# Gravitation question

I’d like to ask the following doubt that came to me after some exercises:

Is it possibile to exactly establish,given a satellite orbiting Earth with a certain $\vec v_{0}$ and distance $\vec R_{1}$ from earth, with $v_{0} \bot R_{1}$,

Whether the satellite is at perigee or apogee?

• If you know what the velocity vector v0 is, and you know the distance vector R1 at that same instant, and you know the v0 is exactly perpendicular to R1 at this instant, seems to me that you've uniquely determined the path of the satellite. You know its total energy, as well as its gravitational potential and kinetic energies at that instant. So, yeah, I would say that you pretty much know all that there is to know about the orbital path of the satellite, including whether it is at its perigee or apogee at that particular instant in time. – Samuel Weir Jul 5 '18 at 21:52

## 1 Answer

If it were in circular orbit, its velocity would be $\sqrt{GM/r}$. If it is in elliptical orbit and going faster than that, it must be at perigee; if slower, at apogee.

• Beat me to it. +1. – David Hammen Jul 5 '18 at 21:57
• Low hanging fruit. – Bert Barrois Jul 5 '18 at 21:59
• not so fast. 1) apogee and perigee are points. Might the satallite not be between the apogee and perigee and still have an orbital speed different from that of a circular orbit. – JMLCarter Jul 5 '18 at 22:02
• 2) the orbital speed is greater than the speed measured perpendicular to R for most points on an elliptical orbit. How is that accounted for? – JMLCarter Jul 5 '18 at 22:03
• Indeed, but the OP does not state that he knows $V_0$ is the orbital velocity. It could be a measurement of the component of orbital velocity perpendicular to $R_1$. In which case this question might be more interesting. – JMLCarter Jul 5 '18 at 22:16