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?
+26367 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
\(\begin{array}{|rcll|} \hline v(t) &=& \sqrt{\dot{x}^2+\dot{y}^2} \\ &=& \sqrt{\Big(-10\cdot \cos(2t)\Big)^2+\Big(-10\cdot \sin(2t)\Big)^2} \\ &=& \sqrt{100\cdot \cos^2(2t)+100\cdot \sin^2(2t) } \\ &=& 10\cdot \sqrt{\cos^2(2t)+\sin^2(2t) } \quad | \quad \cos^2(2t)+\sin^2(2t) = 1 \\ &=& 10\cdot 1 \\ \mathbf{v(t)} &=& \mathbf{10} \\ \hline \end{array} \)
