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

1 Answer
Yahia M.
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(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.

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