An experimental trolley moves along a track. A detailed position–time graph for the trolley is described as follows for the time interval 0 ≤ t ≤ 8 s. • From t = 0 s to t = 2 s the graph is a straight line rising from x = 0 m to x = 2 m. • From t = 2 s to t = 5 s the graph follows a smooth cubic-like curve that first becomes progressively steeper until t = 3.5 s, then becomes progressively less steep, reaching x = 7 m at t = 5 s. The point of maximum slope occurs at t = 3.5 s. • From t = 5 s to t = 8 s the graph is a straight line with negative slope, descending uniformly to x = 3 m at t = 8 s. Label point D at t = 1 s, point E at t = 3.5 s, and point F at t = 6.5 s.
(a) Calculate the average velocity of the trolley during the interval 0 ≤ t ≤ 2 s.
(b) Compare the instantaneous velocities at points D, E, and F, ranking them from greatest to least and justifying each comparison.
(c) Identify the time at which the trolley’s instantaneous acceleration is zero within the interval 2 s < t < 5 s and explain how the graph supports your answer.
(d) Determine the sign of the trolley’s acceleration during the interval 5 s < t < 8 s and justify based on the given information.