newtonian mechanics - If Earth starts moving when attracted by an object, where does the energy come from?

Suppose a body(very big but not bigger than Earth) moves against gravitational force of Earth. The force will do negative work on the body decreasing its kinetic energy. The decreasing energy is converted to potential energy. Now as gravitational force is mutual and also due to Newton's third law of motion, the body will exert force equal to its weight on Earth and will do positive work on it. The Earth,though very big, will accelerate slightly towards the body. As it is accelerating , kinetic energy of it is increasing. According to Law of Conservation of energy, the increasing kinetic energy is not created; it must be transferred to this form from other. So, from where does the Earth get this increasing kinetic energy??? Please help.

[ I think the increasing kinetic energy of the Earth comes from the potential energy of the body-Earth system. Thus it is the kinetic energy lost by the body when ascending upwards against the gravitational force cum the potential energy that supplies the kinetic energy of the accelerating Earth and energy is conserved hence. Am I right? If not, why??]

Answer

Suppose a body(very big but not bigger than Earth) moves against gravitational force of Earth.

Let us take a real situation: a large rocket is fired straight up. It takes kinetic energy from the chemical explosions; from momentum conservation part of the available energy from the explosion moves the earth in the opposite direction.

The gravitational force will slow down the rocket, but viewed in the center of mass system of the two bodies both the earth and the rocket will be slowing down because of their mutual attraction.

The force will do negative work on the body decreasing its kinetic energy.

OK

The decreasing energy is converted to potential energy.

Yes.

Now as gravitational force is mutual and also due to Newton's third law of motion, the body will exert force equal to its weight on Earth and will do positive work on it.

The Earth,though very big, will accelerate slightly towards the body.

Hmm, no. The earth and rocket initially are accelerating away from each other due to the explosion, and then will both start decelerating until the zero momentum is reached for both, and the maximum of potential energy i.e. maximum separation, then they will accelerate towards each other, the rocket will fall .

As it is accelerating , kinetic energy of it is increasing.

According to Law of Conservation of energy, the increasing kinetic energy is not created; it must be transferred to this form from other. So, from where does the Earth get this increasing kinetic energy??? Please help.

From the original explosion. The original explosion released energy divided between earth and rocket, then because of the dynamics it turned into potential energy at the peak of the trajectory, and then because of the unimpeded attraction, turned again into kinetic energy as the earth and rocket accelerate towards each other and meet with a bang.

Edit after comment:

Note that nothing can move against the gravitational pull of each other, as gravitation is always attractive, unless outside energy is provided .

Let us take the hypothetical case that two objects, far away from other gravitational fields are at a distance r, at rest with each other by coincidence of the dynamics of the creation of the cosmos. Then it is true that due to their gravitational attraction their potential energy will turn into kinetic energy as they fall at each other. The energy originally to be found at this position at rest with each other was provided by the original creation of these two objects, i.e. the Big Bang energies in the BB model.