ratio

Two fixed circular rings, in contact with each other, stand in a vertical plane. A ball bounces elastically back and forth between the rings as in Figure 4.33. Assume that initial conditions have been set up so that the ball’s motion forever lies in one parabola. Let this parabola hit the rings at an angle ? from the horizontal. Show that if you want the magnitude of the change in the horizontal component of the ball’s momentum at each bounce to be maximum, then you should pick cos ? = (v 5 – 1)/2, which just happens to be the inverse of the golden ratio