There is no mass for which one ball attracts the earth more than the other ball if the other (fixed mass) ball is less than the mass of the Earth. The ratio of the two forces, the force between the varying mass ball (vmb) and the Earth and the varying mass ball and the fixed mass ball (fmb), is constant°: m_earth/m_fmb.
°Assuming equal distance between the two objects. It's still constant w.r.t. mass if you allow different distances, you just pick up a ratio term of the two distances. Again, you don't 'start' attracting one ball more than the Earth. You're _always_ attracting the Earth more.
Now that I've put it that way, we can see that the mass doesn't matter, since the previously neglected force is _only_ operating between the direction of the two masses, so there is no additional acceleration in the direction of the earth. If you pretend the earth is a infinite plane (which is typical) than we can see that this will have no effect on the fall time. Since the distance to the earth starts being equal, and the accelerations (relative to the earth) are equal, the time will be equal.
If, however, you instead acknowledge that the earth is a sphere, and we really started the two objects at rest at the same radius but different angles, than you have a restricted three body problem and I should probably not try to solve that before I've had coffee.
°Assuming equal distance between the two objects. It's still constant w.r.t. mass if you allow different distances, you just pick up a ratio term of the two distances. Again, you don't 'start' attracting one ball more than the Earth. You're _always_ attracting the Earth more.
Now that I've put it that way, we can see that the mass doesn't matter, since the previously neglected force is _only_ operating between the direction of the two masses, so there is no additional acceleration in the direction of the earth. If you pretend the earth is a infinite plane (which is typical) than we can see that this will have no effect on the fall time. Since the distance to the earth starts being equal, and the accelerations (relative to the earth) are equal, the time will be equal.
If, however, you instead acknowledge that the earth is a sphere, and we really started the two objects at rest at the same radius but different angles, than you have a restricted three body problem and I should probably not try to solve that before I've had coffee.
Thank you very much. It will be a good day.