Why does an orbit increase in altitude with more velocity?

[KSK]

3 minutes ago, Fez said:

Wouldn't energy be conserved if, when you add kinetic energy, it just stays in a circular, faster orbit?  Why does it have to climb?

Because for that to happen, gravity would need to get stronger.  If a body is moving at a constant speed in a circular path, it's always accelerating towards the centre of that circle. Which sounds a little crazy until you remember that acceleration is a change in velocity - that is speed and direction. The body is constantly changing direction as it moves around the circle, hence it's always accelerating. 

Now, the faster the body is travelling, the more it needs to be accelerated to keep it in the same circular path and thus the more force is required. But in our case, the force is provided by gravity and so remains constant (or constant to a reasonable approximation). So going faster in the same orbit doesn't work.

 

[Fez]

OK, so then because there's more horizontal speed, while you're falling at a fixed rate, the trajectory curves less, and climbs away.  Thanks

[K^2]

20 minutes ago, mikegarrison said:

Because energy is conserved. If you want to know why energy is conserved, you need to worry about 17-dimensional strings or Higgs bosuns or some crazy things like that.

Not at all. Energy conservation is consequence of Noether's Theorem, and in Classical Mechanics it is pretty simple. So long as Hamiltonian is time-independent, energy is conserved. For orbital mechanics, it boils down to the statement that if you include all of the sources of gravity in your energy considerations, then total energy is conserved. For example, if you consider just the ship's energy during a fly-by, it is not conserved, because there are moving sources of gravity. If you consider energy of craft and the planet it flies-by, then energy is conserved.