x(t)=2sin(((πt)/3)) y(t)=−3cos(((πt)/3))+4 cm What is the velocity at t=1s


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Question Number 183440 by aurpeyz last updated on 25/Dec/22

$x (t) = 2 s i n (\frac{π t}{3})$ $y (t) = - 3 c o s (\frac{π t}{3}) + 4 c m$ $W h a t i s t h e v e l o c i t y a t t = 1 s$
	Answered by cortano1 last updated on 26/Dec/22
	$v = (dy/dx) , t=1 { (((dy/dt)=π sin (((πt)/3)))),(((dx/dt) = ((2π)/3) cos (((πt)/3)))) :} (dy/dx) = (dy/dt) .(dt/dx) = π sin (((πt)/3)).(3/(2π cos (((πt)/3)) )) = (3/2) tan (((πt)/3)) , t=1 ((dy/dx))_(t=1) = (3/2).tan ((π/3))=((3(√3))/2) cm/sec$
	$v = \frac{d y}{d x}, t = 1$ ${\begin{cases} \frac{d y}{d t} = π \sin (\frac{π t}{3}) \\ \frac{d x}{d t} = \frac{2 π}{3} \cos (\frac{π t}{3}) \end{cases}$ $\frac{d y}{d x} = \frac{d y}{d t} . \frac{d t}{d x} = π \sin (\frac{π t}{3}) . \frac{3}{2 π \cos (\frac{π t}{3})}$ $= \frac{3}{2} \tan (\frac{π t}{3}), t = 1$ ${(\frac{d y}{d x})}_{t = 1} = \frac{3}{2} . \tan (\frac{π}{3}) = \frac{3 \sqrt{3}}{2} cm / \sec$
	Commented by mr W last updated on 26/Dec/22

	$b u t \frac{d y}{d x} i s n o t t h e v e l o c i t y, {i t}^{'} s t h e$ $d i r e c t i o n o f v e l o c i t y .$
	Answered by mr W last updated on 26/Dec/22

	$v_{x} = \frac{d x}{d t} = \frac{2 π}{3} \cos \frac{π t}{3} = \frac{2 π}{3} \cos \frac{π}{3} = \frac{π}{3}$ $v_{y} = \frac{d y}{d t} = π \sin \frac{π t}{3} = π \sin \frac{π}{3} = \frac{π \sqrt{3}}{2}$ $v = \sqrt{v_{x}^{2} + v_{y}^{2}} = \sqrt{{(\frac{π}{3})}^{2} + {(\frac{π \sqrt{3}}{2})}^{2}} = \frac{π \sqrt{31}}{6} \frac{m}{s}$
	Commented by aurpeyz last updated on 26/Dec/22

	$S o f i n a l a n s w e r w i l l b e 2.92 ? u s i n g π = 3.142$ $i n i t i a l l y a t m y f i r s t a t t e m p t i w a s c o n f u s e d$ $h o w t h a t i w i l l u s e π a s 3.142 a n d t h e n a s$ $180 f o r s i n a n d c o s$
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