A particle, $P$, is moving along the $x$-axis. The velocity of particle $P$ at time $t$ is given by $v_{P}(t) = \sin\left(t^{1.5}\right)$ for $0 \leq t \leq \pi$. At time $t = 0$, particle $P$ is at position $x = 5$. A second particle, $Q$, also moves along the $x$-axis. The velocity of particle $Q$ at time $t$ is given by $v_{Q}(t) = (t - 1.8) \cdot 1.25^{t}$ for $0 \leq t \leq \pi$. At time $t = 0$, particle $Q$ is at position $x = 10$. (a) Find the positions of particles $P$ and $Q$ at time $t = 1$. (b) Are particles $P$ and $Q$ moving toward each other or away from each other at time $t = 1$? Explain your reasoning. (c) Find the acceleration of particle $Q$ at time $t = 1$. Is the speed of particle $Q$ increasing or decreasing at time $t = 1$? Explain your reasoning. (d) Find the total distance traveled by particle $P$ over the time interval $0 \leq t \leq \pi$.
A particle, $P$, is moving along the $x$-axis. The velocity of particle $P$ at time $t$ is given by $v_{P}(t) = \sin\left(t^{1.5}\right)$ for $0 \leq t \leq \pi$. At time $t = 0$, particle $P$ is at position $x = 5$.
A second particle, $Q$, also moves along the $x$-axis. The velocity of particle $Q$ at time $t$ is given by $v_{Q}(t) = (t - 1.8) \cdot 1.25^{t}$ for $0 \leq t \leq \pi$. At time $t = 0$, particle $Q$ is at position $x = 10$.\\
(a) Find the positions of particles $P$ and $Q$ at time $t = 1$.\\
(b) Are particles $P$ and $Q$ moving toward each other or away from each other at time $t = 1$? Explain your reasoning.\\
(c) Find the acceleration of particle $Q$ at time $t = 1$. Is the speed of particle $Q$ increasing or decreasing at time $t = 1$? Explain your reasoning.\\
(d) Find the total distance traveled by particle $P$ over the time interval $0 \leq t \leq \pi$.