The total mechanical energy of a satellite in a circular orbit is determined by its mass and the radius of its orbit. For satellites A and B, the ratio of their masses is 3:1, and the radii of their orbits are r and 4r , respectively. The mechanical energy of a satellite in orbit is directly proportional to its mass and inversely proportional to the radius of the orbit.

Satellite A, having three times the mass of satellite B, has a greater gravitational interaction, leading to higher energy. However, its energy is also inversely related to the smaller orbital radius \( r \). Satellite B, with one-third the mass of A, moves in a larger orbit with a radius 4r . which further reduces its total mechanical energy.

Taking both the mass and orbital radius into account, the total mechanical energy of A is significantly larger than that of B. Specifically, the energy of A is 12 times that of B. This ratio arises because the increased mass of A amplifies its energy, while B’s larger orbital radius reduces its energy proportionally. Thus, the ratio of their total mechanical energies is 12:1, reflecting these combined effects.