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\textbf{Vector Functions and Curves}

\textit{\textbf{One variable functions}}
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\textbf{Question}

It is given that the position and velocity vectors of a moving object satisfy
$\underline{v}(t)=2\underline{r}(t)$ for all times $t$. If
$\underline{r}(0)= \underline{r}_0$,
find $\underline{r}(t)$ and $\underline{a}(t)$, the acceleration. Also determine the
path of motion.


\textbf{Answer}

\begin{eqnarray*}
\frac{d \underline{r}}{dt} & = & \underline{v}(t) =
2\underline{r}(t)\\
\underline{r}(0) & = & \underline{r}_0
\end{eqnarray*}
And so,
\begin{eqnarray*}
\underline{r}(t) & = & \underline{r}(0)e^{2t} =
\underline{r}_0e^{2t}\\
\underline{a}(t) & = & \frac{d \underline{v}}{dt} = 2
\frac{d\underline{r}}{dt}\\
& = & 4\underline{r}_0e^{2t}
\end{eqnarray*}

The path is a half-line from the origin in the direction of
$\underline{r}_0$.


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