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Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

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Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#201

Post by thfuran »

dotancohen wrote: post #191
thfuran wrote: post #178

Earlier quoted context omitted.

Holding that block stationary at arms length then. 0 work.

Put that block on a shelf at the same height. 0 work. The fact that your muscles burn ATP just fighting gravity is a feature of biology, not fundamental to the physics involved. If you want to read about another similar example, Google for rocket launch gravity losses.

I think you're missing the point. A lot of basic mechanics actually isn't especially intuitive because things like work simply do not map well to everyday experience. I'm not suggesting that work is defined incorrectly or something.

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#202

Post by dragonwriter »

PunchyHamster wrote: post #164
cubic_earth wrote: post #55

It's easiest to visualize in terms of conversion from potential energy. We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder. And we also know when they fall, by the time they reach the ground and all the potential energy has been converted to kinetic energy, the previously higher ball will have twice the kinetic energy too. But a twice higher ball won't have…

> We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder. ...no ? dropping something 10 times from 1ft is nowhere near energetic/damaging as once from 10tf

That might seem an attractive intuition based on observing damage and having an understandable intuition of how damage should relate to energy, its wrong.

Energy is force × distance. For gravitational potential (and, equivalently, the kinetic energy acquired when that is tranformed into kinetic energy by a fall), the force is the force of gravitational attraction between the body and the object it is falling toward (the object’s weight) and the distance is the distance of the (actual or potential, as appropriate) fall.

The nonlinearity you observe in damage is because damage is a complicated outcome that depends on a lot more than aggregate energy involved.

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#203

Post by SilasX »

drabbiticus wrote: post #146
SilasX wrote: post #89

Earlier quoted context omitted.

Okay but that depends on the intuitions the question is trying to justify, which makes it circular. We also know, for example, that the body uses more than twice as much energy to do twice as much work (because of fatigue on the muscles or whatever the right term is here). In fact it takes positive energy just told a weight at a fixed height, doing zero mechanical work! So you’re actually appealing to even weaker int…

> In fact it takes positive energy just told a weight at a fixed height, doing zero mechanical work! Stacking a weight on top of a table holds it at a fixed height and requires zero mechanical work. The failure in intuition here relates to physiology and the mechanism by which muscles work, not physics. Myosin and actin are constantly cycling through bonding and release during muscle contraction, as this is how the s…

I get that involving a human body complicates the analysis. That was the point: that you can’t appeal to it as a simple example to ground the intuition in other case.

Also:

>Stacking a weight on top of a table holds it at a fixed height and requires zero mechanical work.

I was referring to a human holding it. What would have been a better way to keep you from missing that?

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#204

Post by SilasX »

hibernator149 wrote: post #91
SilasX wrote: post #89

Earlier quoted context omitted.

Okay but that depends on the intuitions the question is trying to justify, which makes it circular. We also know, for example, that the body uses more than twice as much energy to do twice as much work (because of fatigue on the muscles or whatever the right term is here). In fact it takes positive energy just told a weight at a fixed height, doing zero mechanical work! So you’re actually appealing to even weaker int…

All intuitions are wrong, but some are usefull. You have to do the experiment, discover the formula, and then adapt your intuitions accordingly.

What point that I made are you responding to? I was disputing someone’s appeal to a specific intuition for being an unhelpful one to use here. So I obviously get the concept of some intuitions being useful. Did you see the comment being responded to?

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#205

Post by acchow »

cubic_earth wrote: post #55

It's easiest to visualize in terms of conversion from potential energy. We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder. And we also know when they fall, by the time they reach the ground and all the potential energy has been converted to kinetic energy, the previously higher ball will have twice the kinetic energy too. But a twice higher ball won't have…

> We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder.

This assumes that energy increases linearly with distance (given constant force over that distance

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#206

Post by drabbiticus »

SilasX wrote: post #203
drabbiticus wrote: post #146

Earlier quoted context omitted.

> In fact it takes positive energy just told a weight at a fixed height, doing zero mechanical work! Stacking a weight on top of a table holds it at a fixed height and requires zero mechanical work. The failure in intuition here relates to physiology and the mechanism by which muscles work, not physics. Myosin and actin are constantly cycling through bonding and release during muscle contraction, as this is how the s…

I get that involving a human body complicates the analysis. That was the point: that you can’t appeal to it as a simple example to ground the intuition in other case. Also: >Stacking a weight on top of a table holds it at a fixed height and requires zero mechanical work. I was referring to a human holding it. What would have been a better way to keep you from missing that?

> I get that involving a human body complicates the analysis. That was the point: that you can’t appeal to it as a simple example to ground the intuition in other case.

Yeah, fair enough. It's unfortunate that the comment you were responding to involved "caloric intake" suggestive of a biological system when it could just as easily have involved a mechanical pulley. Their intuition would have been stronger phrased as: Within a gravitational field that is approximated as a uniform force field, the amount of energy/work to raise a weight by a fixed distance is independent of the initial position of that weight on a (massless) rope attached to a (frictionless) pulley.

> I was referring to a human holding it. What would have been a better way to keep you from missing that?

Since you are asking, for me, it would helped to have the following inserted text: "In fact it takes positive energy [for biological muscle] just told a weight at a fixed height, doing zero mechanical work!" so that the statement stood on it's own, or else including within your post a more definitive statement to the effect of "intuitions involving the human body complicates the analysis", as you did in this reply. If "that was the point", go ahead and state it.

To be fair, I often have the same problem. It's easy for me to write a lot of text that goes around a statement without actually getting to it.

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#207

Post by Cleonis »

davidivadavid wrote: post #14
digdugdirk wrote: post #9

Earlier quoted context omitted.

It seems that we're exact opposites! But if mathematics is your thing, it might be interesting for you to explore trying to learn things from a lagrangian perspective first? Not sure if it'll help you with gaining an intuitive understanding, but at least it'll be interesting! https://en.wikipedia.org/wiki/Lagrangian_mechanics

Lagrangian / Hamiltonian mechanics, the principle of least action, always seemed neat, in L&L and other places I encountered it, until I tried doing exactly what you're saying: gaining an intuitive understanding. At that point it just never made sense to me and seemed like a gratuitous deus ex machina that happens to work beautifully but for no apparent reason. You won't be surprised to learn I dropped out of my STEM…

About the stationary action concept: Yeah, it looks impenetrable, but here's the thing: there is a way of looking at it from just the right angle, and then becomes transparent.

Part of the story is this: the actual criterion is: the true trajectory corresponds to a point in variation space where the derivative of the action (derivative wrt applied variation) is zero.

In the cases examined when the concept was first introduced I suppose that in those cases the derivative-is-zero point was seen to be a minimum. From there, I suppose, came a supposition that there was some form of minimization at play.

However, within the scope of classical mechanics there are also classes of cases such that at the point in variation space corresponding to the true trajectory the action is at a maximum.

The above, and other aspects, are discussed in a resource that I created.

https://cleonis.nl/physics/phys256/energy_position_equation....

In the resource the mathematics is illustrated with interactive diagrams. Move sliders to sweep out variation. The diagram shows how the kinetic energy and the potential energy respond.

About interpretation: As we know: motion along the true trajectory has the property that at every point in time the rate of change of kinetic energy matches the rate of change of potential energy. As we know: that property is known as the work-energy theorem.

The criterion derivative-wrt-variation-is-zero corresponds mathematically to the property: rate-of-change-of-kinetic-energy-matches-the-rate-of-change-of-potential-energy.

In the resource a two stage process is presented:

- Derivation of the work-energy theorem from F=ma

- Transformation from the work-energy theorem to classical mechanics stationary action

Of course: when you look at the work-energy theorem you wouldn't expect that it can be transformed to classical mechanics stationary action. The transformation consists of multiple steps. In the resource I present it step by step; for each step the logic and consistency is readily recognizable.

For me, having the breakdown into mathematical elements available changed my whole perspective on classical mechanics stationary action.

I hope I can persuade you to check out the resource

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#208

Post by krackers »

electricwallaby wrote: post #154

It helps to re-frame the premise. An object which has a constant force applied will have it's distance increase quadratically with respect to time. Energy is force times distance. Intuition: the energy it takes to lift an object up is proportional to the height you lift it to. So if you apply a constant force, you get a constant acceleration which leads to a quadratically increasing distance. If you accept that energ…

>Energy is force times distance

This is not true though, work is only necessary to change KE. An object can have kinetic energy even when no forces are presently acting on it (of course force was needed to bring it from resting to that state though)

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#209

Post by gorfian_robot »

AngryData wrote: post #181
gorfian_robot wrote: post #88

Earlier quoted context omitted.

have you ever had to split wood?

Ive split wood by hand my entire life to use for heat. Have you? A practiced arm with an axe beats a maul any day of the week. That's why splitting mauls are a modern device and splitting axes have existed since forever. Plenty of information on it online and on youtube, and why there are dozens of expensive specialty handmade splitting axes to buy and just cheap mauls for the rest. Also this post is the physics behi…

sure have and I will take a maul over an axe. an axe is for felling not splitting. generally prefer a wedge and a sledge for splitting. gotta say some of the little electric splitters look sweet.

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#210

Post by cubic_earth »

ThoAppelsin wrote: post #190
mistercow wrote: post #182

Earlier quoted context omitted.

If you're talking about intuitions, you have no firsthand intuitions about lifting effort decreasing with distance to the Earth. We can intuit about constant gravity, and the math of constant gravity works fine for this description. And while the real situation at scale is more complicated, the math is going to come out to the same answer, albeit with extra terms muddying everything up. If someone says that something…

I’m not talking about intuitions; I’m talking against them. The intuition about carrying something 1st to 2nd then to 3rd floor is clearly wrong as evidenced by the example I gave; it is less wrong in smaller scales, but it still is wrong. If the floors were as high as the radius of the Earth, the first one would be three times as hard as the second one. The math doesn’t come out the same. It’s not at all linear, it’…

Simplifying assumptions are allowed when reasoning about physics. What you are saying is interesting, but I think you might be misapplying it. For small heights differences, the difference in gravitation approaches 0. A ball raised .0002 mm off the ground has twice the potential energy of one raised only .0001 mm. But that is never true for speed. An object moving 0.0002 MPH does have 4x the kinetic energy of an object moving 0.0001 MPH.

So yeah, I am zooming in on an exponential and calling it linear, but that doesn't work for speed. And doubling of velocity gives 4x the energy regardless of how tiny the step.

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