The following table gives coordinates of a particle moving through space along a smooth curve.
$\begin{array}{|c|c|c|c|}
\hline \mathbf{t} & x & y & z \
\hline 0.5 & 5.8 & 9.1 & 4.3 \
\hline 1 & 12.6 & 14.9 & 16.8 \
\hline 1.5 & 25.6 & 21.2 & 29.4 \
\hline 2 & 39.2 & 39.5 & 37.9 \
\hline 2.5 & 42.4 & 42.4 & 43 \
\hline
\end{array}$

Find the average velocity over the time interval $[ 0.5,1.5 ]$. Select the correct answer.
A) $\mathbf { v } = 25.1 \mathbf { i } + 19.8 \mathbf { j } + 19.8 \mathbf { k }$
B) $\mathbf { v } = 25.1 \mathbf { i } + 12.1 \mathbf { j } + 19.8 \mathbf { k }$
C) $\mathbf { v } = 12.1 \mathbf { i } + 12.1 \mathbf { j } + 25.1 \mathbf { k }$
D) $\mathbf { v } = 19.8 \mathbf { i } + 12.1 \mathbf { j } + 25.1 \mathbf { k }$
E) $v = 25.1 \mathbf { i } + 19.8 \mathbf { j } + 25.1 \mathbf { k }$

Question

The following table gives coordinates of a particle moving through space along a smooth curve.
$\begin{array}{|c|c|c|c|}
\hline \mathbf{t} & x & y & z \
\hline 0.5 & 5.8 & 9.1 & 4.3 \
\hline 1 & 12.6 & 14.9 & 16.8 \
\hline 1.5 & 25.6 & 21.2 & 29.4 \
\hline 2 & 39.2 & 39.5 & 37.9 \
\hline 2.5 & 42.4 & 42.4 & 43 \
\hline
\end{array}$

Find the average velocity over the time interval $[ 0.5,1.5 ]$. Select the correct answer.
A) $\mathbf { v } = 25.1 \mathbf { i } + 19.8 \mathbf { j } + 19.8 \mathbf { k }$
B) $\mathbf { v } = 25.1 \mathbf { i } + 12.1 \mathbf { j } + 19.8 \mathbf { k }$
C) $\mathbf { v } = 12.1 \mathbf { i } + 12.1 \mathbf { j } + 25.1 \mathbf { k }$
D) $\mathbf { v } = 19.8 \mathbf { i } + 12.1 \mathbf { j } + 25.1 \mathbf { k }$
E) $v = 25.1 \mathbf { i } + 19.8 \mathbf { j } + 25.1 \mathbf { k }$

Quizplus · Accepted Answer

We can use the formula for the average velocity:
$$\mathbf{v}_{\text{ave}} = \frac{\mathbf{r}(t_2) - \mathbf{r}(t_1)}{t_2 - t_1}$$
where $\mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j} + z(t)\mathbf{k}$ is the position vector.

Substituting the given values, we get:
\begin{align*}
\mathbf{v}_{\text{ave}} &= \frac{\mathbf{r}(1.5) - \mathbf{r}(0.5)}{1.5 - 0.5} \
&= \frac{(25.6\mathbf{i} + 21.2\mathbf{j} + 29.4\mathbf{k}) - (5.8\mathbf{i} + 9.1\mathbf{j} + 4.3\mathbf{k})}{1} \
&= 19.8\mathbf{i} + 12.1\mathbf{j} + 25.1\mathbf{k}
\end{align*}

Therefore, the correct answer is (D).

The Following Table Gives Coordinates of a Particle Moving Through