Particle $P$ moves along the $x$-axis such that, for time $t > 0$, its position is given by $x_P(t) = 6 - 4e^{-t}$. Particle $Q$ moves along the $y$-axis such that, for time $t > 0$, its velocity is given by $v_Q(t) = \dfrac{1}{t^2}$. At time $t = 1$, the position of particle $Q$ is $y_Q(1) = 2$. (a) Find $v_P(t)$, the velocity of particle $P$ at time $t$. (b) Find $a_Q(t)$, the acceleration of particle $Q$ at time $t$. Find all times $t$, for $t > 0$, when the speed of particle $Q$ is decreasing. Justify your answer. (c) Find $y_Q(t)$, the position of particle $Q$ at time $t$. (d) As $t \to \infty$, which particle will eventually be farther from the origin? Give a reason for your answer.
Particle $P$ moves along the $x$-axis such that, for time $t > 0$, its position is given by $x_P(t) = 6 - 4e^{-t}$. Particle $Q$ moves along the $y$-axis such that, for time $t > 0$, its velocity is given by $v_Q(t) = \dfrac{1}{t^2}$. At time $t = 1$, the position of particle $Q$ is $y_Q(1) = 2$.
(a) Find $v_P(t)$, the velocity of particle $P$ at time $t$.
(b) Find $a_Q(t)$, the acceleration of particle $Q$ at time $t$. Find all times $t$, for $t > 0$, when the speed of particle $Q$ is decreasing. Justify your answer.
(c) Find $y_Q(t)$, the position of particle $Q$ at time $t$.
(d) As $t \to \infty$, which particle will eventually be farther from the origin? Give a reason for your answer.