant of time.

VISUALIZE Figure 2.2 is the graphical representation.

SOLVE a. The car’s position-versus-time graph is a sequence of three straight lines. Each of these straight lines represents uniform motion at a constant velocity. We can determine the car’s velocity during each interval of time by measuring the slope of the line.

The position graph starts out sloping downward—a negative slope. Although the car moves a distance of 4.0 m during the first 2.0 s, its displacement is

Δ x = x_{at 2.0 s} - x_{at 0.0 s} = - 4.0 m - 0.0 m = - 4.0 m

The time interval for this displacement is $Δ t = 2.0 s,$ so the velocity during this interval is

v_{x} = \frac{Δ x}{Δ t} = \frac{- 4.0 m}{2.0 s} = - 2.0 m / s

The car’s position does not change from $t = 2 s$ to $t = 4 s$ $(Δ x = 0)$ , so $v_{x} = 0$ . Finally, the displacement between $t = 4 s$ and $t = 6 s$ is $Δ x = 10.0 m$ . Thus the velocity during this interval is

v_{x} = \frac{10.0 m}{2 .0 s} = 5.0 m / s

These velocities are shown on the velocity-versus-time graph of FIGURE 2.3 .

b. The car backs up for 2 s at 2.0 m/s, sits at rest for 2 s, then drives forward at 5.0 m/s for at least 2 s. We can’t tell from the graph what happens for $t > 6 s$ .

REVIEW The velocity graph and the position graph look completely different. The value of the velocity graph at any instant of time equals the slope of the position graph.

Example 2.1 brought out several points that are worth emphasizing.

TACTICS BOX 2.1 Interpreting position-versus-time graphs

1 Steeper slopes correspond to faster speeds.

2 Negative slopes correspond to negative velocities and, hence, to motion to the left (or down).

3 The slope is a ratio of intervals, $Δ x / Δ t$ , not a ratio of coordinates. That is, the slope is not simply $x / t$ .

Workbook Exercises 1–3

NOTE We are distinguishing between the actual slope and the physically meaningful slope. If you were to use a ruler to measure the rise and the run of the graph, you could compute the actual slope of the line as drawn on the page. That is not the slope to which we are referring when we equate the velocity with the slope of the line. Instead, we find the physically meaningful slope by measuring the rise and run using the scales along the axes. The “rise” $Δ x$ is some number of meters; the “run” $Δ t$ is some number of seconds. The physically meaningful rise and run include units, and the ratio of these units gives the units of the slope.

2.1 Uniform Motion

FIGURE 2.1 Motion diagram and position graph for uniform motion.

EXAMPLE 2.1 Relating a velocity graph to a position graph

FIGURE 2.2 Position-versus-time graph.

FIGURE 2.3 The corresponding velocity-versus-time graph.

TACTICS BOX 2.1 Interpreting position-versus-time graphs