January 12, 2016

.In the 1600s, Sir Isaac Newton determined 3 principle **Laws of Motion** that any classical body will follow. These laws are:

- An object at rest will remain at rest, and an object in motion will remain in motion unless acted upon by an external force (Law of Inertia)
- The net force acting on an object is equal to the mass of the object multiplied by the acceleration of the object.
- Every action has an equal and opposite reaction. (Conservation of Momentum)

Rockets are some of the best examples of Isaac Newton’s Laws of Motion, and function by making use of the third law of motion.

A rocket is any device that causes itself to accelerate by moving a portion of its mass in the direction opposite the desired direction of motion. The exhaust that the rocket expelled is usually accelerated through a nozzle of some sort to increase the thrust the rocket produces.

In order to understand how a rocket can move by leaving particles behind, it is important to understand the concept of **momentum**. Mathematically, momentum is mass multiplied by velocity, or . In any given system, or set of objects, the total change in momentum is equal to the external forces acting on the system. For a system containing a rocket and its exhaust, such forces would include drag on the rocket from the surround air resisting the motion of the rocket, as well as the weight of rocket and the exhaust. However, for simplicity, these forces will be neglected. Therefore, for the system of the ideal rocket and exhaust, there are no external forces on the system, so there cannot be any change in the total momentum of the system.

__IMPORTANT NOTE:__ No change in total momentum does not mean that there can be no changes in the momentum of the individual components of the system. It simply means that these changes must be equal in magnitude and opposite in direction.

To illustrate how a rocket accelerates, we will start with a rocket of mass M kg that is releasing no exhaust, and is not moving (v = 0 m/s). For reference, a positive velocity will denote motion to the right. We will consider the system of this rocket to be the rocket itself, plus the exhaust of the rocket.

Under the given conditions, the momentum of the system

The rocket then turns on its engine and begins releasing exhaust relative to the rocket of c = -4000 m/s (4000 m/s to the left). The amount of exhaust released is a very small amount dm.

By conservation of momentum, we know that the total momentum of the system must remain at 0, because there are no external forces on the system. Therefore, the momentum of the exhaust () must be equal in magnitude and opposite in direction to the momentum of the rocket. Therefore, . Therefore, the final velocity of the rocket is m/s. By repeating this process constantly, rockets can accelerate to very high velocities.

From this statement of conservation of momentum, it is possible to determine how much energy the rocket can gain from expelling a certain amount of propellant mass. For rockets, this energy is usually expressed as ΔV, or the change in velocity of the rocket. This is because in space, position is not very meaningful, since the rocket is constantly in motion. The path a rocket in space follows is determined by its velocity, however, so velocity is used as a measure of motion in space.

Once again, conservation of momentum states that:

for a closed system with no outside forces. This is just a way of saying what was said before, which is that the change in the total momentum of the system is 0 if there are no outside forces.

We can use some calculus to derive a relationship between the change in mass and the change in velocity of the rocket. First we need to separate the variables onto different sides of the equation.

Then, we can integrate each side of the equation to compare the initial mass and velocity of the rocket to the final mass and velocity.

Evaluating these integrals gives

With a bit of rearrangement, this expression becomes **Tsiolkovsky’s canonical rocket equation**:

where is the gravitational acceleration on Earth, or 9.81 m/s^{2}, and *I*_{sp} is the specific impulse of the rocket.

**Specific impulse** is the most common value used to measure how efficient a rocket is. While each component of the rocket has its own complex efficiencies, such as combustion efficiency for burning the propellants, aerodynamic efficiency in the atmosphere, efficiency of the plumbing, and many more, specific impulse takes all of these into account. Put simply,

For a chemical rocket, or a rocket that produces the high energy exhaust by burning an oxidizer, like oxygen, and a fuel, like hydrogen or kerosene, *I*_{sp} is limited to about 450 to 500 seconds. In exchange, chemical rockets have very high thrusts, and can therefore need to be used for launching rockets off Earth’s surface. Another type of rocket, electric propelled rockets, accelerate charged particles through an electromagnetic field as exhaust. These rockets can have a specific impulse of up to 10,000s, but suffer from having very low thrust. There is no electric propulsion system currently in existence that can lift its own weight against Earth’s gravity.

Every rocket has 5 important groups that must work together to achieve the mission the rocket is designed for. These parts are:

**Propulsion.**This group is responsible for making the rocket go**Structures**. This group is responsible for making sure the rocket can withstand the acceleration of the rocket without damaging the rocket or the payload**Avionics**. This group lets the rocket determine where it is and control its path**Recovery**. This group makes sure the rocket comes back safely.**Payload**. This group operates separate to the design of the rocket, but is the reason the rocket is flying.

I will go briefly in more detail as to what each group’s task is, and some of the technology being used in each.

Propulsion is the part of the rocket that produces and accelerates the exhaust used to make the rocket move. This is done through use of some sort of engine, though how each engine works may be very different. The two most effective forms of propulsion that are known are chemical propulsion, where an oxidizer and a fuel will combust and produce an exhaust, and electric ion propulsion, which accelerates heavy charged particles through an electromagnetic field.

Chemical rockets have limited efficiency, with a specific impulse of at most 450 to 500 seconds, but produce large amounts of thrust very quickly, and so are used when moving the rocket is difficult, such as during launch. Chemical rockets can use liquid propellants, such as on a launch vehicle, or solid propellants, which are used with some launch vehicles or in amateur sounding rockets.

Electric engines are very, very efficient, with specific impulses of up to 10,000 seconds, but suffer from very low thrust. No electric propulsion system currently in existence can lift its own weight against Earth’s gravity. For this reason, electric ion engines are used on probes being sent into deep space, where they can accelerate over a very long time.

All propulsion systems are very complex, and explaining them is beyond the scope of this article. For those who are interested in learning more about these, and are willing to read a collegiate level textbook, Sutton’s *Rocket Propulsion Elements* is an excellent source for learning more about these.

The purpose of the structure of the rocket is to provide enough durability for the rocket to withstand the loads at launch, which can be extremely, while housing the payload and avionics systems safely. They also provide the exterior airframe seen on a rocket, which provides the aerodynamic behavior seen during launch. Unfortunately, this is not *The Martian*, and rockets cannot fly on Earth without a nosecone to greatly reduce drag. What materials are used to build the rocket depend strongly on the scale of the rocket. For small amateur rockets, cardboard tubes and wood are often used. Larger amateur rockets will reinforce cardboard with fiberglass or carbon fiber fabrics to allow the rocket to survive the effects of supersonic flight. Launch vehicles are usually made of carbon fiber or metals like aluminum or titanium.

The avionics team provides the “brain” of the rocket. In the aerospace industry, avionics is referred to as Guidance, Navigation, and Control (GNC). GNC must allow the rocket to know where it is at, using information from GPS, inertial measurement units (IMUs), and other sensors, determine where it wants to go to establish the desired path, whether this is an orbit or to reach a target altitude, and then to control the propulsion to achieve this. The control can be done in a few ways. Many launch vehicles have the engines on a gimbal, which is a fancy name for a pivot, allowing the engines to point in different directions. This can change the angle of the rocket, allowing some control over the flight of the rocket. The engines can also be throttled, changing the amount of thrust they produce and changing how high or fast the rocket moves. Finally, there are sometimes small thrusters near the tip of a rocket that allow the nose to be moved around, steering the rocket.

The recovery team is responsible for making sure those parts of the rocket that can be recovered are returned to Earth safely. This can range from putting a parachute on a small amateur rocket to make sure it falls slowly to the propulsive landings achieved by SpaceX and Blue Origin in the 2015. Recovery systems are some of the most important systems on flights that land on other planets or on manned flights, as these are some of the flights with the highest risks and have the most extreme consequences should the recovery not work.

The payload of a rocket is the reason the rocket is flying. This can be anything from a package of sensors to a satellite that orbits Earth to a crew flying to the International Space Station to a deep space probe. Such payloads must be designed to withstand the forces of launch, but are otherwise made with the final mission in mind. Many of these payloads are small rockets in and of themselves, using ion engines or small chemical thrusters to steer on their flight to their final destination.

When a rocket in flying an atmosphere, it is very, very important that the rocket continues to point in the desired direction, which is generally up. If the rocket is unstable, usually it will tip over, and then catastrophically fail. In theory, stability is determined by the relative positions of the center of gravity of the rocket and the center of pressure.

For any object, the center of gravity, or center of mass, is the point at which the entire mass of the rocket could be concentrated without changing the balance of the rocket. To determine where this point is, the object in question can be balanced on a thin rod, like a pencil or a finger. Wherever the object balances in the center of gravity. The center of gravity can be moved by changing the distribution of mass within the rocket.

Center of pressure is a term that refers to the point that the aerodynamic forces on the rocket are effectively applied at. This includes drag due to the cross-sectional area of the rocket, as well as friction between the skin of the rocket and the surrounding air. To determine the location of the center of pressure, the Barrowman equations can be used, or a simulation program like RockSim or OpenRocket can be used. For a worst-case approximation that will guarantee that a model is stable, the center of pressure can be determined by finding the center of area of the lateral cross-section of the rocket.

For a rocket to be stable, the location of the center of pressure must be behind the location of the center of gravity. By behind, I mean farther away from the nose, such that when the rocket is standing vertically, the location of the center of pressure is below the center of gravity. This difference is usually measured in “calibers”, or rocket body diameters. Usually, a difference of between 1 and 1.5 calibers of stability at launch is desirable. Simulation programs will be able to determine how stable a model is. For small, light rockets, stability can also be determined by tying a string around the center of gravity, and spinning the model around rapidly. If the model points in the direction of travel, the rocket is stable and will fly straight during launch.

For those who are interested in continuing to learn about rockets, below are listed a series of useful resources to explore.

http://www.nakka-rocketry.net/ -- Website for Richard Nakka, who develops experimental amateur rockets. He has most topics relevant to amateur rocketry available on his site.

http://copenhagensuborbitals.com/ -- Copenhagen Suborbitals is an amateur space program. They develop rockets from relatively cheap supplies. Great source of information, and a good source of information on manufacturing

http://www.rocketryforum.com/ -- The Rocketry Forum is a discussion board for the entire amateur rocketry community. Discussions are there about the projects people are doing, how to develop solid rocket motors, and most other topics that amateur rocketeers run into. All experience levels use the forum, from those who have flown only a couple rockets before to those who develop and fly rockets like this for a living, and have been doing so for 40 years.

https://www.apogeerockets.com/education/downloads/Newsletter238.pdf -- A solid article about the ways to determine stability. All major ways of determining the stability of a rocket are compared in this article, with advantages and disadvantages to each.

http://openrocket.sourceforge.net/download.html – Download link for OpenRocket, a free simulation software that can be used to design and test rockets without needing to build any components until a successful design is determined.

*Rocket Propulsion Elements*, [George P. Sutton, Oscar Biblarz]—Great textbook on all forms of propulsion, but quite advanced.

http://www.nasa.gov/audience/foreducators/rocketry/home/#.VqkNECorKUk – NASA’s education page on rocketry. Great place to see how different rockets compare, and to learn more about how they work

https://spaceflightsystems.grc.nasa.gov/education/rocket/bgmr.html -- Another NASA page about rocketry education. It’s a little difficult to find where everything is, but it’s a great source for establishing a background in rocketry.

The following links are a series of videos posted by Apogee Rockets, one of the largest suppliers of rocketry components in the United States. The process shown in these videos is generally the same for all model rockets, whether they are Estes rockets or more advanced high-powered rockets.

https://www.apogeerockets.com/Advanced_Construction_Videos/Rocketry_Video_61: Part 1- Motor Mount Construction

https://www.apogeerockets.com/Advanced_Construction_Videos/Rocketry_Video_62: Part 2- Motor Mount Installation

https://www.apogeerockets.com/Advanced_Construction_Videos/Rocketry_Video_63: Part 3- Fin Installation

https://www.apogeerockets.com/Advanced_Construction_Videos/Rocketry_Video_64: Part 4- Rail Button and Retention Installation

https://www.apogeerockets.com/Advanced_Construction_Videos/Rocketry_Video_65: Part 5- High Power Fin Fillets

https://www.apogeerockets.com/Advanced_Construction_Videos/Rocketry_Video_66: Part 6- E-Bay Assembly Part 1

https://www.apogeerockets.com/Advanced_Construction_Videos/Rocketry_Video_67: Part 7- E-Bay Assembly Part 2

https://www.apogeerockets.com/Advanced_Construction_Videos/Rocketry_Video_68 : Part 8- Priming the Rocket

https://www.apogeerockets.com/Advanced_Construction_Videos/Rocketry_Video_69: Part 9- Sanding the Primer

https://www.apogeerockets.com/Advanced_Construction_Videos/Rocketry_Video_71: Part 10- Finishing and Flying

If you have any questions or comments, feel free to leave a comment in the comments section below.

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