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# help

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A particle is moving so that its position at time $$t$$ is given by the parametric equations \begin{align*} x &= 5\sin(-2t) \\ y &= 5\cos(2t). \end{align*} What is the speed of the particle?

Nov 5, 2019

#1
+23862
+3

A particle is moving so that its position at time  is given by the parametric equations  \begin{align*} x &= 5\sin(-2t) \\ y &= 5\cos(2t). \end{align*}
What is the speed of the particle?

\begin{array}{|rcll|} \hline \begin{align*} x &= 5\sin(-2t) & y &= 5\cos(2t) \\\\ \dot{x} &= (-2)\cdot 5\cos(-2t) & \dot{y} &= 2\cdot 5\cdot (-\sin(2t)) \\ &= -10\cdot \cos(-2t) & &= -10\cdot \sin(2t) \\ &= -10\cdot \cos(2t) \\ \end{align*} \\ \hline \end{array}

speed

$$\begin{array}{|rcll|} \hline v(t) &=& \sqrt{\dot{x}^2+\dot{y}^2} \\ &=& \sqrt{\Big(-10\cdot \cos(2t)\Big)^2+\Big(-10\cdot \sin(2t)\Big)^2} \\ &=& \sqrt{100\cdot \cos^2(2t)+100\cdot \sin^2(2t) } \\ &=& 10\cdot \sqrt{\cos^2(2t)+\sin^2(2t) } \quad | \quad \cos^2(2t)+\sin^2(2t) = 1 \\ &=& 10\cdot 1 \\ \mathbf{v(t)} &=& \mathbf{10} \\ \hline \end{array}$$

Nov 5, 2019
#2
+8813
+2

Thank you heureka!

See http://mathworld.wolfram.com/ParametricEquations.html

There it is explained.

!

Nov 5, 2019
#3
+23862
+1

Thank you, asinus !

heureka  Nov 6, 2019