An incoming comet of mass m is detected at the edge of the solar system. It has velocity v and an impact parameter b (the minimum distance between the centre of the sun and the extrapolation of the comet's initial path) with respect to the Sun (mass M).
a) Sketch the expected trajectory of the comet. Indicate the impact parameter b and the angle of scattering (deflection) on your plot.
b) Show that the total energy of the comet as a function of position vector r from the centre of the Sun may be written as: (see the attachment for the equation).
Find the expressions for J and alpha.
c) Using the result in (b) or otherwise, show that the distance of closest approach to the centre of the Sun is (see the attachment for the equation) where G is the gravitational constant.
This is a model solution as given to Oxford University 1st Physics students studying the Mechanics2 module. The Questions asks to draw a comets trajectory around the Sun based on a number of parameters including the comets initial velocity at great distance (the edge of the Solar System) and its impact parameter. The solution then goes on to prove 2 equations one describing the total energy of a comet at a point r in terms of its radial velocity at that, its angular momentum and gravitational PE the other equation determining the minimum distance of approach of the comet to the Sun.