Global Positioning System (GPS) satellites transmit radio signals used by receiv

Question

Global Positioning System (GPS) satellites transmit radio signals used by receivers on the Earth to determine position with very high precision. The operating principle is that a receiver gets signals simultaneously from several different satellites (typically at least four) and uses them to "triangulate" its position. This method relies on precise knowledge of the time and also the time it takes for a satellite signal to reach the receiving position. Suppose the GPS satellites are placed in orbits for which the period is 12 h. (They are thus not in geosynchronous orbits.)

(a) Calculate the time it takes a GPS signal to travel to the Earth's surface from a GPS satellite that is directly overhead.
s

(b) Calculate the time it takes a GPS signal to travel to the Earth's surface from a GPS satellite that is on the horizon.
s

Explanation / Answer

mv*v / r = GMm/(r*r) ------------1

so v*v/r = GM/r*r

so v = sqrt (GM/r)

time period = 2*pi*r / v = 12h = 12*3600 seconds

so 2*pi* sqrt( r*r*r / GM) = 12 * 3600

where G = 6.67*10^-11

M = mass of earth = 6 * 10^24

so r^3 = 12*12*3600*3600 * 6.67*10^-11 *6 * 10^24 / 4*pi*pi = 7.468*10^23 / 39.478 = 18.9 * 10^21

r = 2.66 * 10^7 m

speed of electromagnetic waves = c = 3*10^8 m/s

so time taken = 2.66*10^7 / 3 * 10^ 8 == 88.79 * 10^-3 seconds.

=

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