Tsiolkovsky's Ideal Rocket Equation

Jared Corbin

The History of Math

Professor C. Seaquist

April 22, 2010

Konstantin Tsiolkovsky is one of the most important 19th and 20th century

mathematicians because he laid the mathematical groundwork for many fields within physics

and most importantly the field of rocket science. He wrote over 500 papers on mathematics

and physics which accelerated the U.S. and Soviet space programs. To understand Tsiolkovsky

one must understand the challenges he faced as a young man, his philosophy, and his

fascination with science fiction. I believe that it is also necessary to have a general

understanding of his greatest work the so called "Tsiolkovsky Ideal Rocket Equation" and how

to apply it to model real world rockets.

Konstantin Eduardovich Tsiolkovsky was born on September 17, 1857 in Ijevskoe,

Ryasan Province, Russia. He was the son of Eduard Ignatyevich Tsiolkovsky a provincial forestry

official and Mariya Ivanovna Yumasheva. Konstantin was the fifth out of eighteen children.

Around the age 10 Tsiolkovsky all but lost his hearing due to complications of Scarlet Fever. This

misfortune disqualified him from attending school but, fortunately, led him to become an avid

reader who read every book in his father's personal library. In spite of his hearing disability and

lack of formal education, Tsiolkovsky managed to educate himself by reading voraciously (The

life of Konstantin Eduardovitch Tsiolkovsky). He taught himself the principles of mathematics

and geometry which served him throughout both his teaching career and scientific studies.

In 1873 at the age of 16 Tsiolkovsky moved to Moscow and further educated himself at

the famous Chertkovskaya Library. While in Moscow Nikolai Fedorovitch Fedorov gave

Tsiolkovsky a job in one of the libraries and also tutored him. Nikolai Fedorov was a philosopher

as well as a believer in Cosmism. Cosmism is a very broad philosophy that used science to

explain the theories of origin, evolution, and future of the universe and human kind. In effect it

was the precursor to modern theoretical astrophysics. The fundamental basis of Cosmism is

that it is mankind's goal to survive in the cosmos. This means that mankind must migrate into

space and colonize other planets and live on space stations (What is Cosmism?). I believe

Fedorov's teaching and his investigations influenced the young Tsiolkovsky to theorize about

space flight and rockets. Tsiolkovsky began to seriously investigate space travel at the age of

seventeen after reading Jules Verne's "From the Earth to the Moon" published in 1865.

Tsiolkovsky was an avid science fiction reader and many of Jules Verne's books influenced his

scientific studies. Tsiolkovsky did not only believe that mankind would reach space but that

mankind would become a space civilization roaming between the stars and galaxies.

In 1879 Tsiolkovsky obtained his teaching certification and became a mathematics

teacher in Borovsk, Kaluga Province from 1880 to 1892. It was during this period that he began

to do serious scientific research. He experimented in hot air balloon construction and theory as

well as theorized about life in space and aerodynamics. It was also during this period that in

August of 1880 Tsiolkovsky married Barbara E. Sokolova and eventually had five children by her

(Zak, Konstantin Tsiolkovsky Slept Here).

In 1892 Tsiolkovsky was promoted and moved to Kaluga, Kaluga Province which was the

provincial capital and taught both mathematics and geometry. While in Kaluga Tsiolkovsky

wrote and published most of his papers. His three most notable papers were "Airplane, or Bird-

like Flying Machine" published in 1894, "Exploration of the Universe with Reaction Machines"

published in 1903, and "The Space Rocket Trains" also published in 1929 (The life of Konstantin

Eduardovitch Tsiolkovsky).

"Airplane, or Bird-like Flying Machine" was a revolutionary paper because it described a

heavier-than-air powered aircraft with an aerodynamic shape. This aerodynamic shape

included an unbraced pair of wings, wheeled landing gear, and a propeller. The paper was well

ahead of its time because being published in 1894 it was nine years before the Wright Brothers'

first heavier-than-air powered flight. The aerodynamics Tsiolkovsky outlines was very advanced

because his ideas were not put into practice until the Interwar Period with the advent of

stressed aluminum aircraft bodies. To test many of his aerodynamic theories Tsiolkovsky built

Russia's first wind tunnel (Tokaty).

Tsiolkovsky published "Exploration of the Universe with Reaction Machines" in 1903 but

it was not released in "The Aeronautical Courier" until 1912. In this paper Tsiolkovsky derived

the Ideal Rocket Equation which solved for the velocity of the rocket. Since this equation is the

main scope of my paper I believe it is necessary to show its derivation (Productions).

Recall that in physics Newton's Second Law of Motion states that the net force is equal

to the mass times the acceleration (∑𝐹 = 𝑚 ∗ 𝑎). If one were to think about this equation in

terms of a rocket it has a constant force applied by the rocket engine and a constantly

decreasing mass because the ejected propellant which means an increasing acceleration. Also

recall that the sum of the forces are also equal to change in momentum divided by the change

in time (∑𝐹 =∆𝑝

∆𝑡). Momentum is the mass times velocity (𝑃 = 𝑚 ∗ 𝑣). With the general

understanding of Newton's Second Law and an equation relating that to momentum we can

derive Tsiolkovsky's Ideal Rocket Equation.

∑𝐹 =∆𝑝

∆𝑡

The change in momentum is the final momentum(Pf) minus the initial momentum (Pi) which

yields.

∑𝐹 =𝑃f − 𝑃𝑖

∆𝑡

𝑃f is equal to the empty rocket mass times the initial velocity plus the gain in velocity plus the

ejected fuel mass times its velocity with respect to an observer on the ground 𝑃f = 𝑚 𝑣 +

∆𝑣 + (∆𝑚 ∗ 𝑉𝑒). Notice that 𝑉𝑒 is with respect to an observer on Earth and for this equation

we need the exhaust velocity relative to the rocket. It can be found by the rocket velocity plus

the exhaust velocity relative to the observer 𝑉𝑒 = 𝑣 − 𝑣𝑒. Now we get 𝑃𝑖 is equal to the total

mass of the rocket plus the mass of the propellant times the velocity of the rocket and

propellant 𝑃𝑖 = (𝑚 + ∆𝑚)𝑣.

𝑃𝑓 − 𝑃𝑖 = 𝑚 𝑣 + ∆𝑣 + ∆𝑚 ∗ 𝑉𝑒 − [ 𝑚 + ∆𝑚 𝑣]

Substituting 𝑉𝑒 for 𝑣 − 𝑣𝑒 yields the following.

𝑃𝑓 − 𝑃𝑖 = 𝑚 𝑣 + ∆𝑣 + ∆𝑚(𝑣 − 𝑣𝑒) − 𝑚 + ∆𝑚 𝑣 = 𝑚 ∗ ∆𝑣 + 𝑣𝑒 ∗ ∆𝑚

Now put this in terms of Newton's Second Law which we defined as ∑𝐹 =𝑃f−𝑃𝑖

∆𝑡.

∑𝐹 =𝑚 ∗ ∆𝑣 − 𝑣𝑒 ∗ ∆𝑚

∆𝑡= 𝑚

∆𝑣

∆𝑡 − 𝑣𝑒

∆𝑚

∆𝑡

Since ∆𝑚

∆𝑡 is a negative rate because the rocket is losing mass it actually looks like the

following.

∑𝐹 = 𝑚 ∆𝑣

∆𝑡 + 𝑣𝑒

∆𝑚

∆𝑡

Notice that ∆𝑣

∆𝑡 is acceleration. If the equation were integrated it would find the velocity of the

rocket. Before integration the ∆𝑣

∆𝑡 term must be isolated. For this equation assume that there

are no forces except the rocket engine. This means no downward force of gravity or drag and

the sum of the forces is 0 so ∑𝐹 = 0.

0 = 𝑚 ∆𝑣

∆𝑡 + 𝑣𝑒

∆𝑚

∆𝑡

𝑚 ∆𝑣

∆𝑡 = −𝑣𝑒

∆𝑚

∆𝑡

∆𝑣

∆𝑡 = −𝑣𝑒 ∗

1

𝑚∗

∆𝑚

∆𝑡

∫ ∆𝑣

∆𝑡 𝑑𝑡 = ∫ −𝑣𝑒 ∗

1

𝑚∗

∆𝑚

∆𝑡 𝑑𝑡

𝑣 = −𝑣𝑒 ln 𝑚

𝑚 + ∆𝑚 = 𝑣𝑒 ln

𝑚 + ∆𝑚

𝑚

In this equation the 𝑚 + ∆𝑚 can be thought of as a the rocket and propellant before it

is expelled to supply the force. This is called the "wet weight." The 𝑚 can be thought of as the

rocket after it has expelled all of its propellant. This is called the "dry weight."

The equation derived above is only good for calculations when gravity and air resistance

are negligible. For a real world application of the equation it is necessary to rework the

equation to include gravity like this 𝑣 = 𝑣𝑒 ln 𝑚+∆𝑚

𝑚 + 𝑔 ∗ 𝑡. This is still just an approximation

because it does not take into account drag due to air resistance. Most rockets travel faster than

the speed of sound which is the point classical aerodynamics break down and are no longer

applicable. In this equation 𝑔 is the acceleration due to gravity which varies with respect to the

distance between earth and the rocket but for all practical applications it is considered to be

9.0867m/s2. In this problem consider the surface of the Earth to be the negative direction so

the acceleration due to gravity will be negative that is -9.8067m/s2. The 𝑡 is the time from

launch until burn out, when the rocket engine runs out of fuel and stops producing thrust.

Noting that 𝑔is negative one can see that the speed of the rocket at burn out in the presence of

gravity will be slower than in space.

Now armed with an equation that at least accounts for gravity it can be applied to a real

rocket. Since Tsiolkovsky was Russian it is appropriate to model a Russian rocket. I chose the

Sputnik 1 launch because it was the first manmade satellite put into orbit. Sputnik 1 was

launched into space by a Russian R-7 rocket which was a two-stage rocket with twenty engines.

That means that at launch only the first stage fires. The R-7's first stage consisted of 16 engines.

After all the fuel was expended for the first stage, the first stage containment vehicle was

released and fell back to earth. Simultaneously the second stage fired to further accelerate the

rocket. Basically this system is a rocket stacked on top of another rocket. Tsiolkovsky originally

developed this idea but at the time he referred to it as "rocket trains" in his 1929 paper "The

Space Rocket Trains." This system allows a rocket to achieve a higher velocity than is possible

with a single stage rocket. This means that for this calculation one must solve for the velocity of

the first stage and then solve for the velocity of the second stage and then sum the two

velocities together.

Variable First Stage Second Stage

Total Mass 242,336.2596 kg 103,355.5574 kg

Mass at Burn Out 103,355.5574 kg 20,075.9983 kg

Exhaust Velocity ~3,462.93 m/s ~3,462.93 m/s

Burn Time 120 s 300 s

(Zak, R-7 Family)

𝑣1 = 3462.93𝑚

𝑠∗ ln

242336.2596 kg

103355.5574 kg − 9.0867

m

s2∗ 120s = 1774.136

𝑚

𝑠

𝑣1 = 3462.93𝑚

𝑠∗ ln

103355.5574 kg

20075.9983 kg − 9.0867

m

s2∗ 300s = 2896.3855

𝑚

𝑠

𝑣1 + 𝑣2 = 1774.136𝑚

𝑠+ 2896.3855

𝑚

𝑠= 4670.5215

𝑚

𝑠

This answer is a rough approximation, but is adequate to illustrate the basic principle.

The calculated speed is about 4.7 km/s and the R-7 Sputnik Rocket actually attained a velocity

of around 7 km/s. The error in this calculation is due to not factoring in drag due to air

resistance, the change in the acceleration of gravity at an object moves farther away from the

center of the Earth, and exhaust velocity error. The exhaust velocity is an approximation

because many factors play into calculating this number such as the specific impulse of the fuel,

the size and shape of the venturi, and actual fuel burn rates (Braeunig). Rocket science is not an

exact science and in many cases no amount of calculation can obtain an exact answer. In almost

all cases the fuel burns for a fraction longer or shorter time than expected. In some cases the

fuel does not fully combust or worse it combusts in the fuel tanks causing a catastrophic

accident (Purdue research helps advance new rocket technology), known as a “radical

distribution of parts beyond the containment envelope.”

Another practical application of the Ideal Rocket Equation is calculating the necessary

amount of propellant for completing a specific rocket maneuver such as accelerating from a low

Earth orbit to a high Earth orbit. The previously derived equation can be re-arranged to

∆𝑚 = 𝑚 + ∆𝑚 ∗ (1 − 𝑒−𝑣/𝑣𝑒). Where ∆𝑚 is the propellant, 𝑚 + ∆𝑚 is the total rocket mass, 𝑣

is the desired rocket velocity, and 𝑣𝑒 is the exhaust velocity of the propellant. Imagine if there

was a small rocket already in orbit above Earth and in order to complete a particular maneuver

in space the rocket must attain a velocity of 4500 𝑚/𝑠. To ensure that the rocket had enough

propellant, a rocket scientist could solve for the percentage of fuel the rocket must contain

(Productions).

Variable Given Terms

Total Mass 𝑚 + ∆𝑚

Mass at Burn Out 𝑚

Exhaust Velocity 4500 𝑚/𝑠

Needed Velocity 1000 𝑚/𝑠

Inserting the numbers into the equation we obtain the following.

∆𝑚 = 𝑚 + ∆𝑚 ∗ (1 − 𝑒−𝑣/𝑣𝑒)

∆𝑚 = 𝑚 + ∆𝑚 ∗ (1 − 𝑒

−1000𝑚𝑠

4500𝑚𝑠 )

∆𝑚 = 𝑚 + ∆𝑚 ∗ .1993

Since 𝑚 + ∆𝑚 is the whole rocket it becomes 1 since we are trying to obtain a percentile.

∆𝑚 = 1 ∗ .1993 = .1993

∆𝑚 = 19.93%

In this particular case only 19.93% of the total mass must be propellant to attain the

necessary velocity of the rocket to perform the maneuver. This is an oversimplified example but

this is similar to the process used at NASA and other space agencies to calculate the required

fuel for orbital maneuvers.

Tsiolkovsky also outlined a rocket engine and it's fuels. Interestingly enough he

theorized that liquid hydrogen and liquid oxygen would be a good fuel. This was well ahead of

its time because both hydrogen and oxygen had yet to be cooled to the point of liquefaction.

NASA uses the same fuel mixture in its space shuttle. In the book "The Mysterious Island" by

Jules Verne, which Tsiolkovsky undoubtedly read, liquid hydrogen and oxygen were used as

fuel. I believe that this is a case of science fiction influencing Tsiolkovsky 's thought direction

and scientific work because these materials had yet to be made. In this paper he also solved for

the escape velocity of earth which he calculated to be about 14 km/s. Recall that the R-7rocket

only achieved an escape velocity of 7 km/s. Sputnik was meant to orbit the Earth therefore it

did not have to escape Earth's gravitational pull.

Tsiolkovsky was a visionary in the field of the mathematical representation of rockets.

Though he never attempted to construct an actual rocket, his work in the field of rocketry

accelerated all subsequent rocket and space programs. Working through his disability gave him

the tools necessary to theorize about rockets and space travel as well as derive his famous

equation "The Tsiolkovsky Ideal Rocket Equation." Living in this age we have seen the advent of

space travel and greater feats of space travel and rocketry are still to come. Someday mankind

may evolve to be a space faring race and Tsiolkovksy's dream may be realized. To quote

Tsiolkovksy himself "The Earth is the cradle of humanity, but mankind cannot stay in the cradle

forever."

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—. R-7 Family. 2007. 17 April 2010 <http://www.russianspaceweb.com/sputnik_lv.html>.