Question concerning math in relation to nature

[sero]

I like to study and read about physics in my spare time. One thing that has always puzled me is how the scientists, particulary physicists and mathematicians can come up with a mathmatical equation for an event in nature.One thing I've tried to study to get a better understanding is the Lorenzt Transformation equation. I don't understand when to use division or multiplication,square roots or some more advanced mathematics like calc plus anything past calc. I took up to calculus in college, but I am still baffled as how it would be used to model or explain a real world event.

1. are there any finite rules for when to use a particular type of mathmatics, or when to use one type with another? for example, how would I know when to use a square root or division ECT ?

 

[pnaj]

Another way to think about this in terms of building models for real life situations. This generally involves trying to relate empirical data (gathered by whatever means) in mathematical relationships that will also predict new data, that can be verified or 'honed' by further experiments. These models may be more or less accurate, but might still serve a purpose, or hint at deeper relationships, etc.

I think the simplest example is proportionality.

For an obvious example just to show the process, you might suppose that the time taken to complete a journey is related to the distance travelled, but what is the relationship?

Obviously, we start with the simplest journey - a straight line - and do experiments. We notice that, for the results to mean anything, we try to 'control' all factors that might affect the experiment, other than the two we are wanting to relate. That means, for example, keeping the motion constant, staying on a flat plane, etc. for all the trials.

After gathering together the data, we notice, empirically, that the distance traveled is equal to some constant number * the time taken.
So, d = k * t , where k is this constant that appeared in the data and ee can now predict how far we'll go for any particular time.
In physical terms, this states that t is directly proportional to d, and k is the constant of proportionality.

To go on with this ...

Someone might then ask what this k actually represents, in the real world, if anything. If we isolate K in the equation, we get k = d/t ... distance over time ... which is what we call (average) speed!


But, in general, this is only really useful over short distances and times, because most things don't even approximate to straight lines for very long.

Calculus was a major step forward in the calculation of motion ... you should really have a crack at it if you want to understand anything to do with motion.

Tell us what you do know about calculus and maybe we can push you on a bit.

 

[Integral]

The key to modeling the real world world is to think about what is changing and how it changes.

For example you can say that the change in a population is porptional to the number of individuals present at any time

$$\frac {\Delta p} {\Delta t} = kp$$

This leads to the standard simple population model.

The goal of modeling is to come up with a DIFFERENTIAL equation. To do this you need a good understanding of the physics which govern you system. If you have that understanding is is ofter pretty straight forward to express the differential equation. Once you have the differential equation you must be able to solve it.

 

[HallsofIvy]

I really wondering if a person who starts off by saying "I don't understand when to use division or multiplication" is going to understand any of this. "When to use division and multiplication", i.e. what division and multiplication mean is one of the things you are supposed to learn in elementary school.

 

[sero]

what division and multiplication mean is one of the things you are supposed to learn in elementary school

obviously, I guess I need to be more clear. I'm looking for some general rules of when to use these types when trying to take a natual event and express it mathmatically. I do know what each mean, did learn them in elementary school, but I wanted to ask in a setting where people that use what I am asking about would understand the question and be able to anwser it so that it would help me get a better feel for it. not just division and multiplication, but more complex things like square roots.

I hope that makes it more clear so that I can get more constructive anwsers then the previous one.

thanks in advance

 

[russ_watters]

Originally posted by Integral
The key to modeling the real world world is to think about what is changing and how it changes.

At its most basic, this is the human ability for pattern recognition. Do an experiment, gather some data, and attach a pattern to the results.

The simplest examples come from Newtonian physics. There were some long held beliefs about motion that no one until Galileo did much experimenting on. For example, most people before Galileo believed heavier objects fall faster than lighter objects. A simple demonstration proved that wrong (Tycho Brahe, I think, used to demonstrate this by dropping food on the floor at parties). That's just one piece of the puzzle of course, but an important one - Newton is the one who put all the pieces together, recognized all the patterns, and formulated the math to describe it.

I guess I need to be more clear. I'm looking for some general rules of when to use these types when trying to take a natual event and express it mathmatically.

Sorry, that still isn't very clear: what do you mean by "when?" What operations you use are whatever operations that allow you to find the pattern in the data. Often you do this by plain, ordinary trial and error.

example:
1:10
2:20
3:30
4:40
5:50
See the pattern? It should be obvious: every number in the second colum is 10x the number in the first. In equation form, v=10t. Guess what - that's acceleration (gravitational, to be specific)!

How about this:
1:5
2:20
3:45
4:80
5:125
See the pattern? This one is tougher. Knowing the first example though may help find the pattern in this one.

Now Newton probably just derived the second one from the first, nevertheless having real data to check is essential.

This is also why math and physics are so closely related.

 

[PrudensOptimus]

Originally posted by sero
obviously, I guess I need to be more clear. I'm looking for some general rules of when to use these types when trying to take a natual event and express it mathmatically. I do know what each mean, did learn them in elementary school, but I wanted to ask in a setting where people that use what I am asking about would understand the question and be able to anwser it so that it would help me get a better feel for it. not just division and multiplication, but more complex things like square roots.

I hope that makes it more clear so that I can get more constructive anwsers then the previous one.

thanks in advance

go do 100 word problems then post and ask for advice.