A particle of unit mass moves under the influence of a central force. By considering the components of the acceleration in polar coordinates prove that the magnitude of the angular momentum is conserved. [You may use . ]
Now suppose that the central force is derived from the potential , where is a constant.
(a) Show that the total energy of the particle can be written in the form
Sketch in the cases and .
(b) The particle is projected from a very large distance from the origin with speed and impact parameter . [The impact parameter is the distance of closest approach to the origin in absence of any force.]
(i) In the case , sketch the particle's trajectory and find the shortest distance between the particle and the origin, and the speed of the particle when .
(ii) In the case , sketch the particle's trajectory and find the corresponding shortest distance between the particle and the origin, and the speed of the particle when .
(iii) Find and in terms of and . [In answering part (iii) you should assume that is the same in parts (i) and (ii).]