## 1 Introduction

I have always been interested in the physics and math behind rocketry and aviation. Aeronautical engineering is my dream job, making it an obvious idea to do some topic related to the same. Recently, I visited the Kennedy Space Centre where I saw the Saturn V rocket, the rocket that was used for all the Apollo missions. While I was bewildered by its very size and scale, what intrigued me the most was haw much of the mass/ scale was the fuel itself rather than the chassis of the rocket. This got me thinking about how the velocity of the rocket would change as, obviously, to gain that velocity, the rocket must expel fuel, making it lighter, once again making the rocket’s velocity faster, creating a positive feedback loop. I decided, then and there that for my Math IA, I would like to explore this concept using physics and calculus and see how close I would be to the right answer as presented by the geniuses that ran the space program.

My goal for this paper is to derive the very important and historic ideal rocket equation through simple physics concepts and formulae and integrate the equation to find the change in velocity in terms of the changing mass of a rocket before first stage separation and compare my results to the historic NASA Apollo 11 mission.

## 2 Data

Toward the end of my paper, my goal is to derive the velocity of the Saturn V rocket through my own calculations. To make sure that my calculations are reliable and somewhat comparative to the NASA given data, I will be using the data about the Saturn V rocket, such as the mass of the rocket, the weight of the fuel within the rocket etc. To simplify the data and, consequently, my calculations, I will only be using the data up to first stage separation rather than the whole trip of the rocket.

The given values by NASA that will be relevant toward the end of my paper are as follows:

- Mass of Rocket with fuel/initial mass = 2290000 kg (NASA)
- Mass of Rocket without fuel/final mass = 130000 kg (NASA)
- Burn time of first stage = 168 s (NASA)
- Mass of fuel lost per second = 18142 kg/s (Space)
- Final Velocity of rocket before stage I separation = 2300 m/s (NASA)

The above-given final velocity of the rocket is what I will be comparing my final experimental value with through a percentage error calculation.

## 3 Definitions

In order to have a real-world application, we must consider real-world parameters and constants in order to obtain an accurate value. A lot of the math that this IA will speak to is heavily related to several laws of physics as a basis. Within this section, I will be defining the various terms that will be important throughout the essay, along with the formulae/equations that will be used for the math.

### 3.1 Newtonian Physics

Among the commonly known 3 Newtonian laws of motion, the math involved in the derivation of the Ideal Rocket Equation hinges upon Newton’s second and third laws of motion.

### Newton’s Second law of Motion:

Newton’s second law of motion can be formally stated as follows: The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object (HowStuffWorks). The above definition gives us a convincing formula that is used in several mathematical discoveries and inventions.

### Newton’s Third law of Motion:

Newton’s third law of motion simply states that “… every action has an opposite and equal reaction” (NASA). This law is fundamental to nearly all fields that have to do with the physical world, but it is especially potent in the science behind rocketry. The fuel that the rocket burns causes a downward force, causing the rocket to motion upward (PCS). With this in place, it’s the rocket equation’s job to find out exactly how, much fuel to burn to cause the reactive force, a force strong enough to lift the mass of the rocket to achieve it’s required velocity, dependent upon its mission statement.

### 3.2 Thrust Force

The thrust force is another force that acts upon a rocket and it is the most important in the flight of a rocket. The thrust force is heavily dependent upon Newton’s third law, as mentioned above. The rockets interior workings allow for the combustion of its fuel, which is allowed to escape from the rear end of the rocket, also known as exhaust, causing a reactionary force: thrust (NASA). This whole action of the rocket’s motion falls under the umbrella term: propulsion.

In Newton’s second law of motion, it states that net force of any system equals the product of m and acceleration. The net force is the total resultant force that acting upon the system, after all vectors have been added (Mansfield). In this paper, I am considering the thrust force as the net force itself as the net force is essentially the final force that is being applied to a object

### 3.3 Conservation of Momentum

The law of conservation of momentum states that “the quantity called momentum that characterizes motion never changes in an isolated collection of objects; that is, the total momentum of a system remains constant” (Britannica). The conservation of momentum is instrumental in creating certain equations that will then be used to derive the Rocket equation.

## 4 Deriving the rocket equation

The above diagram represents the various forces and masses that are acting upon a rocket during its motion.

- Velocity of ejection of the fuel
- Forward velocity the rocket is traveling at
- Mass of Rocket
- = Ejection of mass of fuel as a function of time

As the is the velocity that the fuel leaving the rocket is, we will have to calculate it as a negative and through the law of conservation of momentum, we know that the rate of mass ejection can be written as:

In the above equation, the thrust force is quantified as differential of the velocity of ejection in terms of the ejecting mass, which is again in terms of the change in time.

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According to Newton’s second law of motion:

Acceleration of an object can be found as a differential of the object’s velocity in terms of its changing time, equating:

Substituting this new value of acceleration into Newton’s second law of motion, we get:

Equating the 2 thrust force equations, we can derive the equation:

To get the change in velocity in definite value, I then integrate the above equation. I will be integrating the above equation from an initial state to a final state. As the changing velocity and the changing mass of the rocket are dependent on the fuel in the tank, the initial state, “i”, will be a state when the rocket is full of fuel, and the final state, “f”, will be a state where the mass of the fuel is neglected and the mass of the rocket would be the only contributing mass of the calculation.

To simplify calculations, the negative of as positive by inverting the values in the ln, giving the final equation:

## 5 Integration of the Rocket Equation

To get a more accurate value of the velocity, it would be easier to represent the equation as a function of mass, velocity and time. To make calculations easier, the value of can be omitted as there are far too many variables that impact the velocity that the fuel will be ejected at. To make calculations even more simple, the value of can be written as , where is the constant mass the rocket loses as a definition of time. The final mass of the rocket is only the mass of the chassis of the rocket without the weight of the fuel, and that is the resultant of the mass of the fuel that the rocket loses as a definition of burn time subtracted from the initial mass of the rocket, with the fuel.

Integrating the above equation, in terms of time, with the new variables, we get:

Using integration by parts:

Simplifying the above equation, we get to the final equation:

Note that the final velocity can simply be represented as and not as the change in velocity from a position of rest to a final velocity is the same as the final velocity.

Now, by plugging in the above given values, we get our experimental value to be:

Now that I have an experimental value, I will compare this value to the theoretical value as given by NASA through a percent error calculation:

## 6 Reflection

Seeing as my calculations were based off the “ideal” rocket equation, this equations does not consider for the effects of gravity, lift due to airfoils, drag, wind resistance, the varying amounts of wind resistance as the rocket soars through the atmosphere as the air gets thinner and thinner, the boosts the overall rocket may get due to stage separation etc. The list goes on nearly endlessly, leading to various possibilities and rooms for error, as propagated by the above mention 27% error. More important than the error percentage, is how these calculations and list of problems with these calculations can be solved, because for an error to be solved, the error first needs to be found, and the math that I utilized around the rocket equation allow for the introduction of these various problems, that have been solved for various missions during and beyond the Apollo era.

A major limitation, and arguably the only limitation that births other limitations, is simply the fact that I don’t have all the data, or the data is changing too rapidly for base level calculus that I am working with to work. An example could be the varying effects of gravity at even slightly higher altitudes. While this may seem too tiny to make an impact, for rockets the scale of the Saturn V to escape the Earth’s gravity, the smallest of uncertainties can be the line between a major success or a dismal catastrophe. There are hundreds of more variables such the gravity that I simply cannot fit into.

In conclusion, my IA was able to successfully derive the simple, ideal Rocket Equation, and I was able to successfully convert the above-mentioned equation to fit the data that was publicly available to me by utilizing complex calculus to both derive the equation and integrate the same. The ideal rocket equation was a major starting point for launching humans into space through man-made inventions, and even though the math that I have done is not unique to the studies and missions that occur today, through this math, I am able to reinstate the ground rules that allow for said complexities.

## Works Cited

- Harris, William. “How Newton’s Laws of Motion Work.” HowStuffWorks Science, HowStuffWorks, 8 Mar. 2018, science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion3.htm.
- “Manfield.” Unbalanced Force = Net Force, www.mansfieldct.org/Schools/MMS/staff/hand/lawsunbalancedforce.htm.
- Howell, Elizabeth. “10 Surprising Facts About NASA’s Mighty Saturn V Moon Rocket.” Space.com, Space, 9 Nov. 2017, www.space.com/38720-nasa-saturn-v-rocket-surprising-facts.html.
- NASA. “NASA Flight Manual.” Flight Manual, 1970, history.nasa.gov/afj//ap08fj/pdf/sa503-flightmanual.pdf.
- “Ground Ignition Weights.” NASA, NASA, history.nasa.gov/SP-4029/Apollo_18-19_Ground_Ignition_Weights.htm.