This problem involves placing a satellite of mass m into orbit about a spherical planet (with no atmosphere) of mass M » m and radius p. The mass is raised to altitude h = R – (i.e., to radius R) and given an initial velocity V perpendicular to the radius.
(a) Find the eccentricity E of the resulting orbit as a function of V. For what values of V is the orbit an ellipse, a circle, a parabola, or a hyperbola?
(b) Find the distance of closest and, where appropriate, farthest approach to the planet. For this part ignore the possibility of collision with the planet.
(c) Find the energy needed to raise the satellite into the lowest orbit that fails to collide with the planet (a surface-skimming orbit).
(d) Now suppose V is not necessarily perpendicular to the radius. Find the escape velocity as a function of the angle a between V and the radius. Be careful to avoid collisions with the planet!