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The Rocket Equation (with derivation)


Perhaps most known equation in the engineering world, the name “rocket equation” evokes a mathematical relationship of dominant position. In reality, the “name” of the equation is "Tsiolkovski's equation", due to the Russian scientist, Konstantin Tsiolkovski, who firstly derived it. But what is the reason for its fame and importance?



Before presenting the derivation and characteristics of the equation, it is important that we understand the basic principles behind rocket science. Rockets are engineering projects that originate centuries ago, in the 13th century, in China. The first “rockets” were nothing but arrows fired by gunpowder, developed for military use. Later, with the development of science (mainly with Newton's laws), rocket engineering became more advanced (it should also be noted that fireworks were invented thanks to rocket science in the 18th century). However, it was only at the beginning of the 20th century that rockets as we know were developed.


Like almost every attempt of innovation, failures had to be faced and overcomed to achieve success. After initial trials, physicist Robert Goddard (considered the father of modern rockets) was successful in launching the first American rocket with liquid fuel (see photo 2), in 1937. From then on, improvements in the field of astronautics were carried out. A few decades later, NASA's famous Mercury, Gemini, and Apollo projects were developed (photos 3,4 and 5 show rockets belonging to each project, respectively).



Photo 2



Photo 3

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Photo 5



Okay, we already know a little about the history of rockets, let's turn our attention to their main systems. The structural system refers to the literal structure of the project, that is, the body, nose and "fins" of the rocket (it is basically what we see from the outside). The propulsion system is, not surprisingly, the most important. As we will see later in this article, it constitutes about 80-90% of the rocket (yes, the entire structural system fits a small portion of the remaining percentage). In an extremely simplified way, the propulsion system consists of engines, fuel and oxidant chambers, and channels that allow the interaction between these (such as pumps and valves). The fuel (we will consider liquid hydrogen) mixes with the oxidizer (oxygen) through pumps and valves, in order to generate a combustion reaction (remember the chemistry classes, a combustion occurs when there is a junction of a fuel with an oxidizer), with the burning occuring in the combustion chamber. The gas released by the burning is responsible for providing the thrust/ propulsion to the rocket (see photo 6). The payload system of a rocket is highly variable, depending on the purpose of the mission. In the case of the most famous rocket in history, the Saturn V (see photo 7), the payload system was intended to support the crew for the exploratory mission to the Moon. Extending our example, the Saturn V payload system was composed basically by the Lunar Module (the “part” of the rocket that landed on the Moon), the service module, connected to an engine, and the command module (which was orbiting the Moon during the Apollo 11 mission). See photo 8 for the sections described. The last basic system of rocket composition is the guidance system, composed of on-board computers and radars, promoting stability to the spacecraft.


Photo 6

Photo 7

Photo 8


With the basic components of a rocket in mind, we can now turn to Tsiolkovski's equation. As the rocket burns its fuel with oxygen to generate the thrust, its mass is decreased, since the gas generated is being expelled from the rocket (mass expelled). Tsiolkovski's equation must therefore consider this effect. See photo 9 to 19 for the derivation of the equation in detail.



Photo 9

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Photo 11

Photo 12


Photo 13

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Photo 19

What conclusions do we draw from the derivation of the “rocket equation?” Again, what makes it so important? The simple expression given by v_f = v_0 + v_E * ln⁡ 〖(m_f / m_0)〗 tells us how difficult it is to get a rocket out of the ground and into space. You may have heard about the exponential increase in space transportation cost. Even if only a few dozen kilos are added (in the total composition of the rockets, which usually have a mass in the order of millions of kilos), the cost of the mission is considerably incresead. What would be the reason for this disproportionality? It is not the price of the supposed extra fuel. With certain mathematical manipulations (See photos 20 and 21), the “rocket equation” is able to show that approximately 90% of the rocket is purely “explosive material,” used to generate the thrust necessary for the rocket to reach the desired orbit. Note that everything else we understand as “the rocket”, that is, all additional systems (of physical structure, computers, the crew and/or cargo), is restricted to a mere 10%. Adding more mass would imply an increase of that 10%, which would limit the propulsion system, resulting in a significant decrease in the rocket's range. This delicate balance between the “propulsion mass” (~ 90% of the total mass) and the remaining mass (~ 10% of the total mass) explains the great difficulty of space transportation and the high cost involved.


Photo 20

Photo 21


Imagine if Earth's gravity was greater than the approximate 9.8 m/s ^ (2). For a rocket to "overcome" this gravity, a higher escape velocity would be required, which implies in the need for more fuel. If the current percentage of the “occupation” of the propulsion system is already around 90%, increasing it will decrease the occupance of other basic systems of the rocket (the rocket becomes an explosive totally) or worse, we can barely enter the amount necessary fuel and oxidizer. Let's say we are benefited by the Earth's mass and radius (which determine the value of gravity's acceleration). Photo 22 shows how much of the total rocket composition is “pure explosive” and how much is payload (these data were obtained from the TED lecture by astronaut Don Pettit).


Photo 22

Translation

Foguetes: Rockets

Ônibus Espacial: Space Shuttle

Propelente: "Rocket fuel"

Carga útil: Payload




Photo 1: Cape Photo 2: Robert Goddard's rocket in 1937 Photos 3, 4, and 5: NASA's rockets Photo 6: Rocket's thrust Photo 7: Saturn V Photo 8: Saturn V payload system Photo 9-19: Rocket equation derivation Photo 20 and 21: Algebra for the masses' ratio Photo 22: how much of a rocket is truly a rocket?



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