Thread: How do equations put a man on the moon?

Hi equationists,
I want to know why the equation form is used to put a man on the moon, etc. I mean, what is it about equations that can put a man on the moon,etc. Or perhaps I can ask, is there a form of mathematics that doesn't use equations? I suppose if there isn't then my question would be moot because then equations=mathematics, and that would be like saying, why is a space ship used to take a man to the moon.
So what is an equation trying to do, and how does it get us to the moon,etc.?
Let me grease the wheels: We use a space ship to get to the moon because we are flesh and blood. We use equations to get to the moon,etc. because...
p.s. if it pertains perhaps you can throw in an explanation of what equations are and why they are used. Is it related to solve/dissolve- i.e. a solution where a substance in water is evenly dispersed?

Thank you.

I can't tell what sort of an answer would satisfy you, but the trajectory of the rocket is determined by the equations of gravity. Newton's suffices - you don't need General Relativty.

Just look at the definition of an equation and that should answer your question.

I'm equally confused... if you mean, what equations do we use to figure out the amount of oxygen, the amount of fuel, or the amount of food necessary to put a man on the moon and bring him home, then that may be a question more for chem or bio or physics. All mathematics does is give the ability to express those quantities in whatever relation you want to apply them to. The way I see it, you have equations of chemicals and whatnot to use to figure out how to overcome all of the forces involved.

Originally Posted by LAGoff

Hi equationists,
I want to know why the equation form is used to put a man on the moon, etc. I mean, what is it about equations that can put a man on the moon,etc. Or perhaps I can ask, is there a form of mathematics that doesn't use equations? I suppose if there isn't then my question would be moot because then equations=mathematics, and that would be like saying, why is a space ship used to take a man to the moon.
So what is an equation trying to do, and how does it get us to the moon,etc.?
Let me grease the wheels: We use a space ship to get to the moon because we are flesh and blood. We use equations to get to the moon,etc. because...
p.s. if it pertains perhaps you can throw in an explanation of what equations are and why they are used. Is it related to solve/dissolve- i.e. a solution where a substance in water is evenly dispersed?

Thank you.

In reality, equations per se are not all that important in mathematics. Mathematics is the study of order, and it is formulated in terms of theorems, relatively few of which are stated in terms of equations. Much more common in fact are inequalities. Even when there are useful equations it is the deeper theory and understanding behind the equations that is important.

What is really important is the ability of mathematical models to succinctly describe the behavior of nature and provide accurate predictions. The understandibility of nature is the miracle, according to Einstein, and I think most scientists would agree.

The mathematics of importance in "getting to the moon" was much deeper than just equations. Some key subjects included:

theory of finite element approximations to elasticity

theory of optimal control and calculus of variations

orbital mechanics

thermodynamics of propulsion

Equations don't get us to the moon. People who can understand and apply natural law get us to the moon.

Probably one of the biggest hurdles to learning real mathematics is getting over the impression that comes from high school mathematics that the object of mathematics is to "solve the equation and find the answer". Advanced mathematics is really much more subtle than that and involves understanding and elucidating the logical structure that comes along with studying the order inherent in a subject.

Originally Posted by DrRocket

Probably one of the biggest hurdles to learning real mathematics is getting over the impression that comes from high school mathematics that the object of mathematics is to "solve the equation and find the answer". Advanced mathematics is really much more subtle than that and involves understand and elucidating the logical structure that comes along with studying the order inherent in a subject.

Sometimes I learn something new on this forum. Not often, but it is always worth the wait.

Thank you.

The above replies are excellent but I have a feeling the original poster was looking for something a little more basic.

Let's say you wanted to calculate how much fuel your space ship needed to get to the moon. First you have to get into earth orbit. You know (from Newton's equations) that you have to accelerate your ship to a certain speed to reach orbital velocity. To get to that speed you need a certain acceleration for a certain amount of time. You would use the equation v=at (velocity equals acceleration times time.)

You would need to calculate the acceleration. You would use Newton's equation F=ma, (force equals mass times accleration). The force would be the thrust of the engines and the mass is the mass of your rocket ship plus the fuel it is carrying.

Now that you know the acceleration, you know how much time the rocket has to accelerate to get the the right speed, and you can calculate the time. Then if you know how much fuel the engine burns per second, you would calculate the amount of fuel used. More equations.

This is grossly oversimplified, of course. Dr. Rocket is probably cringing. But I think this might give you some idea of how engineers would use equations.

Originally Posted by Harold14370

The above replies are excellent but I have a feeling the original poster was looking for something a little more basic.

Let's say you wanted to calculate how much fuel your space ship needed to get to the moon. First you have to get into earth orbit. You know (from Newton's equations) that you have to accelerate your ship to a certain speed to reach orbital velocity. To get to that speed you need a certain acceleration for a certain amount of time. You would use the equation v=at (velocity equals acceleration times time.)

You would need to calculate the acceleration. You would use Newton's equation F=ma, (force equals mass times accleration). The force would be the thrust of the engines and the mass is the mass of your rocket ship plus the fuel it is carrying.

Now that you know the acceleration, you know how much time the rocket has to accelerate to get the the right speed, and you can calculate the time. Then if you know how much fuel the engine burns per second, you would calculate the amount of fuel used. More equations.

This is grossly oversimplified, of course. Dr. Rocket is probably cringing. But I think this might give you some idea of how engineers would use equations.

Cringe.

That sort of calculation actually involves some fairly complex mathematics, a lot more than simple equations. But this issue illustrates nicely the point that I was trying to make about mathematics and engineering being about a lot more than equations.

In order to do something like get to the moon or go into orbit, you might want to know what the MINIMUM amount of fuel might be to perform each of several maneuvers.

To get into a spedified orbit what you need is to get to a point on that orbit and then provide precisely the required velocity vector -- both magnitude and direction. The magnitude of that velocity vector tells you something about the energy required at the insertion point, but it tells you little about what it takes to get there.

To get to a point in your orbit you not only have to burn the fuel required to provide the energy to the spacecraft, you also had to burn enough fuel to life the fuel itself during the flight. That is a more complex calculation. So, the question arises as to how to do that burn. Do you burn a lot of fuel all at once, essentially on the launch pad and then coast for a while ? Do you burn it very slowly ?

If gravity were the only consideration, you would burn the fuel very quickly, so that you lift the minimum amount of unburned fuel. But air resistance is also a problem low in the atmosphere, so you need to spread the burn out a bit. Exactly how much to burn and how fast takes a sophisticated computer model of the atmosphere, of the aerodynamics and a sophisticated optimization code as well. It is not simple and it cannot be described by a single equation.

Then you go into orbit, around the earth while you check things out on the spacecraft. But you have been flying essentially straight up in order to minimize time in the atmosphere, so you need to do something to change your velocity direction into something more nearly horizontal, to get an orbit that does not intersect the ground ( an orbit that intersects the ground is called a crash).

So what is the best way to "circularize" your orbit ? The answer to that takes some more sophisticated optimization, in this case applying the methods of optimal control theory. It turns out tha tthe minimum fuel solution is what is called a Hohman transfer, which is an impulsive burn (quickly burn the fuel, essentially all at once) to change the direction of the orbit.

From that point on, in the vacuum of space one proceeds with a series of Hohman transfers. The orbits can be calculated using the method of patched conics, basically Keplerian orbits, derived using the methods of orbital mechanics, another moderately abstract subject in mathematics. And again there you do not have simple equations. There are some equations, but they do not provide directly a relationship between time, position and velocity. So again you need some more sophisticated computer codes to perform the necessary sismulations.

And, believe or not, you DO NOT use F-ma. F=ma is actually wrong. Newton's second law is actually F=dp/dt where p is momentum p=mv. That reduces to F=ma only for the case in which mass is constant, and for rockets mass had better not be constant because you are burning a fuel load that is initially most of the rocket. Also F=ma only applies in an inertial reference frame, basically a frame that is fixed relative to the distant start. For purposes of a moon shot, neither the surface of the earth nor the rocket are adequate for establishing an inertial reference frame. A rocket is clearly an accelerating platform (rockets that don't accelerate are called failures). And the earth is accelerating by revolving around the sun and the surface is accelerating by virtue of the rotationof earth as well. So again you need to use some sophisticated tools to do the calculations.

The point here is that the job of the engineers and scientists is to understand what is going on, and that involves deep understanding of the physics, quie a bit beyond just the equations. It is understanding qualitatively what the dominant effects are that is most important. Only then, and with the aid of sophisticated computer algorithms, do equations enter the picture, and they are not the simple equations that you see in elementary text books. The key to solving those equations numerically lies in understanding approximations, not exact solutions to simple equations. So you eventually find that inequalities are important and the key to solving the real equations.

But the point remains that the actual solvining of equations is a relatively minor aspect of the overall endeavor. The mantra "plug and chug" of many undergraduate engineering students simply will not cut the mustard.

Sort of tangentially touched upon, but not even close to satisfying me. Let me try again: I looked into math books because I wanted to graduate High School or college. I just see equations after equations... My teacher hits me because this is a beast(i.e. mathematical equations) that I can't tame(I was hit by my Algebra teacher in High school). 25 years later after failing college because I couldn't hack math, I want to know what are these equations that "Waterlooed" my career. So I come here to revisit a lost fiend in order to understand the function/root/secret/reason for them. I mean, I just on a lark said that equations could put a man on the moon. I just as easily could have said that they put a man out of college. Try to intuit what I'm saying please! What are equations trying to do? Why are they so useful? Why does equalizing both sides of something do something extraordinary? I'm just trying to get at the root of them so I can understand the root of something that had a great [negative] impact on my life- and so maybe relate to it positively in the future.
I know that you don't have to know this to pass Algebra(or Calculus) but if I could get to their root, perhaps I could feel better for it. I mean 4 is not arithmetic. 2+2=4 is.
What is 2+2=4 trying to say to me(or do in the world). I assume mathematics is just an elaboration of this most simple of equations. I'll leave those things that aren't expressed in equations/equalities to you.

Thank you for your responses
A. Neanderthalis

Equations tell us how things work.

I know I can drive at an average of 60mph on a trip in the UK that is primarily on motorway and dual carriageway.
I want to arrive in Cheltenham by 7.00pm and I am 210 miles away. There fore I need to be on the road and driving by 3.30pm. How do I know that? I used equations.
Required time = 210/60 = 3.5 hrs
Departure time = 7.00 - 3.5

Other equations simply solve different problems?

Is that any nearer to what you are looking for?

Originally Posted by LAGoff

Try to intuit what I'm saying please! What are equations trying to do? Why are they so useful? Why does equalizing both sides of something do something extraordinary?

If I equalize the side of the equation that tells me how much thrust I need over how much time, with the other side of the equation that gives me the amount of fuel, doesn't that tell me how much fuel I need to put in the rocket engine? And wouldn't that help me build a rocket engine to go to the moon?

Originally Posted by LAGoff

What is 2+2=4 trying to say to me(or do in the world). I assume mathematics is just an elaboration of this most simple of equations. A. Neanderthalis

NO !!!!!

Go read my first post again. The whole point is that mathematics is NOT the study of equations. Mathematicians don't go around solving equations.

I'll quote a very early mathematics professor. "Mathematics is the study of any kind of order that the human mind can recognze."

Sometimes in studying that order you find two things that look different and smell different but are really the sam thing --- and THAT is when equations are useful. You start out by noting that two quantities that arise in somewhat ways are really equal, and that fact is used, along with logic, to tell you something more about your problem. Once you have written down an equation, usually all the real work is done and what is left, if anything, is just routine.

I can show you some advanced mathematics texts that are devoid of equations.

Even with physics, in many cases the meaningful work is simply in writing the physics in terms of mathematics. Solving the equations, even difficult equations, is secondary. Most equations that describe real physical situations in detail cannot be solved exactly anyway.

Math is what happens beyond 2 + 2 = 4; beyond fingertips.

The problem with education is that they teach math before they teach you to think. Yet, math requires most exacting, rigorous thinking. Thus, it is very repugnant and foreign to many.

But to those who can, at an early age, make sense of real world, algebraic expression, and geometry; it is a wonderfully simple and expressive language.

Originally Posted by LAGoff

I mean, I just on a lark said that equations could put a man on the moon. I just as easily could have said that they put a man out of college. Try to intuit what I'm saying please! What are equations trying to do? Why are they so useful? Why does equalizing both sides of something do something extraordinary? I'm just trying to get at the root of them so I can understand the root of something that had a great [negative] impact on my life- and so maybe relate to it positively in the future.
I know that you don't have to know this to pass Algebra(or Calculus) but if I could get to their root, perhaps I could feel better for it. I mean 4 is not arithmetic. 2+2=4 is.
What is 2+2=4 trying to say to me(or do in the world). I assume mathematics is just an elaboration of this most simple of equations. I'll leave those things that aren't expressed in equations/equalities to you.

What mathematics does, to begin with (and Dr Rocket, in his first post, provided the best philosophy of mathematics here and nothing I say is intended to contradict any of what he said), is abstract away the actual numbers.

That is, you say:

2 + 2 = 4

And that's fair enough. For a mathematician, however, much more important is the idea of + and =:

Something + Something = Somethingelse

Or, in the more mathematical way:

x + y = z

The idea is to take away the actual numbers, and come up with true statements (actually, models, or axiomatically derived 'proofs') that would hold for any quantities (or even just well-defined ideas) of a particular type.

For instance, for certain groups of ideas

x + y = z

Also implies:

z - y = x

(If this sounds like a jump into equations, a way to make it more concrete is, of course, to replace x,y and z with known quantities, say 2,3 and 5, and confirm this)

And:

z - x = y

This applies, as I said, to certain groups of ideas - not just counting numbers but even, if you will, angles, or clock time (though one must be careful with those and will need additional specifications).

The point is, however, once this 'open' calculation exists (open simply because it applies to any number of actual values and is not limited in its application), it can be used successfully time and time again not only by the engineer:

x + y = z

means that if, with the First Stage I can go x miles above the suface, and the second stage (in space) will take me a further y miles, and the distance to the moon is z miles then, hey, I can get to the moon; but also by the mathematician considering the properties of matrices, or groups or whatever.

It's a tool, and a more universal one than the simple

2 + 2 = 4

- it frees up the + and the = to become a lot more powerful and useful.

I don't know if this helps but...

I think the most important calculation you can make while building a high speed craft is. At what velocity at take off will the craft break your bones. This is a good first calculation.

I noted that how tight your body is up against something means a lot. If you are not tightly packed against the craft before it starts pushing, you could get hurt.

I would suggest doing all the calculations you would like to do, and then do some actual testing on the craft to make sure.

Sincerely,


William McCormick

I think this statement by Dr. Rocket explains it best.

"Sometimes in studying that order you find two things that look different and smell different but are really the sam thing --- and THAT is when equations are useful. You start out by noting that two quantities that arise in somewhat ways are really equal, and that fact is used, along with logic, to tell you something more about your problem. Once you have written down an equation, usually all the real work is done and what is left, if anything, is just routine."

Sorry for not using quotes.

From my personal experience from mathematics and physics, writing the equation down and interpreting is deffinitely more important than actually solving it.

Examples may include something like Schrodinger's Equation where writing down the intial equation and then interpreting what it means afterwards are far more important than doing the steps in between.

So understanding, like Dr. Rocket said, that F=dp/dt, is how equations get us to the moon. But designing the question and then interpreting the solution are where mathematics is most useful. In short the equation is just a means to an ends, and connects the intial phrasing of the question to the eventual interpretation of the answer.

Originally Posted by GenerationE

I think this statement by Dr. Rocket explains it best.

"Sometimes in studying that order you find two things that look different and smell different but are really the sam thing --- and THAT is when equations are useful. You start out by noting that two quantities that arise in somewhat ways are really equal, and that fact is used, along with logic, to tell you something more about your problem. Once you have written down an equation, usually all the real work is done and what is left, if anything, is just routine."

Sorry for not using quotes.

From my personal experience from mathematics and physics, writing the equation down and interpreting is deffinitely more important than actually solving it.

Examples may include something like Schrodinger's Equation where writing down the intial equation and then interpreting what it means afterwards are far more important than doing the steps in between.

So understanding, like Dr. Rocket said, that F=dp/dt, is how equations get us to the moon. But designing the question and then interpreting the solution are where mathematics is most useful. In short the equation is just a means to an ends, and connects the intial phrasing of the question to the eventual interpretation of the answer.

I find you can take and fondle all the equations you want.

But the actual implementing and subsequent test is the only thing that counts. Often the people with the equations do not bet their lives on them. I often can, because I have implemented them.



Sincerely,


William McCormick

Originally Posted by LAGoff

Sort of tangentially touched upon, but not even close to satisfying me. Let me try again: I looked into math books because I wanted to graduate High School or college. I just see equations after equations... My teacher hits me because this is a beast(i.e. mathematical equations) that I can't tame(I was hit by my Algebra teacher in High school). 25 years later after failing college because I couldn't hack math, I want to know what are these equations that "Waterlooed" my career. So I come here to revisit a lost fiend in order to understand the function/root/secret/reason for them. I mean, I just on a lark said that equations could put a man on the moon. I just as easily could have said that they put a man out of college. Try to intuit what I'm saying please! What are equations trying to do? Why are they so useful? Why does equalizing both sides of something do something extraordinary? I'm just trying to get at the root of them so I can understand the root of something that had a great [negative] impact on my life- and so maybe relate to it positively in the future.
I know that you don't have to know this to pass Algebra(or Calculus) but if I could get to their root, perhaps I could feel better for it. I mean 4 is not arithmetic. 2+2=4 is.
What is 2+2=4 trying to say to me(or do in the world). I assume mathematics is just an elaboration of this most simple of equations. I'll leave those things that aren't expressed in equations/equalities to you.

Thank you for your responses
A. Neanderthalis

Dr Rocket put it beautifully and accurately in describing mathematics as the
study of order and relationships, where matters of consistency, completeness,
and reliability (in the real world) are sometimes issues of concern. Have you
ever considered that an equation is simply a formal writing of a 'thought' or a
set of relationships? From simple relationships and ordering to more complex
ones. There is theoretical mathematics where one works within mathematical systems and applied mathematics where one seeks to apply mathematical principles. Early mathematics began in this way using such realities finger counts, the number and appearances in lunar cycles observed over some span of days, and the like. Such systems go back 28,000 years or more! It all begins with
the quest for order, identity, and seeking relationships within the Universe we
ourselves are a part of.

I have a son who has a learning problem which first surfaced in gradeschool,
who also could not make it through college later (forestry). He would not accept
a non-mathematical program he could have coped with much better and would
have made a wonderful social studies teacher, for example, but to this day he remains convinced he can do math in spite of his disability. He found a middle ground where some math was required, but not technical math. He is a govt buyer for the DOD and brilliant at finding things and satisfying technical needs of his customers because of his very strong "social" skills, then at the end of each week he spends hours tallying and accounting (something it only takes others several hours to do). He is proud of his accomplishments and everyone is proud of him.
He is a vital valuable human being in every sense of the word. So maybe you
need to find a middle ground like my son found, suitable to your abilities which
also preserves your soul. Its just a suggestion I make. But I know exactly where
you are coming from because of the experiences in my own family, where my
grandmother for example for a professor of mathematics...

Read Dr. Rocket's posts again. His advice and explanations are brilliant and
compassionate, in my estimation. I hope this helps -

G.

Thanks. I have the soul of a scientist because I am an introvert(and curious); but I don't have the hardware, so being poor at mathematics and logic is a great blow to my self esteem. I mean, math is money, right? They say, "It doesn't COUNT", right? When we think of intelligence, we think of science and logic. It's a great blow to want to think clearly and therefore make a difference in the intellectual climate and find I make a tremendous amount of logical errors and get laughed off the stage. I don't "count" because I cant. O well...

p.s. I didn't understand a thing all of you said about equations, but thanks for trying.

Originally Posted by LAGoff

I mean, math is money, right? They say, "It doesn't COUNT", right? When we think of intelligence, we think of science and logic.
p.s. I didn't understand a thing all of you said about equations, but thanks for trying.

Math is not money. Not by a long shot.

Math is valuable to society. But so are other things.

There are a lot of professions that pay much better than mathematics, science and engineering. Medicine leaps to mind (and yes, medicine is related to but distinct from science). Lawyers commonly make more money than do engineers.

Rock musicians do pretty well and are noted for neither science nor logic.

Artists can do well and receive a lot of recognition. Believe it or not, many if not most research mathematicians consider mathematics to be an art form.

There are lots of things that you might do, even if you eventually find that mathematics and physics are not your particular cup of tea.