Projectile motion with inelastic bouncing back

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This article discusses a scenario/arrangement whose statics/dynamics/kinematics can be understood using the ideas of classical mechanics.
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We consider here a situation where a particle is given an initial speed into the air in some direction and is then left to itself with no forces acting on it other than acceleration due to gravity, until it comes into contact with a solid surface (such as the ground). We ignore air drag.

The particle being released is termed a projectile and the motion it exhibits is termed projectile motion.

We are interested not only in the projectile motion until it hits a solid surface, but also in the projectile motion after it hits the surface, assuming an inelastic collision with a specified coefficient of restitution. We assume that the coefficient of restitution is constant and known in advance for the pair of the object and the surface with which it is colliding, although in practice this is not true. To simplify matters we assume that the object is a sphere (so the angle of collision does not matter) and assume it is not undergoing any rotational motion.