A bicyclist travels along a straight road. Her velocity v(t) in meters per second is recorded every minute. A portion of the data is summarized below:
Time t (min): 0 1 2 3 4 5
Average v(t) on [t,t+1]: 4 6 10 8 5 2
(All velocities are positive in the forward direction.)
(a) Without converting units, write an expression, involving definite integrals, for the bicyclist’s net change in position over the five–minute interval from t=0 to t=5.
(b) Suppose the data analyst accidentally records the second minute of travel (from t=1 to t=2) with the limits of integration reversed, writing ∫21v(t)dt instead of ∫12v(t)dt. Write a corrected expression for the total displacement that incorporates this mistaken term while preserving mathematical accuracy.
(c) Using the average velocities provided, compute the bicyclist’s total displacement in meters. Clearly indicate how the term from part (b) affects your calculation.
(d) Interpret the sign of each term in your expression from part (b) within the physical context of the bicyclist’s motion.