Don't
bother dividing the
project velocity by the length of the iteration or the number of
people. This number isn't any good to compare two project's
productivity.
Each project team will have a different bias to estimating stories and
tasks, some estimate high, some estimate low. It doesn't matter in the
long run. Tracking the total amount of work done during each iteration
is the key to keeping the project moving at a
steady predictable pace.

The
problem with any project is the initial estimate. Collecting lots of
details does not make your initial estimate anything other than a
guess. Worry about estimating the overall scope of the project and get
that right instead of creating large documents. Consider spending the
time you would have invested into creating a detailed specification on
actually doing a couple iterations of development. Measure the project
velocity during these initial explorations and make a much better guess
at the project's total size.