I spent a little time solving the following the problem suggested by
Wilson's earlier question:
A block of negligible extent slides from rest down a
curvilinear surface given by y = (n*x)^1/2. In terms
of n and x0, at what value of x does the block leave
the surface?
I find that it is relatively easy to get a solution for x0 in terms
of n and x and much more difficult but not impossible to solve for x.
I did get a closed form solution that I won't reveal just yet in case
others get bit by the bug. I will, however, mention some interesting
properties of my solution as indicated by the following solution grid
(all values are exact):
n x0 x_leave
1 12.25 1
1 60.5 2
1 168.75 3
Challenge: Can you predict the value of x0 that will yield x_leave =
4, 5, etc?
With my solution it is easy to show that
limit as n goes to infinity of x_leave = (4/9) x0
and that an asymptotic form in the limit n goes to zero is