I thought it was a little odd that he mentioned "impressive for a high school senior in 1969" multiple times throughout -- honestly I would imagine that growing up in the Space Age would have had a massive influence on technically minded folks, reminds me of that movie from a while back called October Sky.OP here. I see your point. But think of what's needed to create this game:
- From high school physics, you know to start with a free body diagram. There are two forces, gravity and thrust. So far, your average high school student with an A in physics should be able to do that.
- Gravity depends on the distance to the center, which of course is changing, that's the whole point of a lander. I mean, you start 120 miles up. You have to realize it doesn't change much, so can be approximated as a constant. But you've been exposed to that in physics class, so maybe you just assume it's a constant.
- How the hell does thrust work as a function of burn rate? Is the exhaust velocity higher when you burn more? In other words, considering 100 lbs/sec vs 200 lbs/sec, when you double the flow of fuel into the engine, and then you burn it, it turns into twice as much fuel in the same volume. Wouldn't it be forced out at twice the speed? Or at least a higher speed? Maybe you think of the universal gas law, PV=nRT. The volume is constant (the volume of the engine), n is doubled, R is a universal constant. So that means P or T changes, or both. Why is T, which is a function of the velocity of the molecules, constant why P is doubled? Why don't both change?
- So you talk to your Dad, who happens to be a physicist. Most high school students, even those getting an A in physics, don't have a physicist for a father who can look up the properties of rocket engines and find the Tsiolkovsky rocket equation. So a high school senior finding the rocket equation is impressive to me.
- To go from velocity to position, you need to integrate. I'm not sure your average A physics senior would think of replace the FLOG() call with a Taylor series and integrating it term by term.
- How many terms of the Taylor series do you need? Does it even converge for you? If he thought of these subtleties, that's impressive. But it's possible that young Jim didn't realize these issues and just uses 5 terms because that seemed like a lot of terms.
- So now you can simulate it in near the moon. Cool! But how do you detect when it hits the ground? You could try to solve for altitude equals zero, and see if there are zero or more solutions. But even if there are solutions, they might be in the past or the future. So instead you decide to look where the velocity is zero, since you know this happens exactly once during the turn. I think that shows some ingenuity there, although I don't know if that was 18 year old Jim's thought process.
- So you try to invert the rocket equation: given a desired delta-V, how much fuel do you need to burn to achieve it? If you try this with pencil and paper and high school math, including Calculus, you keep getting stuck. You don't have the tools to show that it's actually impossible and needs you to introduce a new function, the Lambert W.
- So maybe you give up, or maybe your physicist Dad helps you again. Using your Taylor series, you now have to solve a 5th degree polynomial. So you decide to scrap the 3rd, 4th and 5th term to get yourself a quadratic. Why is it ok to scrap these now, when it wasn't ok when computing the regular dynamics? I'm impressed that he realized he can use different levels of approximation in different circumstances, without it generating some inconsistencies or other problems.
- You somehow figure out how to use the alternate form of the quadratic equation, which means you didn't just look it up and type it in. Possibly impressive.