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{\bf Question}

A particle moves along the curve ${\bf r}  =  (t^2 + 1) {\bf i} +
(3t -2) {\bf j} + (2t^3 - 4t^2) {\bf k}$.  Find the velocity,
acceleration, speed and magnitude of the acceleration at $t=2$.

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{\bf Answer}

\begin{eqnarray*} {\bf r} & = & (t^2 + 1) {\bf i} + (3t -2) {\bf
j} + (2t^3 - 4t^2) {\bf k} \\ {\bf v} = \dot {\bf r} & = & 2t{\bf
i} + 3{\bf j} + (6t^2 - 8t){\bf k} \\ {\bf v} & = & 4{\bf i} +
3{\bf j} + 8{\bf k} {\rm \ \ \ \ \ \ \ \ \ \ at\ } t = 2 \\ {\bf
a} = \ddot {\bf r}& = & 2{\bf i} + (12t - 8) {\bf k} \\ & = &
2{\bf i} + 16{\bf k} {\rm \ \ \ \ \ \ \ \ \ \ \ \ \ \ at\ } t = 2
\end{eqnarray*}



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