Conservation of Energy Problem: Where did it go?

[joelio36]

Conservation of Energy Problem: Where did it go??

Imagine I shoot a boulder vertically into the air with a massive cannon.

Its kinetic energy is converted into gravitational potential, and into KE of air particles (air resistance).

So at m metres above the ground, mgh is the Potential energy

But what happens once this boulder breaks orbit, free from the Earth's gravitational pull?

I think I am missing something but in my head, the Gravitation Potential just disappears, and the object carries on with any KE left.

Thanks for any help, got an exam heavy on conservation of energy in 2 days!

 

[Alewhey]

Imagine I shoot a boulder vertically into the air with a massive cannon.

Its kinetic energy is converted into gravitational potential, and into KE of air particles (air resistance).

So at m metres above the ground, mgh is the Potential energy

But what happens once this boulder breaks orbit, free from the Earth's gravitational pull?

I think I am missing something but in my head, the Gravitation Potential just disappears, and the object carries on with any KE left.

Thanks for any help, got an exam heavy on conservation of energy in 2 days!

The gravitational potential energy does not 'go' anywhere. Contrary to your impression, there is no point in space (except at 'infinity') at which the object is completely free of the Earth's gravitational field - however, since the force is inversely proportional to the square of the distance, at some point we can say that the field is negligible and the boulder appears 'free'. But we still have to put in (some vanishingly small amount of) energy to separate the masses further, and conversely we will gain energy if the masses are brought closer together.

How does this relate to conservation of energy? We define gravitational potential energies to be negative at finite distances (and zero at infinity). This definition means that as an initially-stationary object accelerates towards a mass from infinity, the potential energy 'decreases' (i.e. becomes more negative) whilst the kinetic energy increases (becomes more positive) in such a way that the net energy remains zero - energy is conserved.

Note that your formula GPE = mgh is an approximation only valid for small changes h in radial distance r such as one may achieve when jumping in the air on Earth, but for large changes in r we need to use the full formula GPE = -G/(M.m.r) (note the negative sign!)

 

[Dale]

in my head, the Gravitation Potential just disappears

The PE does not disappear, it goes to a finite maximum limit.