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Rocket Equations: Newton's 3rd Law of Motion

"Houston, Tranquility Base here, the Eagle has landed!" - Neil Armstrong, Apollo 11, July 20, 1969

The Wright Brothers Glider - 1911



This is an extraordinary photo of Wright Brothers Glider in Flight. It was made in 1911.


Historic President Kennedy's 1962 Rice University Space Speech



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!


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!
Or will America make a comeback?
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 ).

delta_rocket_blastoff.png

§ Defining the ideal rocket: 

  1. there are no twisting or turning "moment forces" - i.e., rocket moment force is zero;

  2. thrust acts precisely at the rocket's center of mass [ cm ];

  3. the rocket and its propellant fuel mass are solely involved in upward - direction translational rectilinear motion;

  4. there is a realistic assumption of constant fuel burn rate implying constant thrust ;

  5. , gravity acceleration, possesses negligible variation - i.e., is assumed constant;

  6. 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:

differential_analysis.png

§ Derivation of the rocket equation by algebraic analysis:

algebraic_analysis.png

algebraic_derivation.png


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/Space-formula-of-Konstantin-Tsiolkovsky/ )

Tsiolkovsky.png

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 "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 multi-staged.

§ Tsiolkovsky Rocket Equation ( "Tsiolkovsky formula" ), published 1903:

Tsiolkovsky_formula.pngnatural logarithm function

§ 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!"

american flag


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 equation:

generalized_rocket_equation.png

This generalized rocket equation considers rocket weight by factoring out external gravity force of rocket and fuel weight from F_sub_a.png but keeping drag drag_D.png as part of F_sub_a.png.

(ii). Rocket equation 2:

rocket_equation_2.png

This equation is really a corollary to the above Generalized equation and is easily derived as follows:

derivation_rocket_equation2.png

(iii). Rocket equation 3:

rocket_equation3.png

(iv). Rocket equation 4:

rocket_equation4.pngnatural logarithm base number

(v). Rocket equation 5:

rocket_equation7.png

§ Derivation of rocket equation 3:

derivation_rocket_equation3.pngnatural logarithmic rules

Non - relativistic mass, velocity and time also give the following:

function_of_time.png

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:

derivation_rocket_equation4.pngmathematical definition of ln and e

Therefore,

derivation_rocket_equation4a.png

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,

derivation_rocket_equation4b.png

note: the above rocket equation equation shows that for a much greater u_exhaust_velocity.png, effective exhaust velocity, for a given initial mass m_subzero.png, this equivalently greatly increases m_sub_bo.png, payload mass, and decreases m_sub_propellant.png!!

§ 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 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 ?? seconds before the propellant is totally consumed at final burnout.

a). determining thrust:

determining_thrust.png

b). determining initial vertical acceleration:

initial_acceleration.png

c). determining final acceleration before fuel burnout:

final_acceleration.png

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:

acceleration_360secs.png

f). determining Saturn V moon rocket acceleration at additional 198 seconds, stage 3:

acceleration_500secs.pngfinal acceleration

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:

SSTO.png

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:

TSTO.png


Thrust ( to Weight ) Ratio

§ Define:

thrust_to_weight_ratio.png

§ Corollary:

thrust_weight_corollary.png


Specific Impulse

§ Define:

specific_impulse.png

§ Corollary:

corollary_specific_impulse.png


source: http://www.hq.nasa.gov/pao/History/conghand/propelnt.htm
Some Propellants and their Specific Impulse, Isp, in seconds
Category Propellant Type Isp 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
Analine-Acid
JP-4-Acid
Hydrogen peroxide-JP-4
Medium - energy bipropellants liquid 230 to 260
Hydrazine-Acid
Ammonia-Nitrogen tetroxide
High - energy bipropellants liquid 250 to 270
Liquid oxygen-JP-4
Liquid oxygen-Alcohol
Hydrazine-Chlorine trifluoride
Very high - energy bipropellants liquid 270 to 330
Liquid oxygen and fluorine-JP-4
Liquid oxygen and ozone-JP-4
Liquid oxygen-Hydrazine
Super high - energy bipropellants liquid 300 to 385
Fluorine-Hydrogen
Fluorine-Ammonia
Ozone-Hydrogen
Fluorine-Diborane
Oxidizer-binder 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
Perfluoro-type propellants solid 250 and above


§ Derivations:

(i). Relationship of total impulse and specific impulse:

derivation1.png

(ii). Relationship of specific impulse to rates of fuel consumed and so - called "weight flow":

derivation2.png

(iii). Specific impulse shown here as the constant amount of propellant times the number of seconds required to burn it: 

derivation3.png

(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,

time_traditional_definition.png

But what will immediately follow from the previous mathematics for rocket propulsion and thrust is another derivation for time in terms of time_new_definition.png !!

Here goes:

understanding time

So already we are understanding "time" in terms of the rate of change of variable, non - relativistic ( rocket ) mass!

Finally, 

final_time_derivation.png

Amazing!!

Here is another time derivation using dimensional analysis:

dimensional_analysis.png

Tautology, you say? Maybe. But then again is

time_tautology.png

not also tautology? We could hence resolve t_sub_bo.png, burnout time, as follows: 

time_tautology_less.png

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

rocket_equation5.png

and now we have burnout time

time_new_definition2.png

which gives

rocket_equation6.png


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:

rocket_max_distance.png

2). free flight coasting distance after propellant fuel burnout is:

rocket_distance_coasting.png

3). the total flight distance is:

rocket_total_distance.png

§ Derivation of 1). maximum rocket distance under powered vertical flight:

derivation_of_1).pngreciprocal rule for ln

However,

integration by parts calculus

And therefore,

final_derivation_of_1).pngmaximum rocket distance travelled under powered vertical flight

§ 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.

vertical_coasting_quick_dir.png

§ 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 

planet_gravity.pngsome consequences of Kepler, Galileo and Newton

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 

gravitational_force.png.

In other words,

inverse_force_law.png

And for g, gravity acceleration field on earth, this too varies inversely to earth_radius.png, the distance between earth center and center of any other mass, since

earth_acceleration.png.

That is,

earth_gravity_sea_level.png

So for some mass suspended at a distance above earth's surface, not at sea level, we get

inverse_g_ratios.png

Now let's compare gravity force at earth's surface to gravity force at some arbitrary distance above earth's surface as follows:

f_g_inverse_r.png

Example: A Russian cosmonaut weighs 155 lbs at earth's surface but at 1,200 miles above earth's surface the cosmonaut will experience

cosmonaut_gravity_experienc.png

And the Russian cosmonaut will weigh at 1,200 miles above earth's surface

cosmonaut_weight.png

However, the non - relativistic mass of the cosmonaut will remain constant at

cosmonaut_mass.png

Preliminary Analysis 2: vertical rocket coasting distance using conservation of kinetic energy

By Newton's 2nd Law

newton_2nd_law.png

Let

kinetic_energy_definition.png

and considering that in any vertical rocket flight that T_k.e.png, kinetic energy, will change with respect to time, we therefore obtain

power_work_definition.png

Using vector analysis, we also obtain

power_work_vector_analysis.png

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

change_k.e.png

And we can interpret this at burnout as follows:

final_analysis_rocket_coast.png

Notice the difference between

difference_coasting.png


Rocket Escape Velocity

How position determines work, kinetic energy, and potential energy

Again, we know that

change_k.e.png

which mathematically is equivalent to

mathematically_equivalent.png

and is visually described as "doing work" by moving from point 1 to point 2 according to the following paths:

paths.png

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,

potential_energy_definition.png

Therefore, every position relative to point P has an associated amount of potential energy ; or,

potential_energy_point_P.png

It can also be determined that

conserving_k.e._p.e.png

Another formulation is, thus,

law_conservation2.png

"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,

escape_velocity.png

However, the conditions for escape rocket velocity from earth's gravity bonds occurs when:

conditions_escape_velocity.png

escape_velocity2.png

Previously we know that the gravity force field for acceleration is

planet_acceleration.png

so for planet earth we have

escape_velocity_generalized.png

§ Examples:

1).

earth_velocity_escape.png

2).

sun_escape_velocity.png

§ 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

potential_energy_height.png

§ References:

  1. "Introduction to Space Dynamics", by William Tyrrell Thomson, Dover Publications, 1986 - note: much in these equations was suggested to this author although their finality and completeness is already now included in "Relativity Science 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 at http://www.spacecad.com/.

  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 investigations into the mathematics of a future photon rocket.

  4. "Access To Mars: Earth To Mars Transit - Logistics Alternatives, Part 1", by John K. Strickland, Jr., National Space Society Board of Directors, Presented at the International Space Development Conference, Huntsville, Alabama, May 18 - 22, 2011. You may read Mr. Strickland's biography at http://www.nss.org/about/bios/strickland.html.




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