How are the formulas for potential and kinetic energy derived?

[Avalon_18]

While I was doing some practice questions it hit me "what is energy". 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.

 

[BvU]

Energy is a measure for work: if you do some work to move a book from the bottom shelf to the top shelf, the difference in potential energy of the book is the work you have done. Establishing a zero point for energy is awkward because in the real world all we have is energy differences.

 

[Merlin3189]

An interesting question to which I do not know the answer! My recollection is that I first thought about energy when learning about Joule and work-heat equivalence. But maybe I'd met KE before and not thought much about it.

If you do have a concept of KE, then PE seems obvious. When an object falls, it gains speed and hence KE due to the acceleration caused by gravity. So when an object is high up, it has the potential to gain KE by falling. Or if an object has speed, KE, then it has the potential to rise up against gravity.

Quantitatively the change in KE with height when moving in a gravitational field works out at mgh, so that becomes the PE.

 

[PeroK]

While I was doing some practice questions it hit me "what is energy". 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.

There are many videos online on the subject "what is energy?". Perhaps what they all have in common is that it is not, in general. easy to define. One simple approach is to leave it essentially undefined, but list the different forms it can take: kinetic, potential, chemical, thermal, nuclear, radiation.

 

[Dale]

While I was doing some practice questions it hit me "what is energy". 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.

There are several equally valid definitions used in different branches of physics. In Newtonian mechanics energy is defined as the capacity to do work. In thermodynamics an approach like @PeroK is used, energy is defined as kinetic energy ($\frac{1}{2}mv^2$) and anything that can be converted to kinetic energy. In Lagrangian mechanics it is defined as the conserved quantity associated with time translation symmetry of the Lagrangian.

 

[BeedS]

Do all forms of energy manifest as motion?

 

[PeroK]

Do all forms of energy manifest as motion?

Can you think of any that don't? See the list in post #4.

 

[BeedS]

Can you think of any that don't? See the list in post #4.

No
kinetic = Motion
potential = potential to move
chemical = Motion caused by interactions of atoms / particles
thermal = Motion of heat energy
nuclear = motion of particles and subatomic
radiation = Motion of photons

 

[PeroK]

No
kinetic = Motion
potential = potential to move
chemical = Motion caused by interactions of atoms / particles
thermal = Motion of heat energy
nuclear = motion of particles and subatomic
radiation = Motion of photons

I might question whether potential, chemical and nuclear are directly "motion".

 

[Dale]

Do all forms of energy manifest as motion?

No. KE and thermal energy of an ideal gas are motion. Thermal energy of other materials would have some of its energy in internal degrees of freedom corresponding to motion, but could also have non motion internal degrees of freedom if available. Most other forms of energy are not necessarily manifested as motion, although of course they can all be converted into KE. For example, electromagnetic energy can move (radiation) but you can also have static electromagnetic fields which hold energy without moving.

 

[jtbell]

While I was doing some practice questions it hit me "what is energy". 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.

Does your textbook say anything about this stuff? It would help if we can use it as a starting point for discussion. Is this a grade-school or university textbook? The university level textbooks that I've used all make at least some attempt at this.

 

[A.T.]

While I was doing some practice questions it hit me "what is energy".

A quantity that is conserved in an isolated system over time.

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.

All the formulas for energy where defined such that the total energy is conserved in an isolated system over time.

 

[jbriggs444]

Here is Feynman's explanation: http://www.feynmanlectures.caltech.edu/I_04.html

 

[BeedS]

although of course they can all be converted into KE

That is when work is being done, something is converted to KE. What work doesn't involve moving something?

For example, electromagnetic energy can move (radiation) but you can also have static electromagnetic fields which hold energy without moving.

No work is being done with the static field (PE), when work is done the field will move(KE) the charge or whatever is being moved/worked/energized?

 

[Dale]

What work doesn't involve moving something?

None. $dW=F\cdot dx$

No work is being done with the static field

Sure, but it is still energy. The question was “do all forms of energy manifest as motion” not “do all forms of work manifest as motion”. Work and energy are closely related, but they are not the same thing.

 

[BeedS]

Sure, but it is still energy.

Yes in a stable configuration.

The question was “do all forms of energy manifest as motion” not “do all forms of work manifest as motion”. Work and energy are closely related, but they are not the same thing.

So all forms of work manifest as motion? Judging by your "None. dW=F⋅dx" reply, then yes?
And I need to look into my understanding of work and energy.
Thanks

 

[Dale]

So all forms of work manifest as motion? Judging by your "None. dW=F⋅dx" reply, then yes?

Yes, the dx term is motion of the object on which work is done.

 

[Twodogs]

Good question. I think you should hang on to it for a while. We can measure energy with incredible precision. We can describe its permutations with strict accounting. I don't know that we can say 'what' it is for the time being.

 

[Dale]

I don't know that we can say 'what' it is for the time being.

Sure we can. I gave three definitions for what it is in post 5.

 

[Twodogs]

Sure we can. I gave three definitions for what it is in post 5.

truth -- the quality or state of being true.
• that which is true or in accordance with fact or reality.
• a fact or belief that is accepted as true.

Certainly, you give a workable definition. Regards.

 

[BeedS]

And I need to look into my understanding of work and energy.

Energy = current motion (Since everything is moving, energy is the current motion caused by previous work done)
Work = change of motion (result of changing a objects energy through KE)

Is this better?

 

[A.T.]

Energy = current motion

What about potential energy?

 

[lomidrevo]

What about potential energy?

And we should not forget about the rest energy $E = m c^2$. Quite a good example of a energy not associated with a particle's motion.

 

[BeedS]

What about potential energy?

The object with potential is still created by things moving internally(sum of the objects scaled energy(its mass)). PE is potential motion, the PE object is still moving (KE) and has a potential to change that motion through work done.

 

[A.T.]

the PE object is still moving (KE)

What about two opposite charges at rest? They have PE without moving.

 

[BeedS]

What about two opposite charges at rest? They have PE without moving.

At rest relative to what? Only to the object itself and something with the same motion as the object?. The charges are still moving with the charged objects. The charges are considered part of the objects mass (PE)? Charges do contain mass? Charges do move with the object they are bound to?

 

[jbriggs444]

At rest relative to what?

Yes, every object that is at rest in one frame is moving in another. It does not follow that energy is motion or that rest is a meaningless concept.

Edit: You do realize that energy is a frame-relative concept? If we are talking about energy being present in an object at rest, we are working in a frame of reference within which the energy is non-zero and and the object is at rest.

 

[A.T.]

At rest relative to what?

Relative to the reference frame in which you calculate the energy.

 

[folkethefat]

What is mass?
The dimension of time might be related to energy?

In E=MC2, what does it mean? The energy bound in mass?

 

[lomidrevo]

What is mass?

In relativity, it is a property of an object - a quantity which is invariant for all observers. But mass can be defined in different ways too: https://en.wikipedia.org/wiki/Mass

The dimension of time might be related to energy?

No idea what do you mean by your question.

In E=MC2, what does it mean? The energy bound in mass?

I think this short video provides a nice introductory explanation: