LORAN Long distance radio navigation for aircraft and ships uses synchronized pulses

Chapter 10, Problem 42

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LORAN Long distance radio navigation for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light (186,000 miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations, 300 miles apart, are positioned on the rectangular coordinate system at points with coordinates and and that a ship is traveling on a hyperbolic path with coordinates (see figure). (a) Find the -coordinate of the position of the ship if the time difference between the pulses from the transmitting stations is 1000 microseconds (0.001 second). (b) Determine the distance between the ship and station 1 when the ship reaches the shore. (c) The ship wants to enter a bay located between the two stations. The bay is 30 miles from station 1. What should the time difference be between the pulses? (d) The ship is 60 miles offshore when the time difference in part (c) is obtained. What is the position of the ship?