Velocity in eliptical trajectories

[MMS]

Hello everyone,

A little question bothering me concerning elliptical trajectories:
Say I have a particle at some point with some velocity (both I know - r0, v0) and I need know its minimal velocity so that it reaches some specific r*.
I can easily find this velocity considering conservation of energy.

However, where do the radial and tangential velocities during its motion come into consideration here? Is this calculation right?

Thanks in advance.

 

[Simon Bridge]

Kinetic and potential energies depend on magnitudes not directions - so the components are figures in automatically.
If you use conservation of momentum calculations instead you'll see where the components go.