Why does total energy increase when orbital speed decreases?

[apchemstudent]

The answer to the problem is a), the orbital speed will decrease but the total energy will become larger. How is this? It's kinetic energy has decreased and the gravitational potential energy should also have decreased from this equation

GM(earth)m(object)/r where r has been increased, thus smaller Gravitational potential energy. Can some one explain this? Thanks

 

[Davorak]

Gravitational Potential energy is negative, so when r gets bigger you potential energy gets less negative, or bigger.

 

[Andrew Mason]

The answer to the problem is a), the orbital speed will decrease but the total energy will become larger. How is this? It's kinetic energy has decreased and the gravitational potential energy should also have decreased from this equation

GM(earth)m(object)/r where r has been increased, thus smaller Gravitational potential energy. Can some one explain this? Thanks

The change in potential is positive.

$$U(r) = \int_{R{_0}}^\infty \frac{GMm}{r^2}\hat r \cdot d\vec{s} = \int_{R{_0}}^\infty \frac{GMm}{r^2}dr \hat r$$

$$U(r) = 0 -\frac{GMm}{R_0}$$

So as r increases, U(r) becomes less negative.

AM