Rocket Equations: Newton's 3rd Law of Motion
"Houston, Tranquility Base here, the Eagle has landed!"  Neil Armstrong, Apollo 11, July 20, 1969
This is an extraordinary photo of Wright Brothers Glider in Flight. It was made in 1911.
Listen and Watch President Kennedy's Sept. 12, 1962 Rice University Space Speech
Setting America's Direction for a New Generation of Science and Education!
Can America do this Again?
Will Any Other American President Even Set Such an Ambitious Science and Space Goal?
§ Listen to Neil Armstrong call back to Houston:
Discovery's final rollout  The end of an era!
( thanks to Osama Obama )
The Last Shuttle 3.4 mile, 6 hour travel atop giant crawler to Launch Pad 39A, Sept. 9, 2010  Discovery is prepared for final Nov. 1, 2010 launch. Now the Chinese and Russians!
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photo source: NASA
LAW III
"To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinge upon another, and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, toward the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of the bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made toward contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium."
Source: "Philosophiae Naturalis Principia Mathematica", Sir Isaac Newton, July 5, 1687
Notice: 272 years difference between Newton's 3rd Law promulgation and its full embodiment!
Implementing Newton's 3rd Law of Motion
Newton's 3rd law of motion is as much a philosophic statement of natural law as it is a mathematical proposition. The best implementation for nature's law of action and reaction are the rocket equations of Konstantine Tsiolkovsky ( 1857  1935 ).
§ Defining the ideal rocket:
§ Derivation of the rocket equation by differential calculus analysis:
§ Derivation of the rocket equation by algebraic analysis:
Konstantin Tsiolkovsky: "Rocket Man" and Father of "The Physics of Zero Gravity"
"Earth is the cradle of humanity, but one cannot live in a cradle forever"  Konstantin Tsiolkovsky ( 1857  1935 )
( source: http://english.ruvr.ru/2012_09_17/SpaceformulaofKonstantinTsiolkovsky/ )
Konstantine Eduardovitch Tsiolkovsky ( 1857  1935 ), lone genius and son of a Polish forester and deportee to Czarist Siberia, first theorized in 1896 that a liquid propellant should be the required fuel for rocket thrust in order to achieve maximum range and that the critical determinant for rocket thrust is u, effective velocity of exhaust gases, thus confirming Newton's 3rd Law of Motion. Having been compelled to leave public school at age 10 owing to permanent hearing loss caused by a period of scarlet fever, Tsiolkovsky nevertheless became self  taught in mathematics and physics and thereby established himself as the true pioneering and visionary father of modern astronautics and rocket mathematics.
Tsiolkovsky first published his rocket mathematics in the same year that Orville and Wilbur Wright performed heavierthanair flight ( Kitty Hawk, North Carolina, December 17, 1903 ) with his "The Exploration of Cosmic Space by Means of Reaction Machines", Science Review #5 ( St. Petersburg, 1903 ) together with his classic shorter article "Research into Interplanetary Space by Means of Rocket Power", 1903, subsequently reprinted in book format in 1914. Both of these publications clearly established Konstantine Tsiolkovsky as a space pioneer and visionary and today his memory is thus recognized by both the American NASA and the modern Russian Space Agency.
The essence of Tsiolkovsky's formula is simple enough: in order to achieve "escape velocity", the ratio of fuel mass to the weight of the rocket must be at least 4:1 in favor of the fuel and the rocket must be multistaged.
§ Tsiolkovsky Rocket Equation ( "Tsiolkovsky formula" ), published 1903:
§ The Saturn V Moon Rocket ( "The Moon Rocket" ):
The Saturn V was designed by Wernher von Braun at the behest of President John F. Kennedy on May 25, 1961 to set a goal of landing an American on the moon within a decade and used by NASA for the first Apollo 11 moon landing on July 20, 1969, when Commander Neil Armstrong spoke these famous words back to planet earth: "Houston, Tranquility Base here. The Eagle has landed!"
The massive Saturn V Moon Rocket blasts off July 16, 1969
source: NASA archives
Saturn V Moon Rocket Basic Statistical Characteristics  

mass  3,038,500 kg  
three (3) stage rocket  stage 1:  thrust = 34.02 MN burn time = 150 sec 

stage 2:  thrust = 5.0 MN burn time = 360 sec 

stage 3:  thrust = 1.0 MN burn time = 165 + 335 sec ( 2 burns ) 
The Rocket Equations
Owing to several differing needs, there are several seemingly different types of rocket equations but all actually manifest the same identical underlying rocket mathematics.
§ Summary of the Rocket Equations:
(i). Generalized rocket equation:
This generalized rocket equation considers rocket weight by factoring out fuel weight from but keeping drag as part of .
(ii). Rocket equation 2:
This equation is really a corollary to the above Generalized equation and is easily derived as follows:
(iii). Rocket equation 3:
(iv). Rocket equation 4:
(v). Rocket equation 5:
§ Derivation of rocket equation 3:
Non  relativistic mass, velocity and time also give the following:
Hence, change in rocket velocity is a function of effective exhaust velocity and time change of rocket mass and fuel mass!
§ Derivation of rocket equation 4:
Therefore,
which is the fraction of the initial mass that is expended as reaction mass.
note: a higher propellant mass fraction represents less payload mass delivered.
Also,
note: the above rocket equation equation shows that for a much greater , effective exhaust velocity, for a given initial mass , this equivalently greatly increases , payload mass, and decreases !!
§ Derivation of rocket equation 5: see derivation down below at "Rocket equation and Specific Impulse"
§ Derivation of Tsiolkovsky Rocket Equation ( "Tsiolkovsky formula" ), published 1903:
The "Tsiolkovsky formula" for rocket propulsion strips away any gravitational force effect by setting
and inserting this value into Rocket Equation 3 for initial velocity also set to zero. QED!
Some Hypothetical Rocket Examples
1). A Saturn V moon rocket, 3.04 x 10^{6 }kg, 88.5% of which is propellant fuel, rises vertically by ejecting exhaust thrust gases a constant velocity of 6.3 km/sec and consuming propellant fuel at a constant rate of 5.4 x 10^{3} kg/sec for ?? seconds before the propellant is totally consumed at final burnout.
a). determining thrust:
b). determining initial vertical acceleration:
c). determining final acceleration before fuel burnout:^{∗}
d). determining Saturn V moon rocket acceleration at 150 seconds, stage 1:
e). determining Saturn V moon rocket acceleration at additional 150 seconds, stage 2:
f). determining Saturn V moon rocket acceleration at additional 198 seconds, stage 3:
g). determining Saturn V moon rocket velocity at burnout:^{∗}
^{∗}note : these examples are used in the future upcoming Relativity Science Calculator Mac application
2). SSTO ( single stage to orbit ) rocket:
What this means is that at t_{o}=0, time zero. 78.65% of the initial total mass is propellant mass and that 100%  78.65% = 21.35% of the initial total mass is available for the rocket body, engines, and eventual rocket payload.
3). TSTO ( two stage to orbit ) rocket:
Thrust ( to Weight ) Ratio
§ Define:
§ Corollary:
Specific Impulse
§ Define:
§ Corollary:
Some Propellants and their Specific Impulse, I_{sp}, in seconds  

Category  Propellant  Type  I_{sp} Range ( sec ) 
Monopropellants  
Low  energy monopropellants  liquid  160 to 190  
Hydrazine Ethylene oxide Hydrogen peroxide 

High  energy monopropellants  liquid  190 to 230  
Nitromethane  
Bipropellants  
Low  energy bipropellants  liquid  200 to 230  
Perchloryl fluoride  Available fuel AnalineAcid JP4Acid Hydrogen peroxideJP4 

Medium  energy bipropellants  liquid  230 to 260  
HydrazineAcid AmmoniaNitrogen tetroxide 

High  energy bipropellants  liquid  250 to 270  
Liquid oxygenJP4 Liquid oxygenAlcohol HydrazineChlorine trifluoride 

Very high  energy bipropellants  liquid  270 to 330  
Liquid oxygen and fluorineJP4 Liquid oxygen and ozoneJP4 Liquid oxygenHydrazine 

Super high  energy bipropellants  liquid  300 to 385  
FluorineHydrogen FluorineAmmonia OzoneHydrogen FluorineDiborane 

Oxidizerbinder combinations  
Potassium perchlorate  solid  170 to 210  
Thiokol or asphalt  
Ammonium perchlorate  solid  170 to 210  
Thiokol Rubber 

Ammonium perchlorate  solid  210 to 250  
Polyurethane Nitropolymer 

Ammonium nitrate  solid  170 to 210  
Polyester Rubber 

Double base  solid  170 to 250  
Perfluorotype propellants  solid  250 and above 
§ Derivations:
(i). Relationship of total impulse and specific impulse:
(ii). Relationship of specific impulse to rates of fuel consumed and so  called "weight flow":
(iii). Specific impulse shown here as the constant amount of propellant times the number of seconds required to burn it:
(iv). Time? What is time? In an earlier portion of this mathematical essay, time was philosophically defined as " ... a system of accounting for the relative motion of bodies" of two or more objects. No motion, no objects ... hence no "time" and therefore no "velocity" and certainly no "space".
In other words,
But what will immediately follow from the previous mathematics for rocket propulsion and thrust is another derivation for time in terms of !!
Here goes:
So already we are understanding "time" in terms of the rate of change of variable, non  relativistic ( rocket ) mass!
Finally,
Amazing!!
Here is another time derivation using dimensional analysis:
Tautology, you say? Maybe. But then again is
not also tautology? We could hence resolve , burnout time, as follows:
which is somewhat less of a definitional tautology.
In any event, however, to paraphrase Plato's eternal statement that
"time is the moving image of reality"
is something upon which we can all agree!
Rocket equation and Specific Impulse
We earlier derived rocket equation
and now we have burnout time
which gives
Rocket Distance
The maximum rocket distance in vertical flight is comprised of 1). powered vertical flight and 2). free flight ( coasting ) after propellant fuel burnout.
1). maximum rocket distance under powered vertical flight is:
2). free flight coasting distance after propellant fuel burnout is:
3). the total flight distance is:
§ Derivation of 1). maximum rocket distance under powered vertical flight:
However,
And therefore,
§ Quick and dirty derivation of 2). free flight coasting distance after propellant fuel burnout is:
In this case no account is taken for variable g as an inverse to r, distance between the center of rocket mass and the center of earth mass.
§ Again: derivation of 2). free flight coasting distance after propellant fuel burnout is:
Preliminary Analysis 1: vertical rocket coasting distance using variable gravity
We know that
For planet earth ( as for any other body of mass ), the gravity force F varies inversely to r, the distance between earth center and the center of some other mass, since by Newton's Law of Universal Gravitation we have
In other words,
And for g, gravity acceleration field on earth, this too varies inversely to , the distance between earth center and center of any other mass, since
That is,
So for some mass suspended at a distance above earth's surface, not at sea level, we get
Now let's compare gravity force at earth's surface to gravity force at some arbitrary distance above earth's surface as follows:
Example: A Russian cosmonaut weighs 155 lbs at earth's surface but at 1,200 miles above earth's surface the cosmonaut will experience
And the Russian cosmonaut will weigh at 1,200 miles above earth's surface
However, the non  relativistic mass of the cosmonaut will remain constant at
Preliminary Analysis 2: vertical rocket coasting distance using conservation of kinetic energy
By Newton's 2nd Law
Let
and considering that in any vertical rocket flight that , kinetic energy, will change with respect to time, we therefore obtain
Using vector analysis, we also obtain
What this expresses is that the rate of change of kinetic energy of an object is equivalent to the amount of power of the forces acting upon the object, and that the change in kinetic energy ( not the rate of change! ) is equivalent to the work expended by the forces acting upon the object! Power therefore is work done per unit of time.
Final Analysis: vertical rocket coasting distance using variable gravity and conservation of kinetic energy
We know that
And we can interpret this at burnout as follows:
Notice the difference between
Rocket Escape Velocity
How position determines work, kinetic energy, and potential energy
Again, we know that
which mathematically is equivalent to
and is visually described as "doing work" by moving from point 1 to point 2 according to the following paths:
Now, moving from point 1 to point 2 is equivalent to moving from point 1 to some arbitrary point P plus point P to point 2; or, equivalently, moving from some arbitrary point P to point 1 and minus point P to point 2. Since point P is arbitrarily chosen, some other point, say point Q, may be chosen by the addition ( or subtraction ) of some given constant.
Continuing,
Therefore, every position relative to point P has an associated amount of potential energy ; or,
It can also be determined that
Another formulation is, thus,
"Gravity Work": potential energy made manifest by gravity
More specifically, the work done by gravity is derived as follows:
So, here we observe that the work done by gravity is intimately connected to potential energy derived from earth's gravity field!
§ Derivation of rocket escape velocity:
Now,
However, the conditions for escape rocket velocity from earth's gravity bonds occurs when:
Previously we know that the gravity force field for acceleration is
so for planet earth we have
§ Examples:
1).
2).
§ Postscript:
If in any situation where the vertical distance climbed by any object is not comparable to the radius of the earth such as for a vertically climbing rocket, then
§ References:
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