Say we have the following scenario:

A ball is bouncing forth and back between two parallel walls, located at a distance from each other so that the ball (which is always moving parallel to the normal of the walls) will have to travel one meter to get from one wall to the other. In each bounce the ball will get a new speed by the wall it is bouncing on, which is uniformly distributed between 0 and 1 meter per second. The velocity is of course directed against the other wall from where the ball is located.

At time $\displaystyle t_0$ the ball bounces against one of the walls. What is the expected number of bounces the ball will make after $\displaystyle t_0$ if the ball is stopped after ten seconds?