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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.

$$\underline{r}= a\cos t\sin t\underline{i}+ a\sin^2 t\underline{j}+
a\cos t\underline{k}$$

\textbf{Answer}

Position:
\begin{eqnarray*}
\underline{r} & = & a\cos t \sin t \underline{i} +a\sin^2
t\underline{j} +a\cos t \underline{k}\\
& = & \frac{a}{2}\sin
2t\underline{i} +\frac{a}{2}(1-\cos 2t)\underline{j} -a\sin t
\underline{k}
\end{eqnarray*}

Velocity: $\underline{v}=a\cos 2t \underline{i} +a\sin 2t
\underline{j} -a\sin t\underline{k}$

Speed: $\it{v}=a\sqrt{1+\sin^2 t}$

Acceleration: $\underline{a}=-2a\sin 2t \underline{i}+ 2a\cos 2t
\underline{j} -a\cos t \underline{k}$

Path: the path lies on the sphere $x^2+y^2+z^2=a^2$, on the surface
defined in terms of spherical polar coordinates by $\phi=\theta$, on
the circular cylinder $x^2+y^2=ay$, and on the parabolic cylinder
$ay+z^2=a^2$. Any two of these surfaces can be used to pin down the shape of
the path.


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