SHAWN VICTOR
  • Home
  • Rocketry Projects
    • Horizontal Test Stand
    • Project Quasar
    • COPV Burst Stand
    • Custom Flight Computer MkII
    • Experimental Air Braking
    • Solid Rocket Flight Computer
    • Syncope
  • Personal Projects
    • Persistence of View Globe
    • Hexapod
    • RTOS Race Car
    • OpenBevo
  • Makerstudio Trainings
    • Autodesk Eagle
  • Tutorials
    • Github
    • Electronics Fundamentals >
      • Electricity from an Atomic Perspective
      • Resistor Circuit Analysis
    • Custom Rocket Engines >
      • Injector Orifice Sizing
      • How Rocket Engines Work
      • Choosing Your Propellant
      • Dimensioning Your Rocket
    • DIY Hybrid Rocket Engine >
      • L1: The Basics
    • Semiconductors >
      • L1: Charge Carriers and Doping
      • L2: Diodes
    • Rocket Propulsion >
      • L1: Introduction
      • L2: Motion in Space
      • L3: Orbital Requirements
      • L4: The Rocket Equation
      • L5: Propulsion Efficiency
    • Government 1 >
      • L1: The Spirit of American Politics
      • L2: The Ideas That Shape America
      • L3: The Constitution
    • Government 2 >
      • C1: The International System
      • C2: US Foregin Policy Apparatus and National Interest
      • C3: Grand Strategy I
      • C4: Grand Strategy II
      • C5: The President and Foreign policy
      • C6: Congress in Foreign Policy
    • Control Feedback Mechanisms >
      • L1: Intro to Control Systems
      • L2: Mathematical Modeling of Control Systems
      • C3: Modeling Mechanical and Electrical Systems
    • Electromechanical Systems >
      • L1: Error Analysis and Statistical Spread of Data
    • Rocket Avionics Sourcing

A Recap of the Science and Math

Calculus
Lots of people dread the idea that calculus concepts will be used in hybrid design, but most of the calculus will really just be done by calculators. However, understanding how we will be using calculus to analyse the trajectory of rockets will be our main focus. Let's first start with Differentiation. In high school physics they taught us that acceleration was represented this way:
a = ΔV / Δt     
where the Greek letter Δ represents  "a finite change in". (ΔV happens a lot in rocketry)

No one in high school every really bothered to tell you that this is just a special base of differential calculus, the specialization being that this acceleration is constant over the measured time interval (meaning that for this to be correct,  the velocity would need to be linearly changing). So if the velocity-versus-time graph were to be some exponential graph, then this method of calculating the acceleration would be invalid. If this was the case that the acceleration wasn't constant due to the velocity changing non-linearly over time, then we might rather be interested in the instantaneous acceleration at a particular time. We get this by using a infinitesimally small time interval for our Δt (but not exactly zero). Thus, in differential calculus we use the following equation:
a = dV / d​t       or         a = d/ d​t (v)
 where d signifies a "very small change in". In Physics we would read this as "the time rate of change of velocity is equal to acceleration". One thing to note, is that the notation of d/dt was invented by a mathematician named Leibniz, but Newton would use a notation known as flexion notation where to denote the same thing he would put a dot above the variable. In rocketry we use both notations. 

Newton also needed to create a reverse process to Differentiation known as Integration, in this way we could use the acceleration to calculate the change in velocity. Again, high school physics gave us a specialized equation that assumed acceleration was constant over the time interval.
Picture
Picture
The curly line denotes integration. In Physics we would read this as "Velocity is the time integral of time". If you are looking for an instantaneous velocity value you can use the equation on the left which to find the equation for velocity based on time, then plug in the time value. If on the other hand you are trying to calculate the change in velocity over a specified period of time, you can use the bounded integration equation as shown on the right, specifying the start time and end time. 
The value of this bounded integral is equal to the area under the curve within the bounded interval. We can approximate this value by using the Riemann sum equations. 
Forces
sadasm
Weight and Mass
sadasm
The Rocket Principle
sadasm
Newton's Laws
sadasm
The Kinetic Theory of Gasses
sadasm
Temperature
sadasm
The Mole
sadasm
The Ideal Gas Law: Pressure, Temperature, and Density
sadasm
Mass Continuity and Mass Flux
sadasm
Energy
sadasm
Mixture
sadasm
Types of Rocket Engines
sadasm
The Atmosphere
sadasm
A Suborbital Trajectory
sadasm

  • Home
  • Rocketry Projects
    • Horizontal Test Stand
    • Project Quasar
    • COPV Burst Stand
    • Custom Flight Computer MkII
    • Experimental Air Braking
    • Solid Rocket Flight Computer
    • Syncope
  • Personal Projects
    • Persistence of View Globe
    • Hexapod
    • RTOS Race Car
    • OpenBevo
  • Makerstudio Trainings
    • Autodesk Eagle
  • Tutorials
    • Github
    • Electronics Fundamentals >
      • Electricity from an Atomic Perspective
      • Resistor Circuit Analysis
    • Custom Rocket Engines >
      • Injector Orifice Sizing
      • How Rocket Engines Work
      • Choosing Your Propellant
      • Dimensioning Your Rocket
    • DIY Hybrid Rocket Engine >
      • L1: The Basics
    • Semiconductors >
      • L1: Charge Carriers and Doping
      • L2: Diodes
    • Rocket Propulsion >
      • L1: Introduction
      • L2: Motion in Space
      • L3: Orbital Requirements
      • L4: The Rocket Equation
      • L5: Propulsion Efficiency
    • Government 1 >
      • L1: The Spirit of American Politics
      • L2: The Ideas That Shape America
      • L3: The Constitution
    • Government 2 >
      • C1: The International System
      • C2: US Foregin Policy Apparatus and National Interest
      • C3: Grand Strategy I
      • C4: Grand Strategy II
      • C5: The President and Foreign policy
      • C6: Congress in Foreign Policy
    • Control Feedback Mechanisms >
      • L1: Intro to Control Systems
      • L2: Mathematical Modeling of Control Systems
      • C3: Modeling Mechanical and Electrical Systems
    • Electromechanical Systems >
      • L1: Error Analysis and Statistical Spread of Data
    • Rocket Avionics Sourcing