Introduction

The Kerbal Math & Physics Lab - Explorations in Kerbal Space Program

by Christopher S. Vaughen

Overview with a list of chapters and sections.

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.

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.

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.

Energy and Escape Velocity, Parametric and Polar Equations in Orbit, Tangent to Flight Path and Pitch.

Work Done by Gravity, Great Circle Distance and Bearing, Optimal Staging with Lagrange Multipliers

Solving the Rocket Equation, Modeling the Flight of a Glider

Answers to questions from chapters 1 through 6.

Astronomy and rocket science activities for intermediate algebra.  Please email cvaughen@mc3.edu for solutions.

• 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. 

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.

Part 1: Tangents to the Flight Path and Pitch

Part 2: Mathematics of Navigation and the Rhumb Line

Using a line integral to calculate work done by gravity.  

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. 

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.

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.

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.

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 

Data recorded for elliptical, circular, parabolic and hyperbolic flights from Kerbin.  Example flight data for Vis-Viva Lab Report in Chapter 2.

Enter r1, r2, and gravitational parameter to calculated delta-v to change from circular orbits at radius r1 to r2.

Enter mean anomaly M, and eccentricity e, to solve for eccentric anomaly E with initial guess M.

Using Excel or Google Sheets, with Euler's method, adjust thrust, initial and final mass, drag coefficient, burn time and step size.

Copied from Wolfram Programming cloud, Mathematica code to solve the rocket equation by Euler's method.