Reason Behind Structure of Equations

[prosteve037]

First post!

Hey people! I hope this isn't a topic already posted before. I already used the search function but I couldn't find the question I was looking for :/

Anyways, I was curious as to why equations in physics are set up the way they are.

Let me show you what I mean:

KE = ½mv²

In the kinetic energy equation the mass and velocity squared are multiplied together. Why are they multiplied and not added? Is there a specific reason why?

Now I was thinking proportionality had something to do with this since in some other equations directly proportional factors are multiplied (ie. the masses in Newton's Gravity Equation: F = G (m1m2)/r²)

Also as a side question, what's the reasoning behind the ½ and v²? And I don't mean just the math, what's the history behind it?

Sorry if this is a lot to ask at once. I was just really curious. Hopefully this topic will keep me from posting a new thread for a while :)

 

[tiny-tim]

welcome to pf!

hi prosteve037! welcome to pf!

1/2 mv² comes directly from good ol' Newton's second law (and the chain rule ) …

F = d(mv)/dt = d(mv)/dx dx/dt = v d(mv)/dx = d(mv²/2)/dx …

from which we get the work energy theorem:

F.dx = mv²/2, or work done = change in KE ​

 

[pergradus]

Well it describes a physical connectivity. It's not simply that the kinetic energy can depend on the mass or the velocity, but it depends on both and is proportional to both.

Take the special case of v = 0. If the kinetic energy were something like KE = m + v² then it would be completely independent of velocity at v = 0, but this just isn't true.

And as tiny-tim said, the exact form of the equation is derived from the basic laws of motion.

 

[dlgoff]

Note: When in doubt, look at the units you'll get after the math operations to see if that makes sense.

Check out this from here:

http://hyperphysics.phy-astr.gsu.edu/hbase/units.html"

 

[prosteve037]

Well it describes a physical connectivity. It's not simply that the kinetic energy can depend on the mass or the velocity, but it depends on both and is proportional to both.

Take the special case of v = 0. If the kinetic energy were something like KE = m + v² then it would be completely independent of velocity at v = 0, but this just isn't true.

And as tiny-tim said, the exact form of the equation is derived from the basic laws of motion.

Oh okay I see. Thanks much pergradus! :)

Is there a mathematical term for this "joint interdependence" of variables in an equation? (Like how mass and velocity must both exist in the kinetic energy equation)

In other words, a term for when you need to multiply two variables instead of adding them.

 

[LostConjugate]

Oh okay I see. Thanks much pergradus! :)

Is there a mathematical term for this "joint interdependence" of variables in an equation? (Like how mass and velocity must both exist in the kinetic energy equation)

In other words, a term for when you need to multiply two variables instead of adding them.

They are multiplied because the energy is per unit mass. If you have 5 units of mass you would not want to add the number 5 to your velocity squared. You would want to take your velocity squared and multiply it by 5. Since you have 5 units of mass traveling at your velocity squared your energy is 5 times as much.

The term would just be the product of your kinetic energy and your mass energy. I think the word "product" is what your looking for.

 

[sophiecentaur]

In olden days, there seemed to be much more 'verbal' description of the way things work. In the end, though, when the mechanisms involve complicated relationships, you need Maths just to state these relationships in a concise and accurate way.
There is, however, a weirdness to Maths in that, in itself, it's an abstract thing, full of axioms and abstract rules yet it actually tells us about how the Universe works.
Was Maths there before we 'found it' or did we invent it`?

 

[LostConjugate]

In olden days, there seemed to be much more 'verbal' description of the way things work. In the end, though, when the mechanisms involve complicated relationships, you need Maths just to state these relationships in a concise and accurate way.
There is, however, a weirdness to Maths in that, in itself, it's an abstract thing, full of axioms and abstract rules yet it actually tells us about how the Universe works.
Was Maths there before we 'found it' or did we invent it`?

I would expect that if the universe spoke, it would be in symbols and not in a language made of of words that attempt to generally describe something and have multiple meanings.

 

[sophiecentaur]

I agree. It is humans that are the basic problem. Our minds and consciousness naturally work in an 'organic' way, based on loose personal associations of memories and patterns. To communicate these in a reliable way, we need to have a disciplined common language.
The perception, description and naming of colours is a good example of this. It's all subjective and in our heads but there is enough in common between individuals for us to be able to communicate what one person has seen to another person. The three parameters used in colour reproduction are a sort of common, numerical, language which we can use.

This is where the minimalism of Maths comes in so handy. It seems to be that over a limited scale (not too macroscopic and not too microscopic) some relatively simple Mathematical relationships hold and can be used to communicate and reason about the Science around us and even to make good predictions. This is lucky for us, I guess.

 

[LostConjugate]

I agree. It is humans that are the basic problem. Our minds and consciousness naturally work in an 'organic' way, based on loose personal associations of memories and patterns. To communicate these in a reliable way, we need to have a disciplined common language.
The perception, description and naming of colours is a good example of this. It's all subjective and in our heads but there is enough in common between individuals for us to be able to communicate what one person has seen to another person. The three parameters used in colour reproduction are a sort of common, numerical, language which we can use.

This is where the minimalism of Maths comes in so handy. It seems to be that over a limited scale (not too macroscopic and not too microscopic) some relatively simple Mathematical relationships hold and can be used to communicate and reason about the Science around us and even to make good predictions. This is lucky for us, I guess.

Lucky would be if it did not take us 5 million years to develop. I think we worked hard for it. Cheers

 

[CronoSpark]

I also wondered about these relationships and tried to imagine/make sense of it all. But then again, you guys could be right in that humans are inherently unable to make sense of it without math.

I mean, I can sort of understand mass*velocity is momentum, but once you get into velocity squared, things get a bit complicated to comprehend! How can one visualize "velocity squared" being related with motion energy?

 

[sophiecentaur]

The notion of 'squared' works for Pythagoras. As well as saying :
c² = a² + b²,
you can say that :
the area of a square constructed with the hypotenuse on one side is equal to the sum of areas of the squares on the other two sides.

In that case, the word "square" refers to the area of a shape but the formula actually gives the relationship[ between lengths.

For other, more physical relationships you could always say
KE is the mass times the velocity and times the velocity again, divided by two.
This happens, also, to be half the momentum times the velocity. No 'squared' explicitly - but it's in there implicitly.

There are other quirky things in simple mathematical descriptions of the physical world. For instance, if you draw a Velocity/time graph, the distance traveled is equal to the area under the curve. i.e. a distance is represented by an area.
Or, when an object is stretched, and a stress / strain graph is drawn, the work done is also represented by an area.

No need to get wound up by it. It's just a consequence of our choice of the quantities we choose to describe things.
As Vicky Pollard would say: "get over it" ;-)

 

[tiny-tim]

hi prosteve037!

Is there a mathematical term for this "joint interdependence" of variables in an equation? (Like how mass and velocity must both exist in the kinetic energy equation)

In other words, a term for when you need to multiply two variables instead of adding them.

the term for when you add is "conservation" …

a conservation equation is essentially adding things, and always getting the same result

if the only thing that was conserved was momentum, then we wouldn't have enough information to solve any problem …

imagine that something with known velocity u hits a stationary object, and they move off with unknown velocities v and w …

just conserving u + v + w (or some other scalar multiple of them) isn't enough to find two unknowns, so we need an equation conserving u² + v² + w² (or some other scalar multiple of them)

why not u³ + v³ + w³ (or higher powers) also? because the first two conservation equations are enough to do the job!

 

[prosteve037]

hi prosteve037!


the term for when you add is "conservation" …

a conservation equation is essentially adding things, and always getting the same result

if the only thing that was conserved was momentum, then we wouldn't have enough information to solve any problem …

imagine that something with known velocity u hits a stationary object, and they move off with unknown velocities v and w …

just conserving u + v + w (or some other scalar multiple of them) isn't enough to find two unknowns, so we need an equation conserving u² + v² + w² (or some other scalar multiple of them)

why not u³ + v³ + w³ (or higher powers) also? because the first two conservation equations are enough to do the job!

Thank you tiny-tim!

I wonder if there are any books on the history of mathematical concepts such as this one (conservation).

If there is indeed a book/series of books that cover the development of these mathematical concepts, does anyone any good ones?