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QUESTION

If the point $P$ has position vector
$\textbf{r}=\sin(2t)\textbf{i}-\cos(2t)\textbf{j}+t^3\textbf{k}$
at time $t$, then find the speed of $P$ as a function of $t$.

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ANSWER

$\textbf{r}=\sin(2t)\textbf{i}-\cos(2t)\textbf{j}+t^3\textbf{k}\\
\dot{\textbf{r}}=2\cos(2t)\textbf{i}+2\sin(2t)\textbf{j}+3t^2\textbf{k}$\\
Speed=$|\dot{\textbf{r}}|=\{4\cos^2(2t)+4\sin^2(2t)+9t^4\}^\frac{1}{2}=(4+9t^4)^\frac{1}{2}$




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