Activity 4.1.4.
Suppose that an object moving along a straight line path has its velocity \(v\) (in meters per second) at time \(t\) (in seconds) given by the piecewise linear function whose graph is pictured at left in the following figure. We view movement to the right as being in the positive direction (with positive velocity), while movement to the left is in the negative direction.
(a)
Determine the total distance traveled and the total change in position on the time interval \(0 \le t \le 2\text{.}\) What is the object’s position at \(t = 2\text{?}\)
(b)
On what time intervals is the moving object’s position function increasing? Why? On what intervals is the object’s position decreasing? Why?
(c)
What is the object’s position at \(t = 8\text{?}\) How many total meters has it traveled to get to this point (including distance in both directions)? Is this different from the object’s total change in position on \(t = 0\) to \(t = 8\text{?}\)
(d)
Find the exact position of the object at \(t = 1, 2, 3, \ldots, 8\) and use this data to sketch an accurate graph of \(y = s(t)\) on the axes provided at right in the figure. How can you use the provided information about \(y = v(t)\) to determine the concavity of \(s\) on each relevant interval?
