1.3.2 Displacement

We said that motion is the change in an object’s position with time, but how do we show a change of position? A motion diagram is the perfect tool. FIGURE 1.6 is the motion diagram of a sled sliding down a snow-covered hill. To show how the sled’s position changes between, say, $t_{3} = 3 s$ and $t_{4} = 4 s,$ we draw a vector arrow between the two dots of the motion diagram. This vector is the sled’s displacement, which is given the symbol $Δ \vec{r}$ . The Greek letter delta $(Δ)$ is used in math and science to indicate the change in a quantity. In this case, as we’ll show, the displacement $Δ \vec{r}$ is the change in an object’s position.

The figure illustrates the displacement of a sled between two frames on the x y coordinate system.

FIGURE 1.6 Full Alternative Text

NOTE $Δ \vec{r}$ is a single symbol. It shows “from here to there.” You cannot cancel out or remove the $Δ$ .

Notice how the sled’s position vector $\vec{r}_{4}$ is a combination of its early position $\vec{r}_{3}$ with the displacement vector $Δ \vec{r}$ . In fact, $\vec{r}_{4}$ is the vector sum of the vectors $\vec{r}_{3}$ and $Δ \vec{r}$ . This is written

\vec{r}_{4} = \vec{r}_{3} + Δ \vec{r}

(1.1)

Here we’re adding vector quantities, not numbers, and vector addition differs from “regular” addition. We’ll explore vector addition more thoroughly in Chapter 3, but for now you can add two vectors $\vec{A}$ and $\vec{B}$ with the three-step procedure of 《 TACTICS BOX 1.1.

TACTICS BOX 1.1 Vector addition

To add $\vec{B} to \vec{A}$ :

The figure illustrates two vector arrows labeled vector "A" and vector B.

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1 Draw $\vec{A}$

The figure illustrates a vector arrow labeled vector “A” pointing toward the upper right.

2 Place the tail of $\vec{B}$ at the tip of $\vec{A}$ .

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3 Draw an arrow from the tail of $\vec{A}$ to the tip of $\vec{B}$ . This is vector $\vec{A} + \vec{B}$ .

The figure illustrates three vector arrows labeled vector "A", vector B and vector "A" plus vector B.

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If you examine Figure 1.6, you’ll see that the steps of Tactics Box 1.1 are exactly how $\vec{r}_{3}$ and $Δ \vec{r}$ are added to give $\vec{r}_{4}$ .

NOTE A vector is not tied to a particular location on the page. You can move a vector around as long as you don’t change its length or the direction it points. Vector $\vec{B}$ is not changed by sliding it to where its tail is at the tip of $\vec{A}$ .

Equation 1.1 told us that $\vec{r}_{4} = \vec{r}_{3} + Δ \vec{r}$ . This is easily rearranged to give a more precise definition of displacement: The displacement $Δ \vec{r}$ of an object as it moves from one position ${\vec{r}}_{a}$ to a different position ${\vec{r}}_{b}$ is

Δ \vec{r} = \vec{r}_{b} - \vec{r}_{a}

(1.2)

That is, displacement is the change (i.e., the difference) in position. Graphically, $Δ \vec{r}$ is a vector arrow drawn from position ${\vec{r}}_{a}$ to position ${\vec{r}}_{b}$ .