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

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

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Answer 1 · 2016-04-18T08:31:31

$a_{x} (2) = - 0, 184$ $color(red)(a_x(2)=-0,184)$
$a_{y} (2) = - 1$ $color(green)(a_y(2)=-1)$
$a (2) = 1, 017$ $a(2)=1,017$

Explanation:

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

$a_{x} (t) = \frac{d}{d t} (t sin (\frac{π}{3} t)) = 1 \cdot sin (\frac{π}{3} t) + t \cdot \frac{π}{3} cos (\frac{π}{3} t)$ $a_x(t)=d/(d t)(t sin(pi/3t))=1*sin(pi/3t)+t*pi/3cos(pi/3t)$

$a_{x} (2) = sin (2 \frac{π}{3}) + 2 \cdot \frac{π}{3} \cdot cos (2 \frac{π}{3})$ $a_x(2)=sin(2pi/3)+2*pi/3*cos(2pi/3)$

$a_{x} (2) = 0, 866 + 2 \cdot \frac{π}{3} \cdot (- \frac{1}{2})$ $a_x(2)=0,866+2*pi/3*(-1/2)$

$a_{x} (2) = 0, 866 - \frac{π}{3}$ $a_x(2)=0,866-pi/3$

$a_{x} (2) = 0, 866 - 1, 05$ $a_x(2)=0,866-1,05$

$a_{x} (2) = - 0, 184$ $color(red)(a_x(2)=-0,184)$

$a_{y} (t) = \frac{d}{d t} (2 cos (\frac{π}{2} t) - t)$ $a_y(t)=d/(d t)(2 cos(pi/2t)-t)$

$a_{y} (t) = - 2 \cdot \frac{π}{2} \cdot sin (\frac{π}{2} t) - 1$ $a_y(t)=-2*pi/2*sin(pi/2t)-1$

$a_{y} (t) = - π \cdot sin (\frac{π}{2} t) - 1$ $a_y(t)=-pi*sin(pi/2t)-1$

$a_{y} (2) = - π \cdot sin (\frac{π}{2} \cdot 2) - 1$ $a_y(2)=-pi*sin(pi/2*2)-1$

$a_{y} (2) = - π \cdot sin π - 1 sin π = 0$ $a_y(2)=-pi*sinpi-1" "sin pi=0$

$a_{y} (2) = - π \cdot 0 - 1$ $a_y(2)=-pi*0-1$

$a_{y} (2) = - 1$ $color(green)(a_y(2)=-1)$

$a (2) = \sqrt{{(a_{x} (2))}_{x}^{2} + ({(a_{y} (2))}_{y}^{2})}$ $a(2)=sqrt((a_x(2))^2+((a_y(2))^2))$

$a (2) = \sqrt{{(- 0, 184)}^{2} + {(- 1)}^{2}})$ $a(2)=sqrt((-0,184)^2+(-1)^2))$

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

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

$a (2) = 1, 017$ $a(2)=1,017$

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

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An object's two dimensional velocity is given by v(t)=(tsin(π3t),2cos(π2t)−t)v(t) = ( tsin(pi/3t) , 2cos(pi/2t )- t ). What is the object's rate and direction of acceleration at t=2t=2 ?

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