Understanding the Concepts and Formulas for Energy: Potential and Kinetic

[BvU]

I also don't understand how the formulas for potential and kinetic energy were derived. Was it due to their definitions or is there some reason behind their respective formulas.

Perhaps it's nice to add that in real life the concept of 'force' is much more vivid. Force has something to do with to the change in energy

Some force fields are conservative, hence the introduction of a potential (energy) with somehing like $\Delta {\rm PE} = \vec F \cdot \Delta \vec x\$.
From $$ \vec F = {\Delta\vec p\over \Delta t} \Rightarrow \quad \vec F \cdot \Delta \vec x= m{\Delta\vec v\over \Delta t} \cdot \Delta\vec x = m\;\Delta\vec v \cdot {\Delta\vec x\over \Delta t} = m\;\Delta\vec v \cdot \vec v = \Delta \left ( {1\over 2 }m \vec v^2 \right ) $$ comes the formula for kinetic energy.

It's all differentiation and integration -- which makes math so useful !

[edit] hehe if you do it right

 

[folkethefat]

I do not agree. An "event" is simply a location in four dimensional space-time. It does not involve a change in energy. How could it, since "energy" is a conserved quantity?

In particular, energy is the conserved quantity associated with time translation symmetry by Noether's theorem.

Ok! i see, i was confused about the definition of "event" in a physics context. I thougt it implied a change in the energycontent in a location in space, a defined volume.

 

[folkethefat]

There's no definition that will help when the concept is being refined and used in different contexts. After a while the different contexts provide you with enough experience that you feel comfortable with the concept of energy. Like how we feel comfortable with the concept of money. /

But to ask what "money is" except that its nice to have lots of, , is a legitimate question and there are many theoretical answers for it.
The question about if energy is something more fundamental than the different types of energy, leads beyond the standard model and that's the problem. I Think?

 

[sophiecentaur]

I do not agree. An "event" is simply a location in four dimensional space-time. It does not involve a change in energy.

Strictly, yes perhaps but an 'event' is a word used colloquially to describe 'something happening' and that does imply change because that sort of event has to be observable - which involves some change of Energy state.
I used to use the form of words "Energy is needed for anything to happen" in lower school introductions to Science. It was interesting to note that 'they' would always try to argue with that and to bring up examples that didn't fit. I always managed to explain away their examples. 'Death' would always be one of them.

 

[folkethefat]

Can all sorts/types of energy translate into some other kind? Heat for example.
Is there a hierarchy in this "translation"? Is heat the lowest form or state of energy?

 

[sophiecentaur]

Can all sorts/types of energy translate into some other kind? Heat for example.
Is there a hierarchy in this "translation"? Is heat the lowest form or state of energy?

We could turn this thread into a very practical discussion about Energy - which is quite valid, I think.
Heat Engines (the general term for a device to obtain mechanical work from the flow of heat) work best (higher efficiency) between large temperature differences. Once the work has been 'extracted' from a mass of (say) steam and the steam has condensed, you can sill get some work out of it by using a cold sink at an even lower temperature than 100C BUT NOT MUCH. So you could say that the energy in the condensed steam is a 'lower grade' than that of the superheated steam that is supplied by the original boiler.
The hot water would still be 'useful' for heating but not so much for doing work.
It's basically down hill all the way, though.
Your suggested "hierarchy" is ok as an arm waving term and if you want to talk in terms of relative usefulness. Considerations like possibilities of storing for the various forms of energy are very relevant sometimes. You could perhaps put Mechanical Potential Energy at the top because you can store large amounts of water at height or use a wound spring to store useful energy.

 

[A.T.]

Can all sorts/types of energy translate into some other kind?

Transform spontaneously or be transformed by doing work? Look up:
https://en.wikipedia.org/wiki/Second_law_of_thermodynamics

 

[Merlin3189]

If you think of energy as an accounting term, then all types of energy are equivalent. The energy simply tells you the conversion value, not whether and how a conversion process happens.
You suggest heat may be a low form of energy, but in a power station we use it to generate electricity, a more noble form. We can use that to drive motors and raise objects to give them potential energy, to charge batteries and give them chemical potential energy, produce light and other electromagnetic energy. I don't know if we can put nuclear energy back in the bottle, but I guess if we could, it might involve using large amounts of electrical energy in particle accelerators.

The thing about heat is that it is the kinetic energy of random motions of molecules, lots of them. If two molecules collide, it is normally perfectly elastic, energy is conserved and energy can be transferred from the slower to the faster or vice versa. But it is more likely for energy to be transferred from the faster to the slower. Statistically, when there are large numbers of molecules, energy gets shared out. Energy could not be concentrated in a small number of very fast molecules, because they would very quickly collide with the large number of very slow ones and lose some energy. These ideas lead us down the road of entropy and thermodynamics. Heat doesn't travel from a colder to a hotter (by conduction.) That is probably what you have in mind.

 

[stevieboy]

energy = mass x the square of the fastest velocity in the universe
the square of something is 2 dimensional
times a mass which is described as the 3rd dimension pretruding down from a 2d platform
so is energy not basically a cube or volume?
do we live in the 2d world?

 

[Dale]

is energy not basically a cube or volume?

In geometrized units speed is dimensionless, and the square of speed is therefore also dimensionless. So energy has geometrized dimensions of length, not length cubed.

 

[danachiraq]

Derivation for the equation of Potenstial Energy:
Let the work done on the object against gravity = W
Work done, W = force × displacement
Work done, W = mg × h
Work done, W = mgh

Since workdone on the object is equal to mgh, an energy equal to mgh units is gained by the object . This is the potential energy (Ep) of the object.
Ep = mgh

Derivation for the equation of Kinetic Energy:
The relation connecting the initial velocity (u) and final velocity (v) of an object moving with a uniform acceleration a, and the displacement, S is
v2 - u2 = 2aS
This gives
S = v 2 - u 2 2a

We know F = ma. Thus using above equations, we can write the workdone by the force, F as

W = ma × v 2 - u 2 2a
or
W = 1 2 m( v 2 - u 2 )If object is starting from its stationary position, that is, u = 0, then
W = 1 2 m v 2

It is clear that the work done is equal to the change in the kinetic energy of an object.

If u = 0, the work done will be W = 1 2 m v 2 .

Thus, the kinetic energy possessed by an object of mass, m and moving with a uniform velocity, v is Ek = ½ mv2