Almost certainly the equation your friend saw is wrong. It's easy to mislead people using mathematics because not everybody who reads equations understands what they intend to represent. Math notation is a poor way of conveying concepts to anyone except mathematicians and related geeks. So if someone expresses an idea as an equation, fewer people question it. Equations look like they should make sense and that their authors are brainy for having written them, even if they're complete hogwash.
Skim this article.
www.clavius.org/bibvoron.htmlI wrote it for a similar situation.
Typically the "equations" in this kind of situation look at the engineering values for one kind of rocket (say, a Boeing Delta II) and try to express some purported simple relationship between them and a rocket of a different type based on some simple fact about the moon versus Earth. For example, "If a Delta II rocket is 200 feet tall, then a rocket launched from the moon should be 33 feet tall because that's 1/6 of 200 and the moon has 1/6 the gravity." It doesn't take much ingenuity to see why that statement is false, but you can use the same poor logic to make wrong statements that are harder to detect.
Here's some rocket science without math.
If I take 1 pound of rocket fuel and light it off, it will produce a certain force. Different kinds of fuel produce different amounts of force, but don't worry about that. All you have to know is that a pound of fuel produces a certain amount of thrust.
You remember from basic physics that acceleration depends not just on the force (thrust) but on the mass of what you're pushing. So heavier objects require more thrust to get them going a certain speed than lighter objects.
Here's where life starts to suck.
A rocket is the structure and the payload, but it's also the fuel. If you burn a pound of fuel, that rocket is now a pound lighter. Which means the pound of fuel you burn when tank is mostly full gives you less speed (acceleration, really) than the pound you burn when the tank is nearly empty and your ship is lighter.
One of the key concepts in rocketry is the "mass ratio", which is the ratio of a rocket's mass fully loaded to its empty mass. So if your car weighs 2,000 pounds with its gas tank empty, and 2,200 pounds with a full tank, its mass ratio (MR, or sometimes the Greek letter lambda) is 1.1. The lunar module had a mass ratio of between 2 and 3, and an Earth launch rocket has a mass ratio around 10 or 20. That much fuel is needed because the Earth rocket has to push against greater weight, but also against thick air. You don't need that on the moon.
The notion that a rocket gets easier to push as it expends its fuel is embodied in the algebra of the natural logarithm. The ability of a spaceship to change its velocity is proportional to the natural logarithm of MR. And this holds true for any point in the ship's mission.
If you fill your car up with gas at the beginning of the week (MR = 1.1) and you burn 100 pounds of gas by Thursday so that MR = 1.05, you can re-run the equations with MR=1.05 and get how much speed-changing capacity you have left with that half a tank of gas. It's kinda magic that way.
The rate at which you burn the fuel is also an important value in rocketry, called the "mass flow rate". The symbol for it is an m with a dot over it ("m-dot"), which in physics means the rate at which that quantity changes over time. If your car burns 100 pounds of fuel in 3 days (259,000 seconds) then your mass flow rate (m-dot) is 0.000386 pounds/second. Cars that burn fuel faster might accelerate faster. The same with rockets. If you burn your fuel at a faster rate, you don't get any more
velocity out of that, but you get to that velocity in a shorter amount of time.
And another important number in rocketry is the speed of the exhaust. Remember that rockets work by Newton's third law -- equal and opposite reaction. By "equal" Newton means the conservation of momentum, which is the physics expression for the combination of velocity and mass. It goes back to heavier (actually "more massive" is the proper term) objects being harder to start and stop moving.
You burn 1 pound of fuel in 1 second. So you get a certain amount of thrust applied during that 1 second. But what really happens is that your 1 pound of fuel turns into 1 pound of smoke, flame, and hot gas rushing out the back of the rocket at a certain speed. The faster it goes, the more momentum it has, and the more the rocket has to go in the other direction.
Now back to the 1 pound of fuel. Obviously some fuels are more powerful than others. The term for the strength of fuel is "specific impulse", which describes how much impulse (a force acting for a certain time) you get out of each pound of fuel.
Now physicists, who are whizzes at forces and masses and quantities changing over time and with variable masses and stuff, put these together into the standard rocketry equations. These equations generally bear no relationship to the weird stuff the conspiracy theorists come up with. You can see that many factors apply to whether a rocket will work or not, and the conspiracists generally want to make it very simple to fit their limited understanding: stuff like, "It should have six times the quantity of fuel because the Earth has six times more gravity."
The goal of a rocket in most cases is to put something in orbit. That means (a) getting to a certain altitude, and (b) moving it forward at orbital speed. The orbital speed is determined by how high you want to orbit and the gravity characteristics of the planet in question. But those speeds and altitudes aren't just a matter of a certain amount of Earth gravity divided by a certain amount of moon gravity. Again, it's just not that simple.
If you want to orbit the Earth at an altitude of 200 miles, you have to get your ship 200 miles above the surface moving downrange (not up) at a certain speed that orbital mechanics equations tell you will produce the balancing centrifugal force to Earth's gravity. But if you want to orbit the moon at 60 miles, you can't just take that speed figure for Earth and divide it by a simple number to get your moon orbit velocity. You have to go back to first principles and compute the approprate speed for that altitude and that amount of gravity. Then you can see what kind of propulsion you need to get you to that point and moving that fast in each situation.
And in my article linked above I talk about the other factors for comparing Earth and moon launches, which most conspiracy theorists completely ignore.
Needless to say, people who actually know how to do rocketry equations and know how rockets work can run the numbers for the lunar module and confirm that NASA's claims for how the LM worked are perfectly consistent with what we know would be required for such a ship. There's nothing physically wrong with NASA's claims, but there's almost always something physical wrong with the conspiracy theorists' attempts at rocket science.
