An object moves along the $x$-axis with initial position $x ( 0 ) = 2$. The velocity of the object at time $t \geq 0$ is given by $v ( t ) = \sin \left( \frac { \pi } { 3 } t \right)$. (a) What is the acceleration of the object at time $t = 4$? (b) Consider the following two statements. Statement I: For $3 < t < 4.5$, the velocity of the object is decreasing. Statement II: For $3 < t < 4.5$, the speed of the object is increasing. Are either or both of these statements correct? For each statement provide a reason why it is correct or not correct. (c) What is the total distance traveled by the object over the time interval $0 \leq t \leq 4$? (d) What is the position of the object at time $t = 4$?
An object moves along the $x$-axis with initial position $x ( 0 ) = 2$. The velocity of the object at time $t \geq 0$ is given by $v ( t ) = \sin \left( \frac { \pi } { 3 } t \right)$.\\
(a) What is the acceleration of the object at time $t = 4$?\\
(b) Consider the following two statements.\\
Statement I: For $3 < t < 4.5$, the velocity of the object is decreasing.\\
Statement II: For $3 < t < 4.5$, the speed of the object is increasing.\\
Are either or both of these statements correct? For each statement provide a reason why it is correct or not correct.\\
(c) What is the total distance traveled by the object over the time interval $0 \leq t \leq 4$?\\
(d) What is the position of the object at time $t = 4$?