# What is the instantaneous velocity of an object moving in accordance to  f(t)= (sin2t-cos2t,sin(2t-pi/4))  at  t=(-pi)/3 ?

Apr 12, 2018

$v$=$3.346$ length unit/unit of time
And makes an angle with the positive direction of $x$ axis $\theta$
$\theta = 190.72$ degrees

#### Explanation:

This function is a Parametric function of motion of a particle
which represents the relation between $x , y$ and $t$.
$f \left(t\right) = \left(x \left(t\right) , y \left(t\right)\right)$
By finding the first derivative
$f ' \left(t\right) = \left(x ' \left(t\right) , y ' \left(t\right)\right)$

$f ' \left(t\right) = \left(2 \cos 2 t + 2 \sin 2 t , 2 \cos \left(2 t - \frac{\pi}{4}\right)\right)$

substitute for $t = - \frac{\pi}{3}$

the horizontal component of the velocity of the particle$\left({y}^{0}\right)$
=$- 1 - \sqrt{3}$

the vertical component of the velocity of particle$\left({x}^{0}\right)$
=$\frac{- \sqrt{6} + \sqrt{2}}{2}$

so the magnitude of the velocity of the particle=sqrt((x^0)^2+(y^0)^2

=$3.346$ length unit/unit of time

and its direction can be given through the relation

$\tan \theta = {y}^{0} / {x}^{0}$=$0.18946$

$\theta = 10.72$ degrees

but it's in the third quadrant since both ${x}^{0} , {y}^{0}$ are negative values so $\theta = 190.72$ degrees