# TECH: ( Long) Beating FM To Death (An additional Equivalent Mass)

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
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I do believe your operation time to be in error way on the short side. Also carrying it to 5-6 significant digits is extreme.
Jerry
--
Jerry Irvine, Box 1242, Claremont, California 91711 USA
Opinion, the whole thing. <mail to: snipped-for-privacy@gte.net>
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Well, I did make a mistake there, but the one I identified was the water mass. When I put the motor specs together, I mixed up two water rockets. Took the mass from 1 and everything else form the other.
It may make a tad more sense with the mass being 310 grams.That is a 1-liter bottle, the spout of which is the same size as a 2-liter bottle. It is only 31% full with a starting pressure of 160psi gage. Also, the water is assumed to be at 86 degrees F to lower viscosity.
Point taken on the digits. I swiped a spreadsheet.
Conclusions regarding equivalent masses are, of course, unaffected.
For Alan Jones: I may have to email you my notes on the water rockets, since they are in MSWord equation editor, and I don't have a web site.
Regards
-Larry C.

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Wouldn't a 2-3 liter bottle work better with a restricted nozzle or choke (ie smaller) since the children's water rockets I have flown for decades have tiny chokes (not nozzles)?
Also steam rockets have TINY throats.
Jerry
--
Jerry Irvine, Box 1242, Claremont, California 91711 USA
Opinion, the whole thing. <mail to: snipped-for-privacy@gte.net>
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Tiny chokes on water rockets reduce thrust, which should be high for these things given their great launch masses. More than that, narrow chokes lead to energy loss through viscosity. Viscosity, for its part, is a two-edge sword. If the effects are too great, you lose energy. If they are too small, turbulent flow results, and you could wind up worse off than you without them. Turbulent flow is also mathematically intractable - at least with my mathematical background.
I tried to sail as close to the wind as is safe. Lowered the viscosity by raising the water temperature - but not too much. Used a broad choke but not too broad.
Regards, -Larry C.

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