Determine the ship's location. LORAN

Question

Three LORAN radio transmitters are used by a ship to triangulate its location. Each transmitter, #1, #2, and #3, transmits signals at regular intervals, and the signals travel at the speed of 0.3 kilometers/microsecond (which is the speed of light). The transmitter #1 is located at the origin, transmitter #2 is 100 kilometers north of #1 at (0,100) and #3 is located 400 kilometers north of #1 at (0,400).

The ship receives the signal from transmitter #1 36.5907 microseconds after receiving a signal from transmitter #2 and receives the signal from transmitter #3 302.7756 microseconds after receiving a signal from transmitter #2.

Determine the ship's location.

Answer 1

Since the ship is receiving the signal from transmitter #1 36.5907 microseconds after transmitter #2, you know that the ship is further from transmitter #1.  Since the signal is traveling at 0.3 km/microsecond, you can calculate that the ship is 36.5907 * 0.3 km further from transmitter #1 than from transmitter #2.  Do the same thing with the difference between transmitter #3 and transmitter #2, and now you have two relative distances, given two pairs of transmitters.

 

The difference in the distance describes a hyperbolic curve, where the transmitters are the foci.  The formula of a hyperbola is ((y-k)^2 / a^2) - ((x-h)^2 / b^2) = 1.  In this formula, point (x,y) is the coordinate of the ship, so they are going to remain as your variables, but you can fill in the rest.  Point (h,k) is the center of the hyperbola, which is the midpoint between the two transmitters, and that is easy to determine for each pair of transmitters.  The value (a) is half of the difference in the distance to the ship; when using transmitters #1 and #2, that would be (36.5907 * 0.3) / 2.  The value (c) isn't directly used in the equation, but that's simply half the distance between the foci (transmitters).  You need to get the value (b) using the Pythagorean theorem: a^2 + b^2 = c^2, which is why the value of c is important.

 

Doing this for both pairs of transmitters (1&2 and 2&3), you'll end up with two hyperbolic equations with two unknowns (x,y).  Since the x coordinates of the transmitters are all 0, it should be simple to isolate x^2 in those equations, use substitution to determine y, and then plug that back in to get x.

 

Note that since the three transmitters are on a line, the triangulation is going to be ambiguous for any point that isn't on that line, since there would be equivalent possibilities on both sides of the line (both positive and negative x values).  So don't be surprised if there is more than one possible answer!

 

I hope this helps!