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 [1, 2].
A) $\mathbf { v } = 21.1 \mathbf { i } + 24.6 \mathbf { j } + 26.6 \mathbf { k }$
B) $\quad \mathbf { v } = 21.1 \mathbf { i } + 26.6 \mathbf { j } + 21.1 \mathbf { k }$
C) $\quad \mathbf { v } = 24.6 \mathbf { i } + 24.6 \mathbf { j } + 21.1 \mathbf { k }$
D) $\mathbf { v } = 26.6 \mathbf { i } + 24.6 \mathbf { j } + 21.1 \mathbf { k }$
E) $\mathbf { v } = 21.1 \mathbf { i } + 26.6 \mathbf { j } + 26.6 \mathbf { k }$

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 [1, 2].
A) $\mathbf { v } = 21.1 \mathbf { i } + 24.6 \mathbf { j } + 26.6 \mathbf { k }$
B) $\quad \mathbf { v } = 21.1 \mathbf { i } + 26.6 \mathbf { j } + 21.1 \mathbf { k }$
C) $\quad \mathbf { v } = 24.6 \mathbf { i } + 24.6 \mathbf { j } + 21.1 \mathbf { k }$
D) $\mathbf { v } = 26.6 \mathbf { i } + 24.6 \mathbf { j } + 21.1 \mathbf { k }$
E) $\mathbf { v } = 21.1 \mathbf { i } + 26.6 \mathbf { j } + 26.6 \mathbf { k }$

Quizplus · Accepted Answer

We can use the formula for average velocity:
$$\mathbf{v}_{\text{avg}}=\frac{\mathbf{r}(2)-\mathbf{r}(1)}{2-1}$$
where $\mathbf{r}(t)$ is the position vector of the particle at time $t$.

Substituting $t=1$ and $t=2$:
\begin{align*}
\mathbf{v}_{\text{avg}}&=\frac{\mathbf{r}(2)-\mathbf{r}(1)}{2-1} \
&=\frac{\begin{pmatrix} 39.2 \ 39.5 \ 37.9 \end{pmatrix}-\begin{pmatrix} 12.6 \ 14.9 \ 16.8 \end{pmatrix}}{2-1} \
&=\begin{pmatrix} 26.6 \ 24.6 \ 21.1 \end{pmatrix}
\end{align*}
Therefore, the answer is $\boxed{\textbf{(D)}}$.

The Following Table Gives Coordinates of a Particle Moving Through