A particle moves along the ~axis so that its velocity at time t is given by v( t) - ( t + 1) sin | ~]. At time t - O, the particle is at position ~ - 1. (a) Find the acceleration of the particle at time t - 2. Is the speed of the particle increasing at t - 2? Why or why not? (b) Find all times t in the open interval O t 3 when the particle changes direction. Justify your answer. (c) Find the total distance traveled by the particle from time t - O until time t - 3. (d) During the time interval O ~ t ~ 3, what is the greatest distance between the particle and the origin? Show the work that leads to your answer.
(a) a(2) - vI(2) - 1.587 or 1.588 | 1 a(2) v(2) - ,38in(2) O 2 ] 1 speed decreasing Speed is decreasing since a(2) O and v(2) O . | with reason
(c) Distance - fo3l~ t)|dt - 4.333 or 4.334
:::l:: (distance particle travels while velocity is negative)
Since the total distance from t - O to t - 3 is 4.334, the particle is stll-I to the left of the origin at t - 3. Hence the greatest distance from the origin is 2.265.