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Parametric Curves

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squance:
Im having issues with a problem....

For the following function describing a particle's position at time t, find the equation fo tthe path and state the period of the motion of the path.

u = 2cos(2t)i + 2sin(2t)j

well for the first part I got...

u = 2cos(2t)i + 2sin(2t)j
x(t) = 2cos(2t)         y(t) =2 sin (2t)
x^2 = 2^2 cos^2(2t) y^2 = 2^2 sin^2(2t)
x^2 + y^2 = 4 cos^2(2t) + sin^2(2t) = 1
x^2 + y^2 = 4 (equation of circle with center at (0,0) and radius 2)??

Then for the period of the motion...we choose two values of t and sub them into u, right??

when t = 0...
u = 2cos(0)i + 2 sin(o)j
i + 2j
=(0,2)
when t = 2
u = 2 cos (2X2)i + 2cos(2x2)j......
u = 2 cos4 i + 2 cos (4) j....
(I don't know how to work out the above line...)

Can someone hlep. me plese?? I think the period of the motion is pi or something but I don';tknow how...

Mao:
The equation of the path is correct
is a circle centered (0,0) with radius of 2.

The process to derive the period of motion is easier than you think.

looking at

the period of the circular functions here are

hence the period of u is , as the full period of the two circular functions means u is back to where it started.

squance:
so we don't have to sub t values in into the u equation?? Wait, that only applies when you are trying to find the direction of the curve, right?

Mao:
something or rather...
what do u mean by "direction"??

when you sub t into u you would get a position vector..

squance:
I think the direction of motion???

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