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 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.