First, determine the angular momentum at apogee since the distance and space craft velocity is known at that point.

     (H_o)_A = r × mv = m|r||v| sinθ e_z

         = (16,000 x 10³ m) (18,000 kg)
                                (4,373 m/s) sin90 e_z

         = 1.26 x 10¹⁵ e_z kg-m²/s

Since the gravitational force acting on the Apollo is always directed towards the center of the earth, and the Apollo moves in a plane, it is in plane central-force motion. Thus, angular momentum about the center of the earth is conserved, and the angular momentum at all points in the orbit are the same,

     (H_o)_P = (H_o)_A = 1.26 x 10¹⁵ e_z kg-m²/s

Because the Apollo space craft is in plane central-force motion, the product of its radial distance from the center of the earth and the transverse component of its velocity remains constant, giving

     r_A v_A = r_P v_P

With two unknowns, a second equation is needed. Since gravity is the only force acting on the Apollo, the total energy must be conserved. Thus, the sum of the potential energy and kinetic energy is constant,

Notice, the potential energy terms (P.E.) are negative. When working with potential energy in space, the datum line is assume to be at a large distance from the planet where there is no gravity and PE equals 0.

Solve both equations simultaneously to determine the radius and speed at perigee.

     r_P = r_A v_A /v_P

Substituting,

This reduces to the quadratic equation

which yields the trivial solution (circular orbit)

     v_P = v_A

and the solution for the speed at perigee:

     v_P = 7,000 m/s

Substituting back into the original equation, gives the radius at perigee as

     r_p = 10,000 km