Rocket-Propelled Escape Vehicle

Math That Made the Moon Landing

Dr. Charles Stark Draper
Draper Engineer Bill Robertson
Peter Kachmar
John Houbolt

How Does a Rocketship Fly Through Space?

Pop Quiz: do you remember Newton’s three laws of motion? Flashback to your math teacher scolding you for forgetting your calculator! For most people, we need a little refresher. Like a rocket launch, let’s start with the third one, it’s the most memorable:

3. For every action, there is an equal and opposite re-action. For example, if you punch a wall you'll probably break a knuckle. Sounds familiar right, so do you know the second? Read on!

2. Force is equal to the change in momentum per change in time. For constant mass, force equals mass times acceleration. This one is a bit trickier! For example, if you push a car and a truck with the same amount of force the lighter car will go faster.

And then there’s the first law, which probably also rings a bell:

1. Every object persists in its state of rest or uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it. This is the law of inertia - in space if you hit a golf ball it will travel onward forever, but on Earth, gravity and friction will cause it to stop.  Friction is remarkably absent in space.

Did you pass the test? Two out of three? Good job!

So why bother memorizing these things? The beauty of science is that principles which are proven true can be building blocks for advancing our understanding of the universe. When Sir Isaac Newton wrote his three laws of motion in 1666, it’s hard to imagine he thought about how far they would take humanity. These laws not only allowed fundamental aspects of physics to be built on Earth, but are actually the very same ideas that put the first man on the moon 300 hundred years later. Thankfully the space engineers on the Apollo missions knew them well.

Doc Draper himself used Newton's Laws to develop his first inertial navigation system. His invention of this guidance system is based on the same principles as a spinning top. A rapidly spinning ball resists the force of motion, and can therefore be used as a ‘self contained system’ to measure changes in course and navigate a ship at sea or a rocket in space.

Newton’s laws relate to how a spacecraft travels in space and how the forces enacted upon it (the thrust of its engine boosters and the gravity of the universe) affect it. By understanding how these laws push and pull on an object in space, the engineering team at Draper were able to design a lunar landing concept that relied on the momentum of the Earth and the Moon’s gravitational fields to safely land, rendezvous and return a crew back home. The guidance computers were programmed to navigate between the Earth and Moon using these orbital paths, which abide by Newton's basic laws of motion.

And it's worth mentioning that there were other mathematicians who's equations and theories contributed to navigation and planning of a moon mission including:  1) Kepler's Problem used to calculate the future position of spacecraft using position and velocity. It applies to coasting flight with planetary gravity as the only force acting on the body.  2) Lambert's Problem, used to find the velocity needed to get to a given "target" from the current position and the desired time of flight. The spacecraft and thrust were then oriented in the direction given by Lambert. 3) Kalman, who devised the algorithm which allows the incorporation of measurements from the LM radar or the CM sextant into the current position and velocity of the vehicle making those measurements.

When the first Apollo missions launched, there was little doubt that Newton's basic principles wouldn't still hold true. While each mission was at times dangerous and traveled to worlds unknown, the scientific foundation of how objects move in space never faltered. Thank you Newton!