# Central Force IV

1. Dec 17, 2005

### Nusc

A planet of mass m follows an elliptic path around the Sun with a semi-major axis a and an eccentricity e, using the polar form for the radial position of the planet with respect to the sun:

a) Find both the radial component of the velocity Vr and the angular component of the velocity V theta in polar form

r(t) = r <er>
V(t) = r* <er> + r theta* <e theta>

b) Show that when the planet is located at the perihelion (r=rmax) and at the aphelion (r=rmax) the velocity of the planet has no radial component. Compute the angular component of the velocity at these two points.

r = rmax = a(1+e)
r = rmin = a(1-e)

r* = 0

Vmin = 0 <er> + a(1+e) theta* <e theta>
Vmax = 0 <er> + a(1-e) theta* <e theta>

c)Show that when the orbit is circular, the radial component of the velocity vanishes for all time and the angular component of the velocity is always constant. Give your answers to the above in terms of the constants, m, L, a and e.

How do I start this one provided that the 2 parts above are correct?