You have landed a job with SpaceX as a space physicist. Your first job is launch

Question

You have landed a job with SpaceX as a space physicist. Your first job is launch a satellite. The plan is to lift off from the surface of the Earth and then insert the satellite into an elliptical transfer orbit with its perigee at A and apogee at A'. The spacecraft will be unpowered as it moves between A to A'. At A' a rocket will be fired to increase the speed of the satellite to insert it into a circular orbit around the Earth. Find the change in velocity required at A' knowing that the altitude of the spacecraft at A = 500km.

Explanation / Answer

Mass of earth M = 5.97 * 1024 kg; G = 6.67 * 10-11 N m2 kg-2

When satellite is at A , distance from centre of earth r1 = 6370 + 500 = 6870km = 6.87 * 106 m

When satellite is at A' distance from centre of earth r2 = 32000km = 3.2 * 107 m

Velocity at A is v1 and velocity at A' is v2

Conserving angular momentum

mv1r1 = mv2r2

v1 = 32000/6870 * v2 = 4.66v2

Conserving mechanical energy

1/2 m v12 - GMm/r1 = 1/2 m v22 - GMm/r2

1/2 m (v22 - v12) = GMm (1/r2 - 1/r1)

Substituting for v1

v22 - (4.66)2v22 = 2GM (1/3.2 * 107 - 1/6.87 * 106)

v2 = 1.05 * 10-4 (GM)1/2 = 2.09 * 103 m/s

Orbital velocity v = (GM/r2)1/2 = 3.51 * 103m/s

Change in velocity = v - v2 = 1.42 * 103 m/s

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