Uniform motion is simple, but objects rarely travel for long with a constant velocity. Far more common is a velocity that changes with time. For example, FIGURE 2.7 shows the motion diagram and position graph of a car speeding up after the light turns green. Notice how the velocity vectors increase in length, causing the graph to curve upward as the car’s displacements get larger and larger.

FIGURE 2.7 Motion diagram and position graph of a car speeding up.

If you were to watch the car’s speedometer, you would see it increase from 0 mph to 10 mph to 20 mph and so on. At any instant of time, the speedometer tells you how fast the car is going at that instant. If we include directional information, we can define an object’s instantaneous velocity—speed and direction—as its velocity at a single instant of time.

For uniform motion, the slope of the straight-line position graph is the object’s velocity. FIGURE 2.8 shows that there’s a similar connection between instantaneous velocity and the slope of a curved position graph.

FIGURE 2.8 Instantaneous velocity at time $t$ is the slope of the tangent to the curve at that instant.

What we see graphically is that the average velocity $v_{\text{avg}}=\text{Δ}s/\text{Δ}t$ becomes a better and better approximation to the instantaneous velocity $v_s$ as the time interval $\text{Δ}t$ over which the average is taken gets smaller and smaller. We can state this idea mathematically in terms of the limit $\text{Δ}t\to 0\text{:}$

As $\text{Δ}t$ continues to get smaller, the average velocity $v_{\text{avg}}=\text{Δ}s/\text{Δ}t$ reaches a constant or limiting value. That is, the instantaneous velocity at time $t$ is the average velocity during a time interval $\text{Δ}t$, centered on t, as $\text{Δ}t$ approaches zero. In calculus, this limit is called the derivative of s with respect to t, and it is denoted $ds/dt$.

Graphically, $\text{Δ}s/\text{Δ}t$ is the slope of a straight line. As $\text{Δ}t$ gets smaller (i.e., more and more magnification), the straight line becomes a better and better approximation of the curve at that one point. In the limit $\text{Δ}t\to 0$, the straight line is tangent to the curve. As Figure 2.8 shows, the instantaneous velocity at time $t$ is the slope of the line that is tangent to the position-versus-time graph at time t. That is,

The steeper the slope, the larger the magnitude of the velocity.

EXAMPLE 2.3 Finding velocity from position graphically

FIGURE 2.9 shows the position-versus-time graph of an elevator.

1. At which labeled point or points does the elevator have the least velocity?

2. At which point or points does the elevator have maximum velocity?

3. Sketch an approximate velocity-versus-time graph for the elevator.

FIGURE 2.9 Position-versus-time graph.

FIGURE 2.9 Full Alternative Text

MODEL Model the elevator as a particle.

VISUALIZE FIGURE 2.9 is the graphical representation.

SOLVE a. At any instant, an object’s velocity is the slope of its position graph. FIGURE 2.10a shows that the elevator has the least velocity—no velocity at all!—at points 1 and 3 where the slope is zero. At point 1, the velocity is only instantaneously zero. At point 3, the elevator has actually stopped and remains at rest.

FIGURE 2.10 The velocity-versus-time graph is found from the slope of the position graph.

FIGURE 2.10 Full Alternative Text

b. The elevator has maximum velocity at 2, the point of steepest slope.

c. Although we cannot find an exact velocity-versus-time graph, we can see that the slope, and hence $v_y$, is initially negative, becomes zero at point 1, rises to a maximum value at point 2, decreases back to zero a little before point 3, then remains at zero thereafter. Thus FIGURE 2.10b shows, at least approximately, the elevator’s velocity-versus-time graph.

REVIEW Once again, the shape of the velocity graph bears no resemblance to the shape of the position graph. You must transfer slope information from the position graph to value information on the velocity graph.