Relativity Calculator
"Houston, Tranquility Base here, the Eagle has landed!" - Neil Armstrong, Apollo 11, July 20, 1969
§ Listen to Neil Armstrong call back to Houston: 
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 Issac 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:
(i). there are no twisting or turning "moment forces"; i.e., rocket moment force is zero;
(ii). thrust T acts precisely at the rocket's center of mass [ cm ];
(iii). the rocket and its propellant fuel mass are solely involved in upward y - direction translational rectilinear motion;
(iv). there is a realistic assumption of constant fuel burn rate implying constant thrust T;
(v). g, gravity acceleration, possesses negligible variation; i.e., g is assumed constant;
(vi). in the beginning part of this analysis, the rocket does not escape earth's gravity field. In a later part of this analysis, the rocket does indeed escape earth's gravity field.
§ Derivation of the rocket equation by differential calculus analysis:
§ Derivation of the rocket equation by algebraic analysis:
Konstantin Tsiolkovsky - Rocket Man
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 heavier-than-air flight ( Kitty Hawk, North Carolina, December 17, 1903 ) with his "Exploration of the Universe with 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.
§ 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
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 equation:
This generalized rocket equation considers rocket weight by factoring out external gravity force of rocket and 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"
Some Hypothetical Rocket Examples
1). A Saturn V moon rocket, 3.04 x 106 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 103 kg/sec for 1,010 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 360 seconds, stage 2:
f). determining Saturn V moon rocket acceleration at additional 500 seconds, stage 3:
g). determining Saturn V moon rocket velocity at burnout:
2). SSTO ( single stage to orbit ) rocket:
What this means is that at to=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, Isp, in seconds | |||
|---|---|---|---|
| Propellant | Type | Isp Range ( sec ) |
|
| Monopropellants | |||
| liquid | |||
| liquid | |||
| Nitromethane | |||
| Bipropellants | |||
| liquid | |||
| liquid | |||
| liquid | |||
| liquid | |||
| liquid | |||
| Oxidizer-binder combinations | |||
| solid | |||
| solid | |||
| solid | |||
| solid | |||
| solid | |||
| solid | |||
§ 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" for the relative motion 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
a tautology? We could also resolve 
, burnout time, as follows:
which is somewhat less of a definitional tautology.
In any event, however, 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:
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However,
And therefore,
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§ 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, moving from some arbitrary point P to point t 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:
1). "Introduction to Space Dynamics", by William Tyrrell Thomson - note: much in these equations was suggested to this author although their finality and completeness is now included in "Relativity Calculator - Rocket Equations: Newton's 3rd Law of Motion"
2). For an exciting and very enjoyable rocket experience by way of designing your very own rockets, please go to SpaceCAD - Rocket Simulation Software.
3). "A Transparent Derivation of the Relativistic Rocket Equation", by Dr. Robert Forward, 31st AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, July 10 - 12, 1995, San Diego, CA. This paper provides a concise comparision between the classical rocket equation and the more recent investigation into the mathematics of a future photon rocket.
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