What is the optimum Vsp to get the largest delta_V

expend the least energy E

and/or deliver the largest Payload?Conclusion: the same value of Vsp.

The same value of delta_V/Vsp

Proof

Starting with...


And substituting...

Gives....

Giving the left hand side to be

= [Kinetic Energy into payload and ship] / [Energy into propellant]

The right hand side is just a function Fcn (d, t, u) of three dimensionless numbers (d,t,u). Only d is related to the optimum Vsp. When the value of d is found that maximizes the function, then d=delta_V/Vsp gives the optimum Vsp.
 

Maximizing the term  which is Fcn (d, t, u), maximizes the term . This maximizes P+S, or it maximizes delta_V, or it minimizes E.

When d = 1.59xxxx, and u and t are 0.000, this function is about .647 xxx.

 The optimum Vsp is about 2/3 delta_V (about 1/1.59....).

The propellant is about 3.95 times the payload.

Conclusion:

The same value of delta_V/Vsp


  ( Added 2011-01-14:  apparently Konstantin Tsiolkovsky detailed these key elements towards the end of the 19th century. I was not the first to do so.
       see a copy of Wikipedia article either at Wikipedia or here under Tsiolkovsky     - az )