Question

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

Answer 1

`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(t)=(x(t),y(t))`
By finding the first derivative
`f'(t)=(x'(t),y'(t))`

`f'(t)=(2cos2t+2sin2t,2cos(2t-pi/4))`

substitute for `t=-pi/3`

the horizontal component of the velocity of the particle`(y^0)`
=`-1-sqrt3`

the vertical component of the velocity of particle`(x^0)`
=`(-sqrt6+sqrt2)/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

`tantheta=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

I hope this was helpful.