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 $0 \leq t \leq \pi$.

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.

Two particles, $H$ and $J$, are moving along the $x$-axis. For $0 \leq t \leq 5$, the position of particle $H$ at time $t$ is given by $x _ { H } ( t ) = e ^ { t ^ { 2 } - 4 t }$ and the velocity of particle $J$ at time $t$ is given by $v _ { J } ( t ) = 2 t \left( t ^ { 2 } - 1 \right) ^ { 3 }$.
A. Find the velocity of particle $H$ at time $t = 1$. Show the work that leads to your answer.
B. During what open intervals of time $t$, for $0 < t < 5$, are particles $H$ and $J$ moving in opposite directions? Give a reason for your answer.
C. It can be shown that $v _ { J } ^ { \prime } ( 2 ) > 0$. Is the speed of particle $J$ increasing, decreasing, or neither at time $t = 2$ ? Give a reason for your answer.
D. Particle $J$ is at position $x = 7$ at time $t = 0$. Find the position of particle $J$ at time $t = 2$. Show the work that leads to your answer.

Two points P and Q move on a number line. Their positions $x _ { 1 } , x _ { 2 }$ at time $t ( t \geq 0 )$ are $$x _ { 1 } = t ^ { 3 } - 2 t ^ { 2 } + 3 t , \quad x _ { 2 } = t ^ { 2 } + 12 t$$ Find the distance between points P and Q at the moment when their velocities are equal. [4 points]

At time $t = 0$, two points P and Q simultaneously depart from the origin and move on a number line. Their velocities at time $t$ ($t \geq 0$) are respectively $$v_1(t) = t^2 - 6t + 5, \quad v_2(t) = 2t - 7$$ Let $f(t)$ denote the distance between points P and Q at time $t$. The function $f(t)$ increases on the interval $[0, a]$, decreases on the interval $[a, b]$, and increases on the interval $[b, \infty)$. Find the distance traveled by point Q from time $t = a$ to time $t = b$. (Here, $0 < a < b$) [4 points]
(1) $\frac{15}{2}$
(2) $\frac{17}{2}$
(3) $\frac{19}{2}$
(4) $\frac{21}{2}$
(5) $\frac{23}{2}$