Problem 4 – Hint 4

Let us start with the Runge-Lenz vector as written in the IPhO formula sheet (formula XII-9) in which case its length equals to the ellipticity of the orbit. If we multiply it by k\equiv GMm/L, where m is the satellite’s mass and L – its angular momentum with respect to the centre of the Earth, we obtain \hat z \times \vec v=k(\vec\varepsilon -\hat r), where \hat r denotes a unit vector pointing from the centre of the Earth in the direction of the satellite, and \hat z – a unit vector perpendicular to the orbital plane. Note that k is constant, hence the last equality means that the satellite’s hodograph is a circle (in other words, if we fix the the starting point of its velocity vector, the ending point draws a circle). This is valid for the both satellites. Now, the maximal relative speed can be easily found from the hodographs of the two satellites.

Please submit the solution to this problem via e-mail to