Rivers are typically modelled assuming that the flow is fastest in the middle and stagnant at the shorelines. The relevant formula is , where is the speed of the river in midstream, the river is imagined to flow in the +y direction with banks at , and the speed at any x is . You can verify that the formula gives and . A swimmer attempting to swim across the river (in the positive x direction) will have an effective velocity that is the vector sum of his swimming velocity perpendicular to the shoreline and his drift velocity due to the river current, which varies by position. The true direction of his path across the river makes an angle with the x axis, where . Find a general expression for the swimmer's trajectory across the river and if the river is one mile wide, flowing at 9 mph in the middle, and the swimmer can maintain a speed of 3 mph, how far does he drift downstream by the time he reaches the opposite shore?
