Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2016

A particle of unit mass moves with angular momentum hh in an attractive central force field of magnitude kr2\frac{k}{r^{2}}, where rr is the distance from the particle to the centre and kk is a constant. You may assume that the equation of its orbit can be written in plane polar coordinates in the form

r=1+ecosθr=\frac{\ell}{1+e \cos \theta}

where =h2k\ell=\frac{h^{2}}{k} and ee is the eccentricity. Show that the energy of the particle is

h2(e21)22\frac{h^{2}\left(e^{2}-1\right)}{2 \ell^{2}}

A comet moves in a parabolic orbit about the Sun. When it is at its perihelion, a distance dd from the Sun, and moving with speed VV, it receives an impulse which imparts an additional velocity of magnitude αV\alpha V directly away from the Sun. Show that the eccentricity of its new orbit is 1+4α2\sqrt{1+4 \alpha^{2}}, and sketch the two orbits on the same axes.

Typos? Please submit corrections to this page on GitHub.