The first of these ensures that the ball is moving towards the wall
after the bounce and also implies that x~ nþ140, the second is a
physical constraint, since we require the bounce to occur at a
positive distance from the wall, while the third is required for the
ball to hit the wall before returning to the floor (i.e. yn40 in (1)).
We note that the first condition of (20) is necessary for the third
condition of (20) to hold, assuming that vn,xn40. If the first or third
of the conditions (20) is violated, then the ball will next bounce
again on the floor, not the wall, and the process breaks down.
One can conceive of the limiting situation of a wall impact in
which yn¼0 with the ball grazing the floor without impacting and
interacting elastically with it. From (1) this occurs when
xn¼ 2unvn/g, but to avoid the ball immediately interacting with
the floor we also need from (6) that vw þ
n 40, i.e.