Click "toss the ball" for an animation. Initially, the ball is tossed horizontally with speed 1 m/s from a height of 1.5 m onto a flat floor. The ball has normal and tangential coefficients of restitution of 0.7 and 0.8. (That means the normal and tangential speeds are reduced by factors of 0.7 and 0.8 at each bounce.) It takes 3.10 seconds for the ball to bounce 13 times, and then it has moved horizontally 1.96 m. The formulas below show the ball has finished bouncing at time 3.135, and it is stopped at (1.96, 0). After the first toss, the path and bounce points of the ball are displayed, and change as controls are varied. The bounce time and final position are shown when the floor is flat. If "wavy floor thickness" is positive, the ball lands on a wavy surface, and you can adjust the waviness. It is surprising that if the normal and tangential coefficients of restitution  and  satisfy  and  , then the ball will come to rest in finite time. We compute the stopping time and final ball position for a flat floor. Suppose the ball is tossed onto a flat floor from  , where  and  , with initial velocity  , where  , and assume zero turnaround time. Let  be the ball's velocity at the  rebound and let  be the coordinates of the ball at its maximum height after the  bounce,  . Let  be the time of the first bounce, and let  be the time it takes the ball to fall from  to the floor,  . Then  satisfies  and the vertical velocity at  is  . Thus  and  . Also,  and  . For  ,  ,  ,  ,  , and  . Hence  and  for  . Therefore,  ,  . For  , the ball is at  at time  ,  , and  . It follows that the ball is stopped at  at time  . If the floor is frictionless and  , the ball has ceased bouncing at time  , but continues sliding with speed  .
The bouncing ball program assumes that if the tangential and normal components of velocity are  and  before a bounce, they are  and  after the bounce, where  . We have generalized so the reflected components are  and -  , where we call  the tangential coefficient of restitution and  the normal coefficient of restitution. There may be a reason to assume  , but  is the usual coefficient of restitution, and  seems to be a frictional effect. Of course, friction would affect rotation, which we ignore.
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