Can you explain how/where you derived this expression?
The argument holds in reasonably dense atmosphere, which is the case for most of our launches. The full explanation is involved. The argument goes something like this:
1) In dense atmosphere, one reaches a diminishing return of coast to added velocity when speed exceeds terminal velocity at thrust=2* weight
2) Coincidentally, optimal cruising speed is the same speed.
The attached file is from an unpublished document of mine. It punts a little on the first point, using the Fehskens-Malewicki formula for coast distance.
The derivation of that formula follows (I hope it shows up reasonably well...
COAST PHASE:
Mass dV
--------------------------------------- = dTime
-BurnoutMass*g - ShapeConstant*V2
The Left hand side is integrated from BurnoutVelocity to 0; the RHS from 0 to
CoastTime. Result:
CoastTime = Sqrt{BurnoutMass/(g * ShapeConstant)} *
ArcTan[BurnOutVelocity * Sqrt{ShapeConstant/(BurnoutMass * g)}]
We get CoastDistance from f=ma
-(Weight + DragForce) = BurnoutMass * dV/dTime
-BurnoutMass*g - ShapeConstant * V2 = BurnoutMass * dV/dTime
Using a chain rule hat trick,
dV/dTime = (dV/dDistance) * (dDistance/dTime) = V dV/dDistance
-BurnoutMass*g - ShapeConstant * V2 = BurnoutMass * V dV/dDistance
Separating variables:
-Distance/BurnoutMass = V dV
-----------------------------------
BurnoutMass*g + ShapeConstant * V2
The LHS is integrated from o to CoastDistance; the RHS is integrated
from BurnoutVelocity to 0. Result:
CoastDistance = .5 * (BurnoutMass/ShapeConstant)
ShapeConstant * BurnoutVelocity2
* ln{ ---------------------------------- +1}
BurnoutMass * g
The CoastDistance and CoastTime formulas can be modifued to suit situations where weight exceeds thrust. This is like having a coast phase on a planet where the gravitational constant is a smaller figure, g.
BurnoutMass * g = BurnoutMass * g Thrust
Or
G = g Thrust / BurnoutMass.
Substitute g for g in the coast phase equations, and the math is done. No extra integration needed.
POINT OF DIMINISHING ALTITUDE RETURNS TO ADDED VELOCITY
This is the point of greatest coast altitude, burnout velocity slope. In other words, its the point where the derivative of coast distance with respect to burnout velocity is highest. Barring a corner solution, this would correspond to a point where the second derivative is zero; that is, a flex point. The reader may verify the following:
DCoastDistance/dBurnoutVelocity =
BurnoutMass * BurnoutVelocity /
[ShapeConstant * BurnoutVelocity2 + BurnoutMass * g]
d2CoastDistance/dBurnoutVelocity2 =
BurnoutMass /TERM1
2 * ShapeConstant * BurnoutMass * BurnoutVelocity2 / TERM12
where TERM1 = ShapeConstant * BurnoutVelocity2 + BurnoutMass * g
A flex point exists where where the second derivative is zero. The reader may also verify that is where
FlexPointBurnoutVelocity = SQRT{BurnoutMass * g / ShapeCOnstant}
This happens to be the terminal velocity where thrust is twice weight.
One further differentiation and substitution of this value for BurnoutVelocity yields
ThirdDerivativeAtFlexPoint = -1/(2 * BurnoutMass * g)
Since the third derivative is negative, the flex point occurs at a maximum value for the first derivative, as advertised. This is the point of diminishing altitude returns to added velocity.
View attachment Partial Explanation.docx
