# 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??]

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.

• Sir, if I take a tomato in place of the rocket? When a tomato is thrown upwards,what will happen then? There is no explosion now! – user36790 Oct 23 '14 at 1:12
• exactly the same thing, you provide the energy with the impulse of the throw , taken from the breaking of chemical bonds in the matter that makes you up, and ultimately from the heat of sun that makes things grow that you eat and have available energy. Again, you-and-the-earth initially accelerate downwards and the tomato upwards: from momentum conservation the available energy is shared. (it is the huge mass of the earth that is confusing you, very little motion is needed to give the same momentum as to the tomato). – anna v Oct 23 '14 at 5:00
• Great answer! When the tomato is thrown up, it accelerates momentarily and gains KE from the energy of the thrower. By conservation of momentum & Newton's third law, the thrower-&-earth will accelerate momentarily using the internal energy of the thrower. But due to mutual gravitation, they decelerate & ultimately come to a halt. Their KE thus is converted to PE . When they accelerate towards each other, their PE is converted tn PE. Thus it is all the energy of the thrower. Energy and momentum thus are conserved,right? – user36790 Oct 24 '14 at 8:51
• yes, that is a correct summary – anna v Oct 24 '14 at 10:33
• have you checked en.wikipedia.org/wiki/Conservative_force . the answers you have got are on those lines. – anna v Oct 24 '14 at 12:11

It comes from the work made by the gravitational force of the big object that attracts the Earth. I updated my answer to a previous question from you. But remember that the description in terms of conservation of energy and the use of potential energies is an alternative description (usually simpler) than that using all the forces involved (specially reactive forces from the 3rd law). You can switch back and forth between the two descriptions, but you will become confused if you try to use both simultaneously, as this was not the original intention when developing the new formalism of conservative forces and field potentials.

• So,sir is my intuition right? – user36790 Oct 22 '14 at 18:52
• Your intuition is right, but try not to mix both description together. You should either use the forces description (including the third law) or the potential energy description, which usually ignores it (because it is already implicitly included as potentials) – Wolphram jonny Oct 22 '14 at 19:15
• Just a comment: the rocket example was not the best one, because it has the additional complication that the force that pushes the rocket up also pushes the earth down. So, see Anas's answer that applies to that case. My example was for a simplified case where the force that moves the mass do not interact with the earth (for instance, if the mass is charged and there is an electric force outside earth that attracts the charged mass but do not interact with the Earth. – Wolphram jonny Oct 22 '14 at 20:10