A particle moves along a straight line. Its velocity v(t) (in meters per second) for 0≤t≤8 seconds is described graphically as follows.
• From t=0 to t=2, v(t) increases linearly from 0 m/s to 4 m/s.
• From t=2 to t=5, v(t) decreases linearly to −3 m/s at t=5.
• From t=5 to t=8, v(t) increases linearly to 2 m/s at t=8.
Assume the lines between these key points are straight segments.
(a) Identify the open sub-intervals of 0<t<8 on which the particle moves to the right and the open sub-intervals on which it moves to the left. Provide justification without performing explicit integration.
(b) Describe how the signed area under the velocity graph on [0,8] is related to the particle’s net change in position, and how the total (unsigned) area is related to the total distance traveled.
(c) Without performing any numeric calculation, decide whether the magnitude of the particle’s displacement on [0,8] is less than, equal to, or greater than the total distance it travels on that interval. Clearly explain your reasoning.