Does an orbiting satellite speed up as it falls towards earth

[LotusTK]

Homework Statement

(sorry for spelling of Earth, had to be to be done to fit it in lol)

Does a satellite in orbit around the Earth speed up as it falls towards Earth?

I understand why the satellite speeds up mathematically. If we equate the centripetal force equation and the equation for gravitational force (because the centripetal force on the satellite is the force of gravity) we end up with

v=sqrt( GM/r)

G = gravitational constant, M = mass of Earth, r= radius of circular motion from centre of earth)

So if you decrease r, v increases, and vice versa.

Is it just simply because the gravitational potential energy of the satellite is being converted into kinetic energy? So instead of just increasing velocity directly towards the Earth like when you drop a ball, its tangential/orbital velocity increases?
I picture the satellite just increasing its velocity directly towards Earth, rather than actually increasing the speed with which it orbits. But then the above formula suggests otherwise.

Also, would i be correct in saying:
If the satellite were to maintain the same orbital speed, but was pushed up into a higher orbit without increasing or decreasing its orbital speed, the satellite will no longer be able to maintain its orbit and begin to spiral away from Earth, because the centripetal force requirement to keep the object in that particular orbit at that particular speed is no longer able to be met by gravity?

Thanks

 

[mfb]

Without external force and approximating Earth as spherical symmetric mass, the satellite just keeps orbiting in its orbit. If you push it a bit, it goes to a new orbit. No spirals.


Real orbits with atmospheric drag speed up as their orbit decays, and it is a simple conversion of gravitational to kinetic energy and heating the atmosphere. At some point drag wins, they get much slower and fall back or burn up rapidly.

 

[LotusTK]

Without external force and approximating Earth as spherical symmetric mass, the satellite just keeps orbiting in its orbit.
Real orbits with drag speed up as their orbit decays, and it is a simple conversion of gravitational to kinetic energy and a bit of friction. At some point drag wins, they get much slower and fall back or burn up rapidly.

And is this because of the change of gravitational potential energy to kinetic energy? And when you say speed up, you mean the speed with which it is orbiting, right?

 

[mfb]

And is this because of the change of gravitational potential energy to kinetic energy?

Yes, see the previous post.

And when you say speed up, you mean the speed with which it is orbiting, right?

Sure.