Answer:
(a) The particle is moving to the right in the interval , to the left in the interval , and stops when t = 0, , and .
(b) The equation of the particle's displacement is ; Final position of the particle .
(c) The total distance traveled by the particle is 9.67 (2 d.p.)
Step-by-step explanation:
(a) The particle is moving towards the right direction when v(t) > 0 and to the left direction when v(t) < 0. It stops when v(t) = 0 (no velocity).
Situation 1: When the particle stops.
.
Situation 2: When the particle moves to the right.
Since the term is always positive for all value of t of the interval , hence the determining factor is cos(t). Then, the question becomes of when is cos(t) positive? The term cos(t) is positive in the first and third quadrant or when .
*Note that parentheses are used to demonstrate the interval of t in which cos(t) is strictly positive, implying that the endpoints of the interval are non-inclusive for the set of values for t.
Situation 3: When the particle moves to the left.
Similarly, the term is always positive for all value of t of the interval , hence the determining factor is cos(t). Then, the question becomes of when is cos(t) positive? The term cos(t) is negative in the second and third quadrant or .
(b) The equation of the particle's displacement can be evaluated by integrating the equation of the particle's velocity.
To integrate the expression , u-substitution is performed where
.
Therefore, .
The final position of the particle is .
(c)
.
The total distance traveled by the particle in the given time interval is.