1.4 Velocity

It’s no surprise that, during a given time interval, a speeding bullet travels farther than a speeding snail. To extend our study of motion so that we can compare the bullet to the snail, we need a way to measure how fast or how slowly an object moves.

One quantity that measures an object’s fastness or slowness is its average speed, defined as the ratio

average speed = \frac{d i s t a n c e t r a v e l e d}{t i m e i n t e r v a l s p e n t t r a v e l i n g} = \frac{d}{Δ t}

(1.3)

If you drive 15 miles (mi) in 30 minutes $(\frac{1}{2} h)$ , your average speed is

average speed = \frac{15 m i}{\frac{1}{2} h} = 30 m p h

(1.4)

Although the concept of speed is widely used in our day-to-day lives, it is not a sufficient basis for a science of motion. To see why, imagine you’re trying to land a jet plane on an aircraft carrier. It matters a great deal to you whether the aircraft carrier is moving at 20 mph (miles per hour) to the north or 20 mph to the east. Simply knowing that the ship’s speed is 20 mph is not enough information!

A photo of a few runners competing in a race. — The victory goes to the runner with the highest average speed.

It’s the displacement $Δ \vec{r}$ , a vector quantity, that tells us not only the distance traveled by a moving object, but also the direction of motion. Consequently, a more useful ratio than $d / Δ t$ is the ratio $Δ \vec{r} / Δ t$ . In addition to measuring how fast an object moves, this ratio is a vector that points in the direction of motion.

It is convenient to give this ratio a name. We call it the average velocity, and it has the symbol $\vec{v}_{avg}$ . The average velocity of an object during the time interval $Δ t$ , in which the object undergoes a displacement $Δ \vec{r}$ , is the vector

{\vec{v}}_{avg} = \frac{Δ \vec{r}}{Δ t}

(1.5)

An object’s average velocity vector points in the same direction as the displacement vector $Δ \vec{r}$ . This is the direction of motion.

NOTE In everyday language we do not make a distinction between speed and velocity, but in physics the distinction is very important. In particular, speed is simply “How fast?” whereas velocity is “How fast, and in which direction?” As we go along we will be giving other words more precise meanings in physics than they have in everyday language.

As an example, FIGURE 1.9a shows two ships that move 5 miles in 15 minutes. Using Equation 1.5 with $Δ t = 0.25 h$ , we find

\begin{array}{l} {\vec{v}}_{avg A} = (20 mph, north) \\ {\vec{v}}_{avg B} = (20 mph, east) \end{array}

(1.6)

Both ships have a speed of 20 mph, but their velocities differ. Notice how the velocity vectors in FIGURE 1.9b point in the direction of motion.

The figure consists of two parts labeled ("a") and (b) that illustrate the displacement vectors and velocity vectors of two ships labeled "A" and B.

FIGURE 1.9 Full Alternative Text

NOTE Our goal in this chapter is to visualize motion with motion diagrams. Strictly speaking, the vector we have defined in Equation 1.5, and the vector we will show on motion diagrams, is the average velocity $\vec{v}_{avg}$ . But to allow the motion diagram to be a useful tool, we will drop the subscript and refer to the average velocity as simply $\vec{v}$ . Our definitions and symbols, which somewhat blur the distinction between average and instantaneous quantities, are adequate for visualization purposes, but they’re not the final word. We will refine these definitions in Chapter 2, where our goal will be to develop the mathematics of motion.