Perhaps the most important use of a motion diagram is to determine the acceleration vector $\vec{a}$ at each point in the motion. From its definition in Equation 1.7, we see that $\vec{a}$ points in the same direction as $\text{Δ}\vec{v}$, the change of velocity, so we need to find the direction of $\text{Δ}\vec{v}$. To do so, we rewrite the definition $\text{Δ}\vec{v}=\vec{v}{}_{\text{b}}-\vec{v}{}_{\text{a}}$ as $\vec{v}{}_b=\vec{v}{}_a+\text{Δ}\vec{v}$. This is now a vector addition problem: What vector must be added to $\vec{v}{}_{\text{a}}$ to turn it into $\vec{v}{}_{\text{b}}$? Tactics Box 1.2 shows how to do this.

TACTICS BOX 1.2 Finding the acceleration vector

To find the acceleration as the velocity changes from ${\overrightarrow{v}}_{\text{a}}$ to ${\overrightarrow{v}}_{\text{b}}$, we must determine the change of velocity $\text{Δ}\overrightarrow{v}={\overrightarrow{v}}_{\text{b}}-{\overrightarrow{v}}_{\text{a}}$.

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1 Draw velocity vectors ${\overrightarrow{v}}_{\text{a}}$ and ${\overrightarrow{v}}_{\text{b}}$ with their tails together.

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2 Draw the vector from the tip of ${\overrightarrow{v}}_{\text{a}}$ to the tip of ${\overrightarrow{v}}_{\text{b}}$. This is $\text{Δ}\overrightarrow{v}$ because ${\overrightarrow{v}}_{\text{b}}={\overrightarrow{v}}_{\text{a}}+\text{Δ}\overrightarrow{v}\text{.}$

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3 Return to the original motion diagram. Draw a vector at the middle dot in the direction of $\text{Δ}\overrightarrow{v}$; label it $\overrightarrow{a}$. This is the average acceleration at the midpoint between ${\overrightarrow{v}}_{\text{a}}$ and ${\overrightarrow{v}}_{\text{b}}$.

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Workbook Exercises 21–24

Many Tactics Boxes will refer you to exercises in the Student Workbook where you can practice the new skill.

Notice that the acceleration vector goes beside the middle dot, not beside the velocity vectors. This is because each acceleration vector is determined by the difference between the two velocity vectors on either side of a dot. The length of $\vec{a}$ does not have to be the exact length of $\text{Δ}\vec{v};$ it is the direction of $\vec{a}$ that is most important.

The procedure of 《 TACTICS BOX 1.2 can be repeated to find $\vec{a}$ at each point in the motion diagram. Note that we cannot determine $\vec{a}$ at the first and last points because we have only one velocity vector and can’t find $\text{Δ}\vec{v}$.