For 0≤t≤12 seconds, the velocity v(t) (in m/s) of a particle is given graphically by two pieces:
• On 0≤t≤8, the graph of v(t) is the upper semicircle of radius 4 centered at (4,0).
• On 8<t≤12, v(t) is constant at −2.
The particle’s initial position is s(0)=0 meters.
(a) Evaluate the particle’s displacement on [0,8] by interpreting the appropriate area geometrically.
(b) Determine the particle’s total distance traveled on 0≤t≤12.
(c) At what time t∈[0,12] (if any) does the particle return to its initial position? Justify your answer algebraically.
(d) Describe the particle’s motion (direction and relative speed) on each of the two sub-intervals, relating your conclusions to features of the velocity graph.