# Why the gravitational potential energy at infinity is zero?

If an object is taken from earth's surface to infinity, it's gravitational potential energy becomes zero (always taken as zero), but it doesn't make any sense as energy can never be destroyed so where did that kinetic energy go?

Potential energy doesn't have any physical meaning. Only potential energy difference has physical meaning. You can add a constant to the potential energy at all points, and it will not make any difference. Hence, we have the liberty to choose any point in space as the point with potential energy as 0. Choosing potential energy at infinite as 0 is most convenient for calculations. You may choose any other point to have 0 potential energy, as long as you are consistent about it.

The kinetic energy (assuming, it becomes 0 when it reaches infinity, which means the velocity is equal to escape velocity), when added to the negative potential energy near the Earth, will be equal to zero. At infinity, both the energies are zero. That is, the sum of potential and kinetic energies remain constant.

When the object was on the Earth's surface, its g.p.e. was negative.

When something lifted it off the surface to take it to infinity, it did work on the object, increasing its g.p.e. to 0.

If it fell from infinity toward the Earth, its g.p.e. would become negative again and it would gain kinetic energy. Likely it would subsequently heat up itself and the atmosphere as it fell

In either case, energy is conserved.

Don't assume that the potential energy has decreased in order to reach zero. In fact, it increases to become zero.

Remember that the sign of potential energy can be anything - also negative. It depends on an arbitrarily chosen zero-point. With the choice made at infinity being the zero-point, the potential energy indeed is negative when the object is closer to the Earth.

Since the potential energy increases with distance, the kinetic energy decreases. Which is exactly as expected and upholds the energy conservation law.

There are a lot of quantities in science that can't be defined by themselves.

Like, imagine in space with nothing in it (vacuum) and you put inside of it a sphere of mass $$m$$. It will have an imaginary gravitational field.

But did we need to do anything against any other force to put the sphere?

NO

But to move another sphere in said vacuum, we would have to do some work against the gravitational force of attraction between the 2 spheres.

Definition: Gravitational potential difference between two points is defined as the amount of work done in moving a unit mass from one point to another point against the gravitational force of attraction.

But scientists wanted to define a value for only 1 object, since it's troublesome to put a difference value in applications of G.P.E.

So, scientists took an arbitrary value of G.P.E at infinity to be zero. So G.P.E of object = G.P.E of object at point - G.P.E at infinity = G.P.E of object at point

So, easier to deal with the value. Values can be easily dealt with for multiple objects.

Compare with equation $$(x-y) - (z-y) = x-y+y-z= x-z$$ so the definition is unscathed .

Similarly, in electrochemistry, the standard reduction potential of hydrogen is also taken as zero.

These are merely reference points.

It's simply a matter of definition: the energy at infinity is defined to be zero; but we could have chosen to define the interaction energy at infinity to be five joule, or ten, or whatever other number you like better than zero. Of course the standard definition with zero at infinity comes more natural: if two bodis are infinitely far apart seems reasonable to declare their interaction energy to be zero.

Note that there are no problems regarding conservation of energy: If we have an attraction force, such as gravity a body falling from infinity to some finite distance $$r$$ will speed up, thanks to the attraction force, so it will acquire positive kinetic energy, where does this energy come from? It came from the potential energy of the force! But the potential energy was zero at infinity! How can it get even lower than zero? Easy enough: the potential energy became negative!

An attraction force has negative potential energy at any finite distance $$r$$

No problem about energy conservation whatsoever! But what about repulsion forces? Well same argument, only flip on its head! At it leads us to:

A repulsion force has positive potential energy at any finite distance $$r$$

Remember of course that this two statements only hold true if we accept our definition of zero energy at infinity; this is another reason why putting by definition $$E=0$$ at infinity comes so natural!