Two cyclists, A and B, ride along a straight, flat track that is aligned with the positive x-axis.
• Cyclist A starts at t=0 seconds from x=0 meters and travels with velocity vA(t)=2t+2 meters per second for t≥0.
• Cyclist B starts from the same point 3 seconds later. For B the time variable τ is defined so that τ=0 when B begins. B’s velocity is vB(τ)=4τ+1 meters per second for τ≥0.
Throughout the problem let t denote the number of seconds elapsed since A started (so t≥0).
(1) Express Cyclist B’s position xB as a function of the global time variable t for t≥3. State both the functional form and the value of xB at t=3. Include appropriate units.
(2) For t≥3, determine the exact time t (to three decimal places) at which the two cyclists are at the same position. Justify that this time is the only solution with t≥3.
(3) Calculate the velocity of each cyclist at the meeting time found in part (b). Provide units and indicate the direction of motion relative to the positive x-axis.
(4) Interpret your results from parts (b) and (c) in context. Explain whether the meeting represents B overtaking A from behind or a head-on encounter, and identify which cyclist is traveling faster at the instant they meet.