Well, I just thought about the problem a tad bit longer and noticed
that the problem *does* have a hidden conserved quantity in it. This
quantity is constant for the duration of the whole upward motion and
it is also constant during the whole downward motion. But its value
instantly changes from its upward value to its different downward
value at the moment the motion turns around and stops at the top of
the trajectory. This jumping constant C is defined as
C == (r + s)*exp(s*z) (using the notion of my previous post).
Also, the value of C for ascent is the negative reciprocal of its
value for descent. In particular, for ascent C_a = (r + 1)*exp(z) .
And for descent C_d = (r - 1)*exp(-z) . The values of C_a and
C_d are related by C_d = -1/C_a (because at the top of the trajectory
r = 0 and z is the same limit at each end of the turnaround moment).
Now if we evaluate the C_a constant at only the moment (and height)
of launch we have C_a = r_i + 1 . If we also evaluate the C_d
constant only at the moment of return we have C_d = r_f - 1. Using
the top end tiedown relationship that C_d = -1/C_a means we can
write:
r_d - 1 = -1/(r_i +1) .
A simple bit of algebra on this equation gives the final result that