Sinking sphere

Here's a fun problem for anyone with some time to sink their teeth into.

Consider a sphere of radius R and a smaller sphere of radius r contained
therein. The remaining empty space is filled with water. The smaller
sphere has uniform density, larger than that of water.


(a) ****** (b) ******
**** ++ **** **** ****
** + + ** ** **
* ++ * * *
* * * *
* * * *
* * * ++ *
* * * + + | *
* * * ++ | *
* * * V *
* * * *
* * * *
** ** ** **
**** **** **** ****
****** ******
############################ ############################

The smaller sphere is released from rest from its maximum height (a)
inside the larger sphere and sinks to the bottom. What is its velocity
and acceleration when (b) the centers of the two spheres coincide?

I'll post my solution in a few days. However, I'm sure there are many
different ways to attack this problem: analytical, numerical,
experimental(?). Corrections due to various physical effects, most of
which I've neglected, can also be considered: viscosity, turbulence,
etc.

The original source for this problem:
J. B. Reynolds, Problems for Solution #3607, American Math. Monthly,
v.40, p.243 (1933)

Enjoy.

Igor

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