Working out the kinetic energy of a body due to a gravitational field

[etotheipi]

Homework Statement

Explain how to calculate the kinetic energy of a body brought from rest at infinity to some distance x from another massive body by its gravitational field.

Relevant Equations

PE = -GMm/r

Usually when setting up an energy equation I use the general form, (Initial KE) + (Initial PE) + (Any other work done to the body) = (Final KE) + (Final PE) ...

For this I said the initial GPE and KE are 0, and the work done by the field is GMm/x (derived by integrating a force of -GMm/r^2 from infinity to x)
, so I end up with the equation:

0 + (Work done) = (Final PE) + (Final KE)
GMm/x = - GMm/x + KE

However this gives me a result of KE = 2GMm/x, whilst other sources I have found state that we can simply equate the work done to the kinetic energy. I was wondering what the correct solution would be? Thank you in advance

 

[haruspex]

Any other work done to the body

You have already counted work done by gravity in the GPE change. It is not other work.

 

[etotheipi]

You have already counted work done by gravity in the GPE change. It is not other work.

I see. Would I be right in thinking then that potential energies are just useful ways of keeping track of the work done by their respective fields? It seems as though the maths works this way.

 

[haruspex]

I see. Would I be right in thinking then that potential energies are just useful ways of keeping track of the work done by their respective fields? It seems as though the maths works this way.

Yes.