The very basis of rocket propulsion, and in turn solid rocketry, relies heavily on various physical theories and mathematical formulas. It is what allows the rocket to launch off of the ground and maintain a high velocity for a prolonged period of time.
Understanding the basics behind these concepts allows us to adapt our rockets to the highest efficiency and ensure their safety and stability.

 

Newton's Third Law

Newton's Third Law dictates every action has an opposite and equal reaction. In other terms, if Object A exerts a force on Object B, Object B will exert the same amount of force back onto Object A. 

Rocket propulsion utilizes this concept to boost the rocket upwards-- by pushing itself against the ground with highly pressurized gas (or water, in terms of liquid propulsion), the ground, in turn, pushes against the rocket with an equal and opposite force, thus lifting it off the ground. Using this law, we are able to decipher two important factors when launching: the mass of the rocket and the force it applies on the ground-- for the initial launch to occur, the propulsion system must be capable of creating a downward force greater than the rocket's mass multiplied by the gravitational acceleration (9.81 m/s).  [ Frocket > m*g ]


 

Conservation of Momentum

Conservation of Momentum notes that the total momentum of a system always remains constant. The equation associated with this concept is:

mAvA + mBvB = mAvA' + mBvB'

The left-hand side describes the initial momentum of two objects, [ Note: momentum(p) = mass(m) * velocity(v) ] and the right-hand side describes the final momentum.

 

Applying this to rocket propulsion, visualize the two objects being the rocket and the propellant inside the rocket. When the propellant burns, it becomes tiny particles of pressurized gas shot out of the back of the rocket.

If the rocket is initially at rest, when this occurs, the rocket's momentum will compensate for that downward velocity by pushing itself upwards as a result, allowing the rocket to maintain flight for a longer period of time. To improve the rocket's trajectory, we must consider the velocity of the gas as it exits the rocket and the amount of propellant in the rocket.

 

 

 

 

 

Components of a Solid Propulsion System

Taking these concepts into account, the solid propulsion system must address and optimize its features to these conditions. As a result, we have this:

A.   Forward Closure - caps the motor to prevent the propellant from exiting the wrong end, into the rest of the rocket
B.   Case - encases the motor, closing it off from the rest of the rocket
C.   Liner - maintains the shape of the propellant
D.   Propellant - fuels the motor and burns to become pressurized gas
E.   Nozzle - adjusts the efficiency of the rocket 

 

The nozzle, in particular, is a key part of this system. Though fully functional without it, the rocket would have a dramatic drop in efficiency without this piece, as it is what adjusts the pressure of the gas-- improving the thrust and speed of the rocket without needing to heavily alter the propellant or any other features.

 

There are two parts of the nozzle to take note of: the converging section (A.) known as the nozzle throat and the diverging section (B.)  known as the nozzle divergence.

A. The nozzle throat chokes the flow of gas through the nozzle, thus pressurizing the exhaust. Due to this high pressure, the velocity of the motor is greatly increased.

 B. The nozzle divergence generates thrust by expanding the exhaust to ambient pressure. This subsection controls the shape and angle of the gas as it exits, preventing it from shooting out in all directions and constricting its range

 

Without this piece, all the gas created by the propellant would exit the motor without much pressure, and would not generate enough force to support the rocket's full weight and propel it upwards.

 

 

Calculating Thrust

Thrust is the rocket's propulsive force, and is what allows it to launch itself off of the ground. To calculate thrust we use this formula:

T = Cf (t) * At * P0(t)

  • T = Thrust
  • Cf (t) = Thrust Coefficient (approximately constant)
  • At = Throat Area (m²)
  • P0(t) = Chamber Pressure (Pa)

The thrust coefficient stays relatively constant and the throat area may be measured however, the chamber pressure, the force the walls of the rocket exert onto the space within it, is still unknown. Therefore, we must use another formula:

P(t) = (Kn * c* * ρp * a)1/(1 - n)

  • P(t) = Chamber Pressure (Pa)
  • Kn = Propellant Burn Area to Throat Area Ratio (AB/At)
  • c* = Characteristic Velocity (found experimentally)
  • ρp = Propellant Density 
  • a = Propellant Ballistic Coefficient
  • 1/(1 - n) = Propellant Ballistic Exponent

In particular,  (c*ρpa)1/(1 - n)  becomes approximately constant while Kn varies with time, thus making calculations much simpler.

Essentially, Kn affects Chamber Pressure which then affects Thrust.

 

*Characteristic velocity may also be found through this formula:

All variables are the same and mp is the mass of the propellant.

 

 

Specific Impulse

Specific impulse determines the impulse per weight of propellant. It may also be found to determine the efficiency of a rocket, as shown here:

Isp = Cf * c*/g

Isp = Specific Impulse

Cf = Thrust Coefficient

c* = Characteristic Velocity

g = Gravitational Acceleration (9.81 m/s)

 

 

 

These fundamentals form the basis of Solid Rocketry. It is important that you refer back to these formulas when designing parts or formulating propellants in the future to ensure the safety of your team and the success of your project.

 

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