Toy models: The jumping pendulum

We consider a simple pendulum consisting of a mass attached to an inextensible string of negligible mass. For small or large initial velocities, the motion of the pendulum is along a circle. When given sufficient but not too large an initial velocity, the mass will reach a certain height and leave the circle. After such a jump, it will follow a parabolic path until the string is again fully extended and the motion is again constrained by the string. We assume that the radial component (along the string) of the velocity of the mass instantaneously vanishes when the string becomes taut and that the mass loses some of its energy in the shock and resumes its circular motion. What is the dynamics of such a pendulum? Can it jump more than once? How many times can it jump?