Newtonian approach for the Kepler-Coulomb problem from the point of view of velocity space

The hodograph of the Kepler-Coulomb problem, that is, the path traced by it s velocity vector, is shown to be a circle and then it is used to investiga te the properties of the motion. We obtain the configuration space orbits o f the problem starting from initial conditions given using nothing more tha n the methods of synthetic geometry so close to Newton's approach. The meth od works with elliptic, parabolic and hyperbolic orbits; it can even be use d to derive Rutherford's relation from which the scattering cross Section c an be easily evaluated. We think our discussion is both interesting and use ful inasmuch as it serves to relate the initial conditions with the corresp onding trajectories in a purely geometrical way uncovering in the process s ome interesting connections seldom discussed in standard treatments.