Achieving Circular Orbit

Question:

A satellite is placed into orbit at a velocity of 6km/s parallel to the Earth's surface. Determine the proper altitude of the satellite above the Earth such that its orbit remains circular. What would will happen to the satellite if its velocity is 5km/s when placed tangentially into orbit.

Answer:

In a circular orbit Newton's second law leads us to the relationship
G*M*m/r²=m*v²/r, or v²=G*M/r for a circular orbit. This yields
r=G*M/v²

In this problem v=6e3m/s so
r=6.673e-11*5.976e24/6e3²=1.108e7m
which corresponds to 4.7e6m orbital altitude.

If the initial velocity were 5km/s instead of 6km/s tangential at this altitude, the satellite would have only (5/6)², or 69% of the kinetic energy of the first case. Due to the mismatch between the kinetic energy and potential energy at that altitude, the satellite would slide down the potential energy well, gaining kinetic energy as it falls. The orbit then would be elliptical with its aphelion at the insertion altitude. If the perihelion is below the surface of the Earth it will crash. You should try to determine if this will happen. Refer to satellite speed at extremes.

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