Why is velocity squared in the equation for kinetic energy?

[RJ Emery]

I believe it was Newton who first proposed kinetic energy as being proportional to mass times velocity. Liebniz further refined the relationship by stating that the velocity should be squared. That was later confirmed by Willem 'sGravesande and Emilie du Châtelet. The squaring also is a basis for E=mc².

My question is: why squared?

I don't deny the correctness of the two relationships, but if fitting an equation to data, I would be both suspicious and intrigued by the fact that the exponent is exactly two. Not 0.976233 or 2.343983 or 3.599992 but exactly 2.000000. If one looks at other equations in physics, things are not so simple or elegant.

What is so unique about squaring here? Where does it come from?

 

[robphy]

Kinetic Energy arises from the work-[kinetic]energy theorem.
I describe it to my students as some interesting quantity that appears for the initial state and for the final state when computing the work done by the net force.

 

[russ_watters]

You can derive it easily enough from Newton's laws of motion. Since work is force times distance and acceleration is force over mass, applying a force causes a quadratic increase in work with time.

Please bear in mind that the squaring isn't like multiplying by a constant - it would be incredibly unusual for velocity to be raised to a 2.3 power or something like that.

I don't have a reference, but I don't think you are right about Newton's derivation of energy. It isn't some arbitrary thing and if he used the same definition (the word follows the mathematical definition, not the other way around), there is no good reason why Newton would have gotten it wrong. The difference/relationship between momentum and energy (and the rest of the pieces of Newtonian physics) are all tied together and all relatively straightforward.

 

[RJ Emery]

Kinetic Energy arises from the work-[kinetic]energy theorem.
I describe it to my students as some interesting quantity that appears for the initial state and for the final state when computing the work done by the net force.

My lack of knowledge here can be traced to learning algebra-based physics. Can you suggest a comprehensive calculus-based physics textbook, one that would include an introduction to relativity and quantum theory and whatever else comprises modern physics?

If not in the calculus-based text itself, where could I find the derivation of the algebraic formulae used in first and second year physics?

 

[christianjb]

It's a good question.

Why do some parts of physical law apparently involve integer arithmetic? It's not obvious why we don't live in 3.2 spatial dimensions or that some future theory of energy couldn't produce the formula E= X Y^2.0000023.

I was always intrigued that exactly 6 circles (in a hexagon) fit around an inner circle. Yes, that can be proved from basic axioms in geometry, but it's still philosophically interesting. (We may think it's trivial in 2D, but what about the 87 dimensional case?)

We could all do to question these things from time to time. Think of the integers that go into your favorite theories, and ask yourself- how do we really know that these are integers? Is there any way of making sense of the equation if we used 2.00001? etc.

We used to think that dimensions were integers- now we've got fractal dimensions which can have non-integer values along a continuum. In some sense the spatial dimension is now regarded as a variable in some situations.

We used to think that energy, charge and light were continuous. Now we know that they come in discrete packets.

 

[HallsofIvy]

I would also point out that the "square" in mv² is necessary to make the units match. "Force" has units of "mass times distance divided by time-squared" or kg m/s² in kms (from F= ma). "Work-Energy" is defined as Force times Distance so it has units of "(mass times distance divided by time-squared)(Distance)= mass time (distance divided by time)squared: (kg m/s²)(m)= kg (m/s)².

 

[christianjb]

I would also point out that the "square" in mv² is necessary to make the units match. "Force" has units of "mass times distance divided by time-squared" or kg m/s² in kms (from F= ma). "Work-Energy" is defined as Force times Distance so it has units of "(mass times distance divided by time-squared)(Distance)= mass time (distance divided by time)squared: (kg m/s²)(m)= kg (m/s)².

That's a good point. Any conceivable eqn with energy on the LHS has to have an expression involving units ML^2T-2 on the RHS.

I think similar dimensional analyses are responsible for much of the integer powers we see in physics equations.

But, being especially perverse, why is it that energy should have dimensions involving integer powers to begin with? Are there any meaningful physical equations involving non dimensional powers of units?

 

[RJ Emery]

I would also point out that the "square" in mv² is necessary to make the units match. "Force" has units of "mass times distance divided by time-squared" or kg m/s² in kms (from F= ma). "Work-Energy" is defined as Force times Distance so it has units of "(mass times distance divided by time-squared)(Distance)= mass time (distance divided by time)squared: (kg m/s²)(m)= kg (m/s)².

At the time Newton formulated mv, the left hand side was considered to be "energy," as force as we now know it came later. It is not that the units of the LHS ultimately dictated what the RHS needed to be, rather it was the RHS dictating what "energy" becoming force would be.

 

[russ_watters]

Do you have a reference for that, RJ?

 

[RJ Emery]

Do you have a reference for that, RJ?

It should be in David Bodanis' E=mc²: A Biography of the World's Most Famous Equation, 2005. I would have to re-borrow the book from my local public library to find the relevant pages. In any event, it was the experiments of Willem 'sGravesande in Holland who proved Newton's assertion to be incomplete. 'sGravesande's work was later confirmed and extended by Emilie du Châtelet. Leibniz said the relationship should be mv², and it was du Châtelet who confirmed that.

When were these experiments? I do not recall. It would be interesting to look at Newton's original Principia Mathematica and not a corrected version to see what Newton originally wrote if the dates are otherwise in agreement.

We do get into the Newton vs Leibniz entanglement here. Newton may have discovered a lot before others, but he often did not publish his findings.

 

[russ_watters]

In any case, this is a pretty nonsensical discussion. It is a little like seeing the statistic that the average family has 2.4 children and asking why no family actually has 2.4 children (or why doesn't the Earth have 1.2 moons?). You're asking why a velocity times a velocity produces a velocity squared.

You're asking if there is some magic in the math without actually looking at the math. In your first post, you mention curve fitting - these equations were derived, they were not found by curve fitting. And even if they had been, the scientists who found them wouldn't have thought it a coincidence, they would have looked for a mathematical reason for it...and easily found it.

Work is force times distance. It is useful to look at work from the other side and see how much work it takes to accelerate a mass to a certain speed (or view that quantity as a property of the moving object). The kinetic energy equation can easily (algebra only, no calculus required) be found using only f=ma, the definitions of speed and distance, and a little logic.

 

[RJ Emery]

In any case, this is a pretty nonsensical discussion. ...

Thanks for your help. You can move on now.

 

[arildno]

Leibniz used a concept of "vis viva" that was twice our kinetic energy.

It was some discussion whether the stopping length of an object dropped inta some soft material like putty ought to be proportional to the velocity by which the object hit the material (i.e, proportional to impact strength), or whether it should be thought proportional to the square of the velocity 8i.e proportional to energy).

For the materials examined, energy proportionality was seen to give the best fit.


This is in accordance with the idea that the resistive force from the material is roughly constant throughout the material, an insight that is by no means easy to establish otherwise.

 

[Sean Torrebadel]

First, Newton described mv as momentia or something like that in his principia. Galileo was the one who determined that momentum, mv, was conserved. E=mc squared has to do with the transformation of matter into energy. Kinetic energy is 1/2 mv^2 and not mv^2 and is determined using simple dynamics

 

[Andrew Mason]

Leibniz used a concept of "vis viva" that was twice our kinetic energy.

It was some discussion whether the stopping length of an object dropped inta some soft material like putty ought to be proportional to the velocity by which the object hit the material (i.e, proportional to impact strength), or whether it should be thought proportional to the square of the velocity 8i.e proportional to energy).

The concept of energy evolved. It was developed in the 19th century when physicists began to realize the relationship between work and heat. We tend to overlook this kind of thing because we are taught physics in high school but not the history. The http://www.physics.odu.edu/~kuhn/PHYS101/VisViva.html" is a very interesting part of physics history.

AM

 

[russ_watters]

Thanks for your help. You can move on now.

If you make an attempt to learn (perhaps by reading and understanding my post...) why these things are the way they are, you'll see it, understand it, and thank me sincerely...

There is only so much I can do here, though. I've tried to push you toward looking at the math, but I'm sure you haven't, which is why you are making snide remarks instead of understanding that the question is exactly equivalent to "why does $$V*V = V^2$$ instead of $$V$$^1.4?" Well, here's the math:

$$w=f*d$$ <- definition of work
$$d=V*t$$
$$V=a*t/2$$ (for an object under a constant acceleration force)
$$f=m*a$$

Therefore, by substitution:
$$w=1/2 m*a*t*a*t$$
or, since $$a*t=V$$...
$$w=1/2 m*V^2$$

So, last time: There is no coincidence here. Just simple math.

 

[russ_watters]

The concept of energy evolved. It was developed in the 19th century when physicists began to realize the relationship between work and heat. We tend to overlook this kind of thing because we are taught physics in high school but not the history. The http://www.physics.odu.edu/~kuhn/PHYS101/VisViva.html" is a very interesting part of physics history.

AM

That is a very interesting article and I think you for it (I am interested in the history of science), but I do want to point out that the trials and tribulations of the people who invented/discovered these theories isn't terribly relevant to how they eventually fell into place. It strikes me from the article that the basic problem at the time was more about conventions in terminology and expression of the ideas than about understanding the ideas themselves. Bringing me back to the OP, it doesn't matter what words Newton used to describe it, it seems pretty clear to me from the article that Newton (and everyone else involved) understood the significance of the square function of velocity (and of time in acceleration). It needs to be remembered that it is realizing the significance of this and the connection between the equations of motion that led to the discovery of calculus. And now that these associations are understood, it becomes counterproductive to try to understand the theories by examining the historical confusion of those who created them. We're past it.

 

[Izzhov]

If you make an attempt to learn (perhaps by reading and understanding my post...) why these things are the way they are, you'll see it, understand it, and thank me sincerely...

There is only so much I can do here, though. I've tried to push you toward looking at the math, but I'm sure you haven't, which is why you are making snide remarks instead of understanding that the question is exactly equivalent to "why does $$V*V = V^2$$ instead of $$V$$^1.4?" Well, here's the math:

$$w=f*d$$ <- definition of work
$$d=V*t$$
$$V=a*t/2$$ (for an object under a constant acceleration force)
$$f=m*a$$

Therefore, by substitution:
$$w=1/2 m*a*t*a*t$$
or, since $$a*t=V$$...
$$w=1/2 m*V^2$$

So, last time: There is no coincidence here. Just simple math.

You're kind of missing the point of this discussion. It's a philosophical one, not a mathematical one.

RJ wasn't asking why $$v \times v = v^2$$. He was asking why the universe works out so neatly that you end up multiplying v by itself in the first place. If we lived in a universe of 3.2 spatial dimensions or something, the math would probably work out much differently, and not be as clean. You're essentially answering his question by saying, "It is the way it is because when physicists observed the universe they saw it was this way, so they did some math and this is what happened," which wasn't really what he was asking, I think.

So, in effect, it is kind of a coincidence that the math works out so clean, if you consider the way the universe was formed to be a coincidence.

 

[russ_watters]

This is a physics forum, not a philosophy forum and you can't have 3.2 spatial dimensions. That's nonsensical. This is a simple mathematical issue and can't be turned into philosophy just by arguing about it.

 

[christianjb]

This is a physics forum, not a philosophy forum and you can't have 3.2 spatial dimensions. That's nonsensical. This is a simple mathematical issue and can't be turned into philosophy just by arguing about it.

That's a little strong and aggressive.

You can have non-integer dimensions in fractal theory. There may be other theories in which non-integer dimensions exist. I don't know, but I wouldn't call the speculation 'pointless'.

 

[Izzhov]

This is a physics forum, not a philosophy forum and you can't have 3.2 spatial dimensions. That's nonsensical. This is a simple mathematical issue and can't be turned into philosophy just by arguing about it.

Also, this wasn't really an argument before you turned it into one. It was a discussion.