x-t-2sin-pit-3-y-t-3cos-pit-3-4-cm-What-is-the-velocity-at-t-1s-

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 = \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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