What is the instantaneous velocity of an object moving in accordance to $f (t) = (2 sin (2 t + π), t cos 2 t)$ $f(t)= (2sin(2t+pi),tcos2t)$ at $t = \frac{π}{3}$ $t=pi/3$ ?

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Answer 1 · 2018-03-07T20:10:16

$f'(pi/3)=(3 + 2 sqrt(3) π)/(12 - 6 π)$

We have the path of an object as a function of time given by $f(t)=(2sin(2t+pi),tcos2t)$ .

This is the parametric form of the Cartesian equation with

$x=2sin(2t+pi)$
$y=tcos2t$

We want to find the instantaneous velocity of the object at $t=pi"/"3$ . This is equivalent to evaluating the first derivative of $x$ at $t=pi"/"3$ .

The way to find the first derivative of $x$ using these parameters is given by the following formula

$dy/dx=(dy"/"dt)/(dx"/"dt)$

$dx/dt=4cos(2t+pi)$

$dy/dt=cos2t-2tsin2t$

$rArr dy/dx= (cos2t-2tsin2t)/(4cos(2t+pi))$

We now plug in $t=pi/3$ to get

$f'(pi/3)=(3 + 2 sqrt(3) π)/(12 - 6 π)$

What is the instantaneous velocity of an object moving in accordance to f(t)= (2sin(2t+pi),tcos2t) f(t)=(2sin(2t+π),tcos2t) f(t)= (2sin(2t+pi),tcos2t) at t=pi/3 t=π3 t=pi/3 ?