Question

A spacecraft of mass m = 1900 kg is moving on a circular orbit about the earth at a constant speed v = 5.12 km/s. [Given: Mass of the earth M = 5.98 times 10^24 kg, radius of the earth R = 6.37 times10^6 m, gravitational constant G = 6.67 times 10^-11 N.m^2/kg^2.] a. Determine the radius r of the circular orbit. b. What is the period T of the orbit? c. The satellite, by firing its engines, moves to a new circular orbit of radius r' = 2r/3. What is the new speed v' of the spacecraft in its new orbit? d. What is the change of the gravitational potential energy Delta U of the satellite as it goes from the old to the new orbit? e. What is the change in the total mechanical energy Delta E of the satellite as it goes from the old to the new orbit?

Answer 1

Under stable condition F gravitational = F centrifugal G Mm/p2 = mv^2/r sigma, GM/r = V^2 therefore r = GM/V^2 = 6. 67 x 10^11 x 5. 98 x 10^24/(5. 12 x 10^3) ^2 = 1. 5 x 10^7 m = 15000 Km 6 = > + S/V = 2pir/v = 2pi x 1. 5 x 10^7/(5. 12 x 10^3) = 1. 84 x 10^4 S C GM = V^2P = V^12 P^1 = const sigma, (V^1/V) ^2 = P/p^1 sigma V^1 = V squareroot p/p^1 = V squareroot p/2r/3 = V squareroot 3/2 therefore V^1 = squareroot 3/2 x 5. 12 = 6. 27 km/s d. delta u = gMm (1/p^1 - 1/r) p^1 = 2/3P = GMm/p (p^0/p^1 - 1) = GMm/r (3/2-1) = GMm/2p = 6. 67 x 10^-11 x 5. 98 x 10^24 x 1900/2 x 1. 5 x 10^7 J = 2. 52 x 10^10 J e change in total konetie energy, delta T = 1/2 m (V^12-V^2) = 1/2 x 1900 x (6. 27^2 - 5. 12^2) x 10^6 J = 1. 24 x 10^10 J Change in total Mechemical energy delta E = delta U + delta + = (2. 52 + 1. 24) x 10^10 J = 3. 76 x 10^10 .