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{\bf Question}

Consider the vector ${\bf r}(t) = a \cos \omega t {\bf i} + a \sin
\omega t {\bf j} + a \omega t {\bf k}$

Find: (a) the velocity vector; (b) the acceleration vector; (c)
the speed.

\vspace{.25in}

{\bf Answer}

$${\bf r}(t) = a \cos \omega t {\bf i} + a \sin \omega t {\bf j} +
a \omega t {\bf k}$$ [This is helical motion]

\begin{description}
\item[(a)]
${\bf v}(t) = \frac{d{\bf r}}{dt} = (-\omega a \sin \omega t, a
\omega \cos \omega t, a\omega)$
\item[(b)]
${\bf a}(t) = \frac{d^2{\bf r}}{dt^2} = (-\omega^2 a^2 \cos \omega
t,- a \omega^2 \sin \omega t, 0)$
\item[(c)]
\begin{eqnarray*} |{\bf v}(t)| = \mathrm{speed}  & = & [\omega^2 a^2 \sin^2 \omega t +
\omega^2 a^2 \cos^2 \omega t + \omega^2 a^2]^{\frac{1}{2}} \\ & =
& \omega a[1+1]^{\frac{1}{2}}\\ & = & \sqrt2 \omega a
\end{eqnarray*}
\end{description}


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