A lab workbook for math and physics students on topics in rocket science and orbital mechanics from algebra through differential equations inspired by the computer game Kerbal Space Program.
Written by Christopher S. Vaughen
Embedded Files
Introduction
Introduction
The Kerbal Math & Physics Lab - Explorations in Kerbal Space Program
by Christopher S. Vaughen
Intro_and_Contents.pdfIntroduction and Contents
Introduction and Contents
Overview with a list of chapters and sections.
CH1_Algebra.pdfChapter 1: Algebra
Chapter 1: Algebra
Gravity and Acceleration, Thrust-to-Weight Ratio, The Acceleration and g-Force Lab Part 1: The Thumper Launch in KSP, Orbital Speed, Orbital Period, Computing the Mass of a Planet or Moon, Kinetic and Potential Energy in a Circular Orbit.
CH2_PreCalc_Trig.pdfChapter 2: Pre-Calculus & Trigonometry
Chapter 2: Pre-Calculus & Trigonometry
Linear and Angular Speed, Great Circle Navigation, The Vis-Viva Lab, Delta-v for a Hohmann Transfer, A Trip to the Moon, The Rocket Equation, Kepler's Third Law of Planetary Motion, Orbital Inclination.
CH3_Calc1.pdfChapter 3: Single Variable Calculus
Chapter 3: Single Variable Calculus
Maximum Aerodynamic Pressure, Acceleration and g-Force Lab Part 2: A SpaceX Falcon 9 Launch, Differential in Velocity, Solving Kepler's Equation with Newton's Method, Distance at Engine Cut-Off.
CH4_Calc2.pdfChapter 4: Calculus with Parametric & Polar Equations
Chapter 4: Calculus with Parametric & Polar Equations
Energy and Escape Velocity, Parametric and Polar Equations in Orbit, Tangent to Flight Path and Pitch.
CH5_Calc3.pdfChapter 5: Multi-variable & Vector Calculus
Chapter 5: Multi-variable & Vector Calculus
Work Done by Gravity, Great Circle Distance and Bearing, Optimal Staging with Lagrange Multipliers
CH6_DE.pdfChapter 6: Differential Equations
Chapter 6: Differential Equations
Solving the Rocket Equation, Modeling the Flight of a Glider
KSP_lab_answers.pdfSolutions
Solutions
Answers to questions from chapters 1 through 6.
Rocket Science and Astronomy Labs for Intermediate Algebra - no answers.pdfExtra: Lab Activities for Intermediate Algebra
Extra: Lab Activities for Intermediate Algebra
Astronomy and rocket science activities for intermediate algebra. Please email cvaughen@mc3.edu for solutions.
Videos
Videos
- SIMIODE presentation, February 2022: https://youtu.be/JRNuNtETuHw
- Teaching Math & Physics with KSP, June 2021: https://youtu.be/GC1jZ3qIVdQ
- Teaching Math & Physics with KSP, Fall 2020: https://youtu.be/WTVmKVEhb3s
- ICTCM 2020 Conference sponsored by Pearson, March 2020 https://youtu.be/rO8v6YvvbVw
Video playlist on calculating initial bearing for great circle path on any spherical planet or moon. Part 1 applies the formula, part 2 is the derivation.
Solving the rocket equation with Euler's method. Derivation of the rocket equation.
Using Kerbal Space Program as a virtual lab to verify conservation of energy and angular momentum.
The Vis-Viva Lab - with Real Life Data!
The Vis-Viva Lab - with Real Life Data!
A study of conservation of energy and angular momentum. Examples include the International Space Station, the near-Earth asteroid Apophis, the Tesla Roadster and Oumuamua.
A derivation of the formula for escape velocity using integration, examples include KSP, Mars 2020 and the Apollo 11 flight to the Moon.
Discussion of mean, eccentric and true anomaly, examples include calculating position of the Tesla Roadster launched by SpaceX.
Investigation with Geogebra, Desmos and Kerbal Space Program of the relation between orbit inclination, launch azimuth and launch site latitude.
Parametric and polar equations for an elliptical orbit, delta-v for translunar injection, verifying Kepler's law of areas and computing the mass of the moon.
Using a line integral to calculate work done by gravity.
Optimal Staging with Lagrange Multipliers
Optimal Staging with Lagrange Multipliers
Derivation of formulas to determine theoretical optimal mass for each stage of a multi-stage rocket. Examples comparing theoretical model with Falcon 9, Starship, Saturn V and a model built in Kerbal Space Program.
Derivation of a system of differential equations to model the flight of a glider. The system of equations is solved with Mathematica using Wolfram Programming Lab and compared with flights in KSP.
Modeling the Flight Dynamics of Gliders
Modeling the Flight Dynamics of Gliders
Derivation and numerical solution of the differential equations that model the flight of a glider. Examples with the computer game Geo-FS, Microsoft Flight Simulator and Kerbal Space Program.
Euler's Method and the Rocket Equation
Euler's Method and the Rocket Equation
Inspired by the movie Hidden Figures, this video provides a derivation of the rocket equation and an analysis of the solution using Euler's method.
YouTube playlist by Mike Aben. Includes thrust-to-weight ratios, deriving and applying the rocket equation, deriving and applying the vis-viva equation, calculating delta-v budgets, calculating delta-v costs, building relay networks, determining signal strength, understanding and using the Oberth effect, calculating time in dark while in orbit, determining electricity needs and more.
Desmos and GeoGebra Interactive Graphs
Desmos and GeoGebra Interactive Graphs
Graph of Eccentric, Mean and True Anomaly.
Desmos interactive illustrates relationship between launch azimuth, latitude and orbit inclination
GeoGebra interactive illustrates relationship between launch azimuth, latitude and orbit inclination.
Adjust speed, graph elliptical or hyperbolic trajectories, displays eccentricity, specific energy and equations in rectangular and polar coordinates. Graphed in GeoGebra. This version assumes initial 100 km parking orbit around Kerbin.
Adjust semi-major axis and eccentricity for an elliptical orbit, graph in polar and rectangular form. Graph in Desmos.
Geogebra interactive with real numbers to scale for Earth and Moon transfer orbit. Shows velocity, energy, angular momentum, and altitude.
Edit semi-major axis, visualize orbit, velocity slope and true anomaly. Graph in GeoGebra.
Graph of the Kerbin/Mun System with spheres of influence for Kerbin and the Mun. Adjust semi-major axis and eccentricity of trans-munar orbit, in Desmos.
Compare areas swept out during elliptical orbit on two intervals of the true anomaly.
Plot of position and velocity vectors in an elliptical orbit with flight path angle and angle between position and velocity
Plot of flight path angle during elliptical orbit, with eccentricity as parameter c.
Log/Log plot of semi-major axis and period for planets in the Kerbal System.
Excel and Mathematica Files
Excel and Mathematica Files
heading_distance_calculator.xlsxGreat Circle and Rhumb Line Bearing and Distance Calculators
Great Circle and Rhumb Line Bearing and Distance Calculators
Input point A and B latitude and longitude, and planet/moon radius to find bearing and distance.
Rhumb Line calculator is here: RL Heading Calculator
vis_viva_flight_report.xlsxExample Vis-Viva Flight Report Data
Example Vis-Viva Flight Report Data
Data recorded for elliptical, circular, parabolic and hyperbolic flights from Kerbin. Example flight data for Vis-Viva Lab Report in Chapter 2.
hohmann_calculator.xlsxCalculate Delta-v for Hohmann Transfer
Calculate Delta-v for Hohmann Transfer
Enter r1, r2, and gravitational parameter to calculated delta-v to change from circular orbits at radius r1 to r2.
Newtons_method_calculator.xlsxSolving Kepler's Equation with Newton's Method
Solving Kepler's Equation with Newton's Method
Enter mean anomaly M, and eccentricity e, to solve for eccentric anomaly E with initial guess M.
eulers_method_launch_nonlinear_drag.xlsxSolving the Rocket Equation with Euler's Method (Excel)
Solving the Rocket Equation with Euler's Method (Excel)
Using Excel or Google Sheets, with Euler's method, adjust thrust, initial and final mass, drag coefficient, burn time and step size.
KSP_solving_rocket_equation.pdfSolving the Rocket Equation (Mathematica code)
Solving the Rocket Equation (Mathematica code)
Copied from Wolfram Programming cloud, Mathematica code to solve the rocket equation by Euler's method.
Additional Resources
Additional Resources
KSP Beginner's Series - by Mike Aben
KSP Beginner's Series - by Mike Aben
Produced in 2020
KSP Beginner's Series - by Mike Aben
KSP Beginner's Series - by Mike Aben
Produced in 2016-2017
includes segments from "Let's Do the Math" series
Questions or Comments? Please contact me at cvaughen@mc3.edu
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