You hold your hands out in front of you and catch a bowling ball
dropped from a foot above your hands.  No problem.  If, instead, you
were to hold your hands about half an inch above the floor, and try
the same catch, you would have a painful experience, but Why?? (please
explain)

I'll give you a hint. What happens in that 1/2 inch as you capture the ball?

I know that this seems simple but I want to know from you why would it
hurt you dealing with the distance that you had before and now.

Perhaps you should try this with a real bowling ball and observe what
happens when your hands are that close to the floor. ;-)

This question has two answers, a simpler one, and a more complex one.
I'll give the simpler one first: What you need to consider is not that
the ball is falling, but the momentum it has at the time you catch it.
If you catch a bowling ball half a second after it is dropped, it will
be falling at about 5m/s (remember that, air resistance aside, objects
in free fall accelerate at 9.8m/s^2). If you catch the ball after it
has been falling for 5 seconds, it will be traveling at about 50m/s.
So why talk about the speed at which the ball is traveling? Well as I
said, the key here to your hand being crushed is momentum. Momentum is
equal to the mass of the object times the speed of the object. The
bowling ball weighs about 6 Kg (about 14 lbs), which is pretty heavy.
Multiply that by the velocity it is traveling at (which keeps
increasing the farther the ball drops)and its momentum increases the
farther (longer) the ball is left falling. Imagine you dropped that
bowling ball off of a skyscraper, and it falls on a car. Liken this to
hitting a parked car when you are traveling at 5mph and 55mph. Thats
the difference between a fender-bender and a hospital stay. Your
momentum is 11 times what it was at 5mph when you are going 55mph. So,
simply, the farther the ball is allowed to fall, the faster it is
traveling when it hits your hand, and the more momentum it has as a
result.

Now: an addendum for the more complicated version. Take everything
that I said above, and add on an additional consideration: your hands
have plenty of room with which to stop the ball when you catch it high
up, but only a half inch when you catch it at the floor. As you are
catching the ball, you are exerting a force opposite to the path of
the ball. You are pushing up, the ball is moving down. When you catch
the ball high up, your hands may exert force on the ball for about a
second of time, using a foot of vertical space with which to do so.
(Imagine catching the ball - your arms travel down as they do. If you
hold your hands rigidly when you catch it, allowing no give, they will
hurt.) When you catch the ball at the floor, you cannot slow the ball
down over a long time - you have to slow it down immediately, and the
floor will help, as your hands/bowling ball smack into it. This is
similar to diving 50 feet onto a safety net, rather than a concrete
floor. The safety net will slow you down gradually, using some
vertical space to do so (you can envision the net bending where you
land.) The concrete slows you down all at once, and it will hurt. To
sum up this whole concept, how you feel the force of the impact is how
much force you feel at any one time. Is the entire force of the
bowling ball felt by your hand in 1/4 of a second, or over 1 or 2
seconds? In order to moderate the amount of force felt at a given
time, your hands need to be able to slow the ball down gradually,
requiring space. You have this when you catch the ball high up. You
don't when you catch it against the floor.

Now: couple this second part with the first part of the answer, and
you have the whole story.

That was the long answer.  The short one is that in the first example,
your hands also descend until the motion of the ball is stopped, but
you take this for granted from a lifetime of experience.

Great answer by muehlbauer .... so well explained!

I think muehlbauer's way of thinking about this is pretty much the
same as I would think about it, but here is my explanation anyway:
From Newton's second law we have f=ma.  Some algebra will give you
FT=MV.  MV = mass*velocity=momentum. FT+Force*Time=Impulse.  So to get
the bowling ball to stop, you must apply a force for a certian amount
of time equal such that it equal's the bowling balls momentum.  If you
are really strong, you will apply a lot of force and thus the time it
takes to stop the bowling ball will be small.  I think this is
actually quite complicated overall because 1) To determined whether or
not your hands hit the floor requires not only the time as determined
by the force applied, but also the velocity your hands are moving
down, which is also determined by the bowling ball's momentum at
impact.  Also, the bowling ball is actually accelerating due to
gravity and is thus applying a force(it's weight) even after you've
stopped it, so you must continue to apply a counter force even after
you've stopped it.

the reason that you will be hurt is very simple. when things fall they
accelerate at a rate of 9.8 meters/sec^2 so if you drop the ball it
starts at rest and accelerates until it reaches a velocity. now as the
ball falls its kinetic energy increases ((1/2)mass*velocity^2) so does
its momentum (P(momentum)=mass*velocity). now as both of these things
increase the work needed to stop the bowling ball increases as well.
now work is equal to force*distance so   unless you can supply tons of
force to stop the ball from smashing your hands into the floor, then
you need distance to make up for the lack of force. For example if the
ball weighing 5.4 kilograms (12 lbs) falls from a height of 1.8 meters
(6ft) off the ground then according to the equation Potential energy
before(Mass*gravity*height)+kinetic energy
before((1/2)mass*velocity^2)=potential energy after + kinetic energy
after you can find that the speed of the ball just before it hits the
ground is about 5.9 meters per second. so to find the work needed to
stop the ball use the equation work = change in kinetic energy. so the
work needed is 94 joules. now saying you can apply a force of about
622 newtons by lifting something, you should be able to apply about 1
forth of that force if your hands are a few inches off the ground. so
you can apply about 150 newtons to the ball now the distance needed to
stop this ball is .62 meters or 62 centimeters or about 24 inches. so
you can see that the reason you cant stop the ball before it smashes
your fingers into the ground is because one cant apply the amount of
force needed in the right amount of time.

So I tried to catch the bowling ball while sitting down on a chair.
Big problem! My arthritic left elbow gave way. Bowling ball nearly hit
my left foot. Cried out loud in agony. Very paiful experience.

After some time, Motrin and rest, I went to the second test. Hands
about half an inch above the floor. So I bent down to position my
hands that low. Ouuuccchh! Untold painful experience! My lower back
locked even before I tried to position them hands......

More Motrin, and now cuss words, and tried to blame it on you, mocha28-ga.
If I don't have a physical theraphist son, you will hear from me.
(The son just treated me as usual. He is used to his wacky Dad already.)

TICBOL-GA,
I didn't mean for you to try this experient out jsut to think about
the situation and how you would feel and what you think would happen. 
So, know I guess I know from your experience not to do it, but my son
was going to try it out and I didn't want him to crush his little
fingers since his hands were close to the floor.  I want to thank
everyone for the responses, but this question does give you something
to think about.

your cant asorb the impact as well from .5 inch.  and wouldent you
hand hit the ground from half an inch.