2 ENG1005 81 2023 WORKSHOP & PROBLEM SET PURSUIT PROBLEMS Pursuit problems involve determining the trajectory needed by one object to intercept an- other. This could be a rocket carrying astronauts to the international space space station, a missile launched at an aircraft, or a police car in pursuit of a fleeing criminal. In the next two workshops, you will explore the following pursuit problem: consider a canal of width w > (), see diagram below. 4 Y, P Relative to the ry-coordinate system indicated on the diagram, assume that the water in the canal is flowing in the positive y-direction with a speed s 2 0 and that a swimmer enters the canal at the point p = (w, 0). The swimmer then swims towards the point q = (0, 0) always facing in the direction of q. Letting (r,y) = (x(t), y()) denote the position of the swimmer at time t and v() = (#, #) their velocity, your objective is to determine the trajectory of swimmer as they move through the canal and attempt to get to the point q on the other side. You may assume that swimmer can swim at a constant speed c > 0 in still water and swims at this speed in the canal. In the above diagram, v, denotes the velocity of the swimmer in still water while vw is the velocity of the water in the canal. PROBLEMS 1. Express v in terms of s, c and d. [3 marks] 2. Use your formula for v from problem 1 to express the slope of the curve that describes the swimmer's trajectory in terms of x, y, c, s. This will yield a first order differential equations of the form dy = /(I, y) for an appropriate function /(x, y) that also depends on the constants s and c. [5 marks] 3. Set s = 0 in the differential equation you found in problem 2 and solve it to determine the trajectory of the swimmer in a canal with still water. Does your answer make sense?" [5 marks]ENG1005 81 2023 WORKSHOP & PROBLEM SET 3 4. If the speed of the swimmer is much greater than the flow of the water in the canal then c s; mathematically, this can be expressed by the limit e -+ co. To investigate this situation, let e - co in the differential equation you found in problem 3 and solve the resulting equation to obtain the trajectory of the swimmer. Does your answer make sense? [3 marks]