QUESTION 4 [Total marks: 20] This question concerns the application of relativity theory to situation resembling the Global Positioning System (GPS). A pair of satellites are in orbit above the Earth. The satellites travel at constant speed u and height H above the surface of the Earth, and are separated by a distance L. At t = 0, the coordinates of the satellites are X₁ = 0, Y₁ = H and X₂ = L, Y2 H. The curvature of the Earth is assumed to be negligible. Each satellite carries a synchronised clock, which at t = 0 transmits a light signal that encodes both the position of the satellite and the time on the clock at the moment the signal was emitted. 4(a) A ground station equipped with a similarly synchronised clock receives these signals. The difference between the time the signals are received and the time of transmission gives the distances between the satellites and the ground station as R₁ = cAt₁ R2 = cat2. Show that the x coordinate of the ground station is x = L² - R²₁₂+ R² 2L 4(b) Relativistic effects cause the satellite clocks and the ground clock to run at differ- ence rates. When these effects are weak, the time elapsed on the satellite clocks (in the primed frame) relative to the ground clock (the unprimed frame) is approximately: gH u² tzt 1+ c2 2c2 Explain the significance of the terms in this equation. [5 Marks] 4(c) Suppose that the signals are transmitted at a time t > 0 so that the encoded time is t'. Show that, if the ground station assumes that all the clocks are synchronised, there will be an error in the computed a coordinate given by ctT (R1 - R2) (H 2c2 8x≈ L [5 Marks] 4(d) If H20000 km, u = 14000 km/hr, g = 10 m² s-1 and (R₁ - R2)/L~1, how long will it take for the position error to exceed a kilometre? [5 Marks]