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\begin{center}
\textbf{Vector Functions and Curves}

\textit{\textbf{One variable functions}}
\end{center}

\textbf{Question}

Find the velocity, speed and acceleration of the particle with
position given by $\un{r}(t)$ at time
$t$. Also determine the particles path.

$$\underline{r}= at\cos\omega t\underline{i}+ at\sin\omega
t\underline{j}+ b\ln 4\underline{k}$$

\textbf{Answer}

Position: $\underline{r}=at\cos \omega t \underline{i} +at\sin\omega t
+b\ln t\underline{k}$

Velocity: $\underline{v}=a(\cos\omega t -\omega t\sin\omega
t)\underline{i}+ a(\sin\omega t +\omega t \cos\omega t)\underline{j}
+(b/t)\underline{k}$

Speed: $\it{v}=\sqrt{a^2(1+\omega^2t^2)+(b^2/t^2)}$

Acceleration:
\begin{eqnarray*}
\underline{a} & = & -a\omega(2\sin\omega t+ \omega \cos
\omega t)\underline{i}\\
& & +a\omega (2\cos\omega t -\omega\cos\omega
t)\underline{j}\\
& & -(b^2/t^2)\underline{k}
\end{eqnarray*}

Path: a spiral on the surface $x^2+y^2=a^2e^{z/b}$.

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