A particle of unit mass moves with angular momentum in an attractive central force field of magnitude , where is the distance from the particle to the centre and is a constant. You may assume that the equation of its orbit can be written in plane polar coordinates in the form
where and is the eccentricity. Show that the energy of the particle is
A comet moves in a parabolic orbit about the Sun. When it is at its perihelion, a distance from the Sun, and moving with speed , it receives an impulse which imparts an additional velocity of magnitude directly away from the Sun. Show that the eccentricity of its new orbit is , and sketch the two orbits on the same axes.