Ship Movement Problem
A ship measures a time difference of 0.000108 seconds between the signals of two LORAN signals sent from the stations 100 miles apart along a coastline. If the ship maintains this time difference, where will the ship land on the coastline? (Assume that the signal moves at the speed of light [186,000 miles per second]) Round answer to nearest whole number.
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1 solution
I'll begin by graphing this situation on a Cartesian plane. I'll put station one at ( − 5 0 , 0 ) and station two at ( 5 0 , 0 ) . The distance from the ship to station one and the distance from the ship to station two have a constant difference, so the ship's path lies on a hyperbola with the stations as foci.
The ship is on a path of the hyperbola and when it lands on the shore it will be at the vertex of the hyperbola. To figure out where it lands, we can thus calculate the position of the vertex with this equation: d = r t
Substituting r = 1 8 6 , 0 0 0 s e c o n d m i l e s and t = 0.000108 seconds, we have d = ( 1 8 6 , 0 0 0 s e c o n d m i l e s ) (0.000108 seconds)\ = approximately 20 miles.
Remember though that d 1 − d 2 = 2 a , where a is the distance of the vertex from the center of the hyperbola. So 2 a = 2 0 , or a = 1 0 .
The answer is 10 miles.