Zion Wheeler

2022-07-01

Given: $x\left(t\right)=3sin\left(t\right)-3,y\left(t\right)=t-1$ for 0 is less than or equal to t is less than or equal to 2pi.
How do you find the speed of the particle at $t=3$?

Donavan Mack

Expert

Step 1

$\stackrel{\to }{v}=\stackrel{\to }{r}\prime \left(t\right)=\left(\begin{array}{c}x\prime \left(t\right)\\ y\prime \left(t\right)\end{array}\right)=\left(\begin{array}{c}3\mathrm{cos}t\\ 1\end{array}\right)$
Step 2
speed, $\stackrel{.}{s}=\sqrt{\stackrel{\to }{v}\cdot \stackrel{\to }{v}}$
$=\sqrt{\left(\begin{array}{c}3\mathrm{cos}t\\ 1\end{array}\right)\cdot \left(\begin{array}{c}3\mathrm{cos}t\\ 1\end{array}\right)}$
$=\sqrt{9{\mathrm{cos}}^{2}t+1}$
${\stackrel{.}{s}}_{t=3}=\sqrt{9{\mathrm{cos}}^{2}\left(3\right)+1}$

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