Consider a river of uniform width w that flows eastward at a constant speed vr. A boat can move at speed vb relative to the water, where vb>vr. Two different strategies are commonly discussed for crossing the river.
• Strategy I: Aim the boat so that it arrives at a point directly opposite the launch point (zero downstream displacement).
• Strategy II: Point the boat perpendicular to the banks and accept any downstream displacement that occurs.
Using symbolic reasoning (no numerical substitution), respond to the following prompts and leave all answers in terms of the symbols vb, vr, and w.
(a) For Strategy I, derive an algebraic expression for the crossing time t1.
(b) For Strategy II, derive an algebraic expression for the downstream displacement d2 and the crossing time t2.
(c) Compare t1 and t2 algebraically to determine which strategy yields the shorter crossing time. Provide a clear justification based on your derived expressions.
(d) Discuss one practical circumstance (other than time minimization) in which Strategy I would be preferable, and one circumstance in which Strategy II would be preferable. Support each choice with physical reasoning drawn from your results.