When I look at the velocities of elliptical orbiting satellites the radial velocity (k in the figures) increases from zero magnitude at periapsis, to a maximum at the latus rectum, then back down to zero at the apoapsis. This describes a rate of increase opposite the direction of gravity that changes over time.
From periapsis to the latus rectum, the satellite to planet distance increases at an increasing rate. Then from the point of the latus rectum to the furthest point, apoapsis, this distance increases at a decreasing rate.
The change in distance describes a net acceleration away from the planet until the satellite reaches the latus rectum where the net acceleration reverses direction - in the same direction as gravity. This occurs despite the fact that gravity is in one direction and is decreasing in magnitude throughout the periapsis to apoapsis satellite journey.
What is the math that we use to calculate the radial velocity?
When the satellite crosses the latus rectum, the radial velocity k decreases. Fig 4 reveals that the rate of change in distance from the planet decreases at that point which agrees with the change of radial velocity k. There is a point of inflection at the latus rectum which means there would be a mathematical change of sign. Is there math out there that describes this change of sign?
What is the physics we use to describe why the radial velocity does what it does? I’m not looking for a geometrical answer here. We should be able to describe this physically like we do with any other thing in motion experiencing a force and acceleration. I just can’t find any mention of it in my searches.