Two experimental shuttles move along the same straight test track. Shuttle M begins at the origin at t=0 seconds with velocity vM(t)=4sin(0.2t)+1 meters per second for t≥0. Shuttle N is located 90 meters ahead of the origin and travels in the same direction (positive x–axis) with acceleration aN(t)=0.04te−0.05t meters per second squared for t≥0. Its initial velocity is 3 m/s at t=0. Assume both models hold for all relevant times.
(a) Obtain an explicit formula for the velocity vN(t) of Shuttle N and show that vN(t)>0 for all t≥0.
(b) Derive expressions for the positions sM(t) and sN(t), each measured from the origin, choosing the constants of integration so that the given initial conditions are satisfied.
(c) Determine all critical points of the function D(t)=sN(t)−sM(t) for t≥0 and classify each as a local minimum, local maximum, or neither.
(d) Prove that exactly one global minimum of D(t) exists for t≥0 and find the time tmin at which it occurs.
(e) Evaluate D(tmin) and interpret its value in the context of the shuttles’ motion. State whether the shuttles ever occupy the same position at the same time and justify your conclusion.