Summary: The Fehskens-Malewicki equations can be extended for the case of high mass fractions by using *separate* equivalent masses for cutoff velocity and cutoff altitude. The masses are:
EquivalentMassForCutoffVelocity = PropellantMass/Ln(LaunchMass/BurnoutMass)
EquivalentMassForCutoffAltitude = 0.5*PropellantMass / (1 - Ln(LaunchMass/BurnoutMass)* (BurnoutMass/PropellantMass))
===== Body: Last week, I posted an equivalent mass for use in the Fehskens-Malewicki equations. That was taken from the equation
(Thrust/EquivalentMass)*BurningTime = Ve*Ln(LaunchMass/BurnoutMass)
The result is that
*** EquivalentMassForCutoffVelocity = PropellantMass/Ln(LaunchMass/BurnoutMass) ***Also posted a digital solution to a water rocket trajectory and the FM solution, and showed an improvement when the equivalent mass was used.
It turns out my digital solution had a small error. The corrected F-M solution is still much better than the uncorrected FM solution, but the cutoff altitude is not quite as good. Neither FM solution is entirely satisfactory. It turns out, a different equivalent mass should be used for cutoff altitude.
When the rocket equation is integrated with respect to time, the result is
CutoffDistanceInSpace = Ve*BurningTime * (1-(BurnoutMass/PropellantMass)* Ln(LaunchMass/BurnoutMass)
One can get a cutoff distance equivalent mass in a parallel way:
CutoffDistanceInSpace = 0.5*(Thrust/EquivalentMass) *BurningTime*BurningTime
Equating the two right-hand sides:
*** EquivalentMassForCutoffAltitude = 0.5*PropellantMass / (1 - Ln(LaunchMass/BurnoutMass)* (BurnoutMass/PropellantMass)) ***Here are corrected trajectory computations for a hypothetical water rocket.
Impulse 15.86411 newton-seconds Water Mass 367.917 grams Operation Time .024468 seconds
Trajectory Computations (Assumptions: Flat thrust curve and constant air density)
Launch mass = 519 grams Diameter = 8.25 cm Cd = .65 Digital New FM Old FM YTot 190.50 ft 190.91 169.61 YCutoff 1.70 ft 1.70 1.88 VCutoff 173.27 ft/sec 173.27 154.19
Changing the launch mass to 418 grams (Natural launch mass) Digital New FM Old FM YTot 141.82 ft 142.99 120.81 YCutoff 2.44 ft 2.45 2.70 VCutoff 296.73 ft/sec 296.71 220.51
The above done without drag : Digital New FM Old FM YTot 1396.19 ft 1396.22 766.37 VCutoff 2.45 ft 2.45 2.71 VCutoff 299.38 ft/sec 299.38 221.60