Let be polar coordinates in the plane. A particle of mass moves in the plane under an attractive force of towards the origin . You may assume that the acceleration a is given by
where and are the unit vectors in the directions of increasing and respectively, and the dot denotes .
(a) Show that is a constant of the motion. Introducing show that and derive the geometric orbit equation
(b) Suppose now that
and that initially the particle is at distance from , moving with speed in a direction making angle with the radial vector pointing towards .
Show that and find as a function of . Hence or otherwise show that the particle returns to its original position after one revolution about and then flies off to infinity.