The next View from the Pennines looks at some fairly recent work on visco-elastic collisions based on Newton's cradle. This is an executive toy with five (or so) ball-bearings hanging from a frame so that they line up perfectly when hanging down. If one of the end balls is lifted back and released it makes a pendulum swing, strikes the line and an impulse is instantaneously propagated along the line to the other end, and that ball swings up and the cycle repeats.
That is the standard physics textbook explanation anyway, and of course it is wrong! Experiments show that all the balls move a little and there is a complicated long term evolution of the behaviour.
As part of the article I made a simple model to deal with the three halves power law of the (Hertzian) interactions of the balls whilst in contact, with a simplifying assumption that the time of interaction is constant (I had to beg an extra day from the IMAs patient editor, Rebecca Waters to make the calculation). Of course, no sooner had I sent off the final version than I realized that -- almost certainly (I haven't checked the details) -- my analysis is much more accurate than I had thought and the 'assumption' is in fact completely justified.
I'll have to go through it more carefully, but if true it provides a nice example of a hybrid system for which it is possible to obtain an exact return map without being able to solve for the full details. One unknown parameter is introduced, but the quantitative details are pretty much independent of this parameter anyway!
This peachy version of Newton's Cradle is from Caroline Savva's website:
It makes me smile.