# An object's two dimensional velocity is given by v(t) = ( tsin(pi/3t) , 2cos(pi/2t )- t ). What is the object's rate and direction of acceleration at t=2 ?

Apr 18, 2016

$\textcolor{red}{{a}_{x} \left(2\right) = - 0 , 184}$
$\textcolor{g r e e n}{{a}_{y} \left(2\right) = - 1}$
$a \left(2\right) = 1 , 017$

#### Explanation:

$v \left(t\right) = \left(t \sin \left(\frac{\pi}{3} t\right) , 2 \cos \left(\frac{\pi}{2} t\right) - t\right)$

${a}_{x} \left(t\right) = \frac{d}{d t} \left(t \sin \left(\frac{\pi}{3} t\right)\right) = 1 \cdot \sin \left(\frac{\pi}{3} t\right) + t \cdot \frac{\pi}{3} \cos \left(\frac{\pi}{3} t\right)$

${a}_{x} \left(2\right) = \sin \left(2 \frac{\pi}{3}\right) + 2 \cdot \frac{\pi}{3} \cdot \cos \left(2 \frac{\pi}{3}\right)$

${a}_{x} \left(2\right) = 0 , 866 + 2 \cdot \frac{\pi}{3} \cdot \left(- \frac{1}{2}\right)$

${a}_{x} \left(2\right) = 0 , 866 - \frac{\pi}{3}$

${a}_{x} \left(2\right) = 0 , 866 - 1 , 05$

$\textcolor{red}{{a}_{x} \left(2\right) = - 0 , 184}$

${a}_{y} \left(t\right) = \frac{d}{d t} \left(2 \cos \left(\frac{\pi}{2} t\right) - t\right)$

${a}_{y} \left(t\right) = - 2 \cdot \frac{\pi}{2} \cdot \sin \left(\frac{\pi}{2} t\right) - 1$

${a}_{y} \left(t\right) = - \pi \cdot \sin \left(\frac{\pi}{2} t\right) - 1$

${a}_{y} \left(2\right) = - \pi \cdot \sin \left(\frac{\pi}{2} \cdot 2\right) - 1$

${a}_{y} \left(2\right) = - \pi \cdot \sin \pi - 1 \text{ } \sin \pi = 0$

${a}_{y} \left(2\right) = - \pi \cdot 0 - 1$

$\textcolor{g r e e n}{{a}_{y} \left(2\right) = - 1}$

$a \left(2\right) = \sqrt{{\left({a}_{x} \left(2\right)\right)}^{2} + \left({\left({a}_{y} \left(2\right)\right)}^{2}\right)}$

a(2)=sqrt((-0,184)^2+(-1)^2))

$a \left(2\right) = \sqrt{0 , 034 + 1}$

$a \left(2\right) = \sqrt{1 , 034}$

$a \left(2\right) = 1 , 017$