Physics planetary motion

Question

Satellites that make up the Global Positioning System, or GPS, orbit at an altitude of 2.12 ✕ 10^7 m. Find the orbital period of a GPS satellite.

Answer 1

mv²/(r + h) = GmM/(r + h)²; v² = GM/(r + h). On surface of Earth mg = GmM/r²; GM = gr²

v² = gr²/(r + h); v = r√g/(r + h). Period T = 2π(r + h)√(r + h)/g/r = 2π(r + h)^3/2g ^-1/2/r = 2π(6.38·10⁶ + 2.12·10⁷)^3/2·9.8^-1/2/(6.38·10⁶) s = 45566 s = 12.66 h