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Describe the motion of a particle with position $ (x, y) $ as $ t $ varies in a given interval.

$ x = 5 + 2\cos \pi t $, $ \; y = 3 + 2 \sin \pi t $, $ \; 1 \leqslant t \leqslant 2 $

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$x=5+2 \cos \pi t, y=3+2 \sin \pi t \quad \Rightarrow \quad \cos \pi t=\frac{x-5}{2}, \sin \pi t=\frac{y-3}{2}, \quad \cos ^{2}(\pi t)+\sin ^{2}(\pi t)=1 \Rightarrow$$\left(\frac{x-5}{2}\right)^{2}+\left(\frac{y-3}{2}\right)^{2}=1 .$ The motion of the particle takes place on a circle centered at (5,3) with a radius $2 .$ As $t$ goesfrom 1 to 2 , the particle starts at the point (3,3) and moves counterclockwise along the circle $\left(\frac{x-5}{2}\right)^{2}+\left(\frac{y-3}{2}\right)^{2}=1$ to$(7,3) \text { [one-half of a circle }]$

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Describe the motion of a p…

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