Even though this question has so far received a downward voting
score, it is something that occasionally crops up, so it seems
worth offering an answer that might perhaps help the questioner
and others similarly placed.
It looks as if the questioner is trying to get just one single figure
for the speed of the moon in its orbit, relative to the earth
(treating the earth as if stationary, as the question says).
One should bear in mind that in reality the moon is continually
varying in its speed and in its distance relative to the earth.
So what the question would be aiming for, with just one single speed
result, has to be some kind of a mean or average value, as the
example figures suggest.
There are many different ways of getting to mean values, of different
kinds, for quantities that change in intricate ways. Any such mean
or average must be considered a kind of approximation, and despite
the possible appearance of many digits (perhaps just for the sake of
maintaining numerical consistency with a source), there is no real
exactness about it. Here is a suggestion for such an approximation:--
Perhaps the simplest and most drastic simplification to make,
to approximate the moon's orbital speed, is to treat the problem
as if the moon moves around the earth at its mean angular rate, and
at its mean distance. That would be motion in a circle.
On this basis, the calculation is much simpler and easier than
the details given in the question.
The two basic essential numbers can both be taken from a
useful paper from researchers at the Paris Observatory,
("ELP 2000-85 - A semi-analytical lunar ephemeris adequate for
historical times", by Michelle Chapront-Touzé and Jean Chapront,
[http://adsabs.harvard.edu/abs/1988A%26A...190..342C]).
The mean distance earth-moon in the cited source is 385000.52899 km,
and the moon's mean angular rate of motion around the earth, in the
reference-frame of the fixed stars (not that of the slowly-moving equinox
point), was (at 1 Jan 2000):
1732559343.19572 arc-seconds per century of 36525 days.
Converting the units, that gives 0.549014926 degrees per hour,
or in radians (units of the radius) per hour 0.00958211810,
and then the corresponding speed (km/h) in the circular orbit
would just be
radius (km) x speed (radians per hour)
i.e. about 3689.12 km/hr, pretty close to the number quoted by the
questioner, by whatever route that was obtained, which I can't identify from
the question.
If anything more exact is wanted, then there seems little point in
looking for some differently-composed mean, because the different
simplifications and approximations needed to reach it would in
practice be just as artificial, as the simplifying supposition
that the moon moves in a circle.
To illustrate that, one need only consider the following --
The real motion of the moon is neither exactly in a circle nor an
ellipse: but it can be approximately modeled by a kind of movable
and plastic ellipse, one that -- apart from its further perturbations --
basically would have the earth at one focus and the moon's angular
speed inversely proportional to the square of the varying earth-moon
distance. On top of that, the motion shows mainly the following kinds
of fluctuation:-
-- the mean major-axis direction of the ellipse slowly rotates, once
in about 8.8 years,
-- but with librations in the moving direction of the axis, so that it
continually wags to and fro in advance and in retard of the average
rotation, on a cycle of nearly 7 months,
-- and an eccentricity that fluctuates, again on a cycle of the same
period of nearly 7 months, but about a quarter-turn out of phase with
the librations in the direction of axis,
-- and then a further speed-up and slow-down, so that the speed
increment is maximum at new- and full-moon, and the slowing is maximal
at the quarters, while the figure of the moon's orbital path is a little
'squashed' along the line joining earth and sun, so that the 'squashing'
of the path brings the moon closer to the earth at the new and full, and
farther away at the quarters.
Beyond that, the motion is as if confined to a slowly-fluctuating plane,
but that has little effect on the orbital speed the subject of this question.
In light of all the fluctuations, it is apparent that for anything
seriously approaching exactness in stating the moon's orbital speed, one
would need to specify a date and time, or times, and make a calculation
of the moon's position and motion relative to the earth, valid
for the calculated times only.
For many purposes, the information in the ELP2000-85 paper cited and linked
above would be entirely adequate for estimation of every aspect of the moon's
motion around the earth for several centuries, though the calculation would
be laborious.
Another and most exact approach would be to read out Chebyshev
coefficients enabling calculation of the moon's positions and velocities
over time, from recent datafiles of solar-system ephemerides placed
online at the Jet Propulsion Laboratory
[e.g. the set at ftp://ssd.jpl.nasa.gov/pub/eph/planets/Linux/de430/].
Although much of the calculation is already done with those data, the
business of correctly handling the data-file format and processing the
Chebyshev coefficients might be hardly less laborious in practice than doing
the full calculation from the analytical paper.
I hope that might help and also place the simplified problem in a useful perspective.
