Gravity and Law of Conservation of Energy

[Dominus Atheos]

Let's say you have a stellar body, say... a large ball of cosmic dust, with earth-equivalent gravity. And then let’s say you have another stellar mass, say, a smaller ball of cosmic dust. Let’s say they exist outside any galaxy, and do not revolve around anything, and their only movement has been away from the location of the Big Bang. Let’s say that the only gravity to ever act on them has been the gravity they took to form. Let’s say that both started out as molecule sized cosmic dust, and that a couple of molecules formed together, creating enough gravity that it pulled in more cosmic dust, and that this happened twice, once for each ball. Now let’s say that they meet, and that both have enough gravity that they pull each other toward themselves, and that they hit each other, with virtually all their force coming from the gravity pulling them toward each other. My question is:

Were does this energy come from? I was always taught that the energy from gravity going toward the thing exerting the gravity was the same as the energy spent going away from the thing. I.E. Earth, as I was always taught, like the energy exerted from a ball falling and denting a car was just the energy spent getting into the air. So were does the energy come from from a stellar object hitting another stellar object? How does this not violate the Law of Conservation of Energy?

[Rihannsu Science Officer]

They both exert gravity on each other, they're both falling, same as the original two molecules that created one of those stellar bodies in the first place.

[Surlethe]

Let's say you have a stellar body, say... a large ball of cosmic dust, with earth-equivalent gravity. And then let’s say you have another stellar mass, say, a smaller ball of cosmic dust. Let’s say they exist outside any galaxy, and do not revolve around anything, and their only movement has been away from the location of the Big Bang.

Quick point: the Big Bang had no specific location from which everything expands; rather, the universe itself is expanding, which indicates there is no set point.

Let’s say that the only gravity to ever act on them has been the gravity they took to form. Let’s say that both started out as molecule sized cosmic dust, and that a couple of molecules formed together, creating enough gravity that it pulled in more cosmic dust, and that this happened twice, once for each ball. Now let’s say that they meet, and that both have enough gravity that they pull each other toward themselves, and that they hit each other, with virtually all their force coming from the gravity pulling them toward each other. My question is:

Were does this energy come from? I was always taught that the energy from gravity going toward the thing exerting the gravity was the same as the energy spent going away from the thing. I.E. Earth, as I was always taught, like the energy exerted from a ball falling and denting a car was just the energy spent getting into the air. So were does the energy come from from a stellar object hitting another stellar object? How does this not violate the Law of Conservation of Energy?

The two bodies will meet at their center of mass, so each one's potential energy is going to be the amount of energy required to raise each away from the system's center of mass.

[drachefly]

They started in a high energy state -- separated. That's where it came from...

[Cykeisme]

Indeed. That's why gravitational potential energy is a negative.. they begin in a high energy state due to their separation, zero at infinity.. and as they come together, the amount of potential energy decreases (into a negative value) as it's converted into kinetic energy, accelerating the objects toward each other. And.. uh.. stuff.

I have it in my head, but I'm not too good at.. spilling it.

[Kuroneko]

As stated above, energy is conserved because of the difference in the gravitational potential energy balances out the difference of the kinetic energy. However, it is probably unwise to think of potential energy as something made to preserve the law of conservation of energy in some ad hoc manner. The mechanical energy (kinetic+potential) is simply a conserved quantity that is useful for predicting physical phenomena. In general, if the acceleration is a radial F(r), it is a fairly simple calculus exercise to show that v²/2 - Int[F(r)dr] is a constant (try it), as well as the specific angular momentum l = r².'. In the case of gravity, F(r) = -k/r², but in general it doesn't make a difference whether or not F(r) is inverse-square; potential energy will be its integral no matter how it behaves.