If one can jump 10 feet from one point on the ground to another at the
same plane, how far would that same person jump out if they were
jumping off a 10 story building (in terms of how far out onto the
street they'd get).  Or put another way, is there a formula for
calculating the diminishing horizontal energy due to the vertical
effects of gravity?

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Clarification of Question by hank1971-ga on 01 Mar 2005 11:15 PST

Thanks for the ideas.  The mathmatical equation is probably the most
helpful, but so are the supporting comments.  So, I have some
follow-ups:

If it's only the ground that stops the horizontal momentum, that would
suggest that there's always an element of horizontal no matter how
high the jump - but that doesn't seem logical.  Surely, you're falling
straight down after some time.  Is it friction with "air" that causes
the horizontal momentum to stop completely in real life?  If so, it is
possible to include the air friction in the calculation and establish
just where that 10 foot jumper lands in the real life 10-story plunge?
 If it's really negligible - like the difference of less than a foot -
then it doesn't effect my question.

Thanks again!!

Hank

We can use projectile motion to solve your problem.
We use the first information (jumping 10 feet) to determine his
velocity of jump. The formula is v = sq root (d x g)
where v = velocity of jump, d = horizontal distance and g = 32 ft/sec2.
(We have assumed that he jumped at 45 degrees to achieve his furthest
distance of 10 feet).

   Thus v = sq root (10 x 32)
          = 17.89 ft/sec
This gives his velocity of jumping off.

We now go to his 10 storey jump. We assume that he jumped horizontally
with a velocity of 17.89 ft/sec.
   The formula to use is D = 17.89 x sq root (2h/g)
where D = horizontal distance from building, h = height of building
(assume 10 storey = 100 feet), g = 32 ft/sec2.
   Substituting, D = 17.89 x sq root (2x 100/32)
                   = 44.725 feet away from the building.

pkuanko gave you the kinetic, mathematical answer, but something else
struck me about this question.  The vertical effects of gravity have
absolutely no effect on the horizontal engery of the jumper.   Indeed,
gravity defines what horizontal means.  Something is horizontal if
gravity does not effect it.  Think of a spherical marble sitting on a
flat table.  The table is horizontal if gravity does not move the
marble.

In the question, the horizontal distance covered by the jumper is
limited by the amount of time he is in the air and his velocity in the
horizontal direction.  It is the friction of contact with the ground
that stops his horizontal movement, not the effects of gravity.  When
he is in the air, the friction resisting his forward motion is
negligible.

Ah, yes Phil, but on level ground, the jumper is going to land with a
lot forward momentum  - the sand thrown forward by what used be be
called broadjumpers as this momentum is stopped, your "friction of
contact with the ground ..."
From the tenth floor, his forward momentum will not be abruptly
stopped, letting him continue his projectile path further, until he
... uh  "reaches" the ground, when his fall may be almost vertical  -
no skid marks.
Any stuntmen out there?  
But I kind of doubt Phuanko's 44.75 feet in practice, due  to air
resistance in the horizontal, but I don't expect that was a
consideration in your question.

The other poster was correct that the horrizontal motion is not
hindered by anything but air.  However the vertical motion is changing
rapidly.  So in relationship, it soon looks like a vertical drop
because the vertical portion of motion becomes so much greater than
the horizontal portion.
This isn't a scale model at all, but an idea of how the jump would go:
  ---
--   --
       --
         -
          -
          -
           -
           -
           -
            -
            -
          SPLAT
Another consideration is the mostion of the body while in the air... 
A good long jumper has a very obvious motion that gets the most
horizontal distance out of his jump.  However in a more vertical jump
perhaps a slightly declined superman pose might get the air resistance
to work a bit more in your favor to get more distance :)  I've seen
parashuters take this pose to move horizontally across the air quite a
ways before opening their shute... of course i'd be too scared to take
on such a brave pose if the ground were only 10 floors away.

This is fun.  Of course, we are all forgetting that the jumper never
reaches the ground.  According to Zeno's paradox, he first must travel
half the distance to the ground, then half of the remaining distance,
and so on. Since he never runs out of half distances, he never reaches
the ground.

If time could only consistantly slow down at the rate required so that
we would never realize the fact that the jumper is moving more than
half the distance to the ground in any measurable amount of time then
I'd completely agree with you Phil :)

But unfortunately time is a constant and that half distance would very
quickly become so quick that it's humanly impossible to only realize a
the half distance occuring.  Maybe in God's eyes he never reaches the
ground... hmmmmm