The distance to the moon can be computed as follows.

We know that:

OK, so acceleration in a circle is $v^2/r$ or $\omega^2r$ where r is the radius of the orbit, and $\omega$ is the rotational velocity. That's 27.3 days divided by $2\pi$ giving 1/375402 radians/second. Hence acceleration in the orbit is $(1/375402)^2r$ where r is the (unknown) distance from the centre of the Earth to the Moon.

But acceleration due to gravity is $9.8(r/R)^{-2}$ where R is the radius of the Earth, so equating these we get:

so This gives an answer of 382517km, which is amazingly close to the figure quoted on WikiPedia of an average centre-centre distance of 384,403km.

Accurate to 0.1%.

Finally, Orbital Velocity is given by $v=\omega{r}$ , and that now works out as 382517/375402 km/s, or almost exactly 1.02km/s.

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