A particle moves in the $xy$-plane so that its position at any time $t$, $0 \leq t \leq \pi$, is given by $x(t) = \frac{t^2}{2} - \ln(1+t)$ and $y(t) = 3\sin t$. (a) Sketch the path of the particle in the $xy$-plane below. Indicate the direction of motion along the path. (b) At what time $t$, $0 \leq t \leq \pi$, does $x(t)$ attain its minimum value? What is the position $(x(t), y(t))$ of the particle at this time? (c) At what time $t$, $0 < t < \pi$, is the particle on the $y$-axis? Find the speed and the acceleration vector of the particle at this time.
A particle moves in the $xy$-plane so that its position at any time $t$, $0 \leq t \leq \pi$, is given by $x(t) = \frac{t^2}{2} - \ln(1+t)$ and $y(t) = 3\sin t$.
(a) Sketch the path of the particle in the $xy$-plane below. Indicate the direction of motion along the path.
(b) At what time $t$, $0 \leq t \leq \pi$, does $x(t)$ attain its minimum value? What is the position $(x(t), y(t))$ of the particle at this time?
(c) At what time $t$, $0 < t < \pi$, is the particle on the $y$-axis? Find the speed and the acceleration vector of the particle at this time.