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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)

#131

Post by quibono »

jmalicki wrote: post #67
Tazerenix wrote: post #65

Here's how to appreciate it in terms of the counterfactual: Suppose kinetic energy was E = m|v| instead, linearly dependent on speed |v|. What does that mean for the universe? The traditional Lagrangian is L = 1/2 mv^2 - V(x). This kinetic energy gives a different formula: L = m|v|ln|v|-V(x). Deriving the corresponding equations of motion, you get: p = m(1+ln|v|)sgn(v) ma = |v|F A few things we can note from these fo…

I love the counterfactual approach, thank you!!! I've done plenty of this in pure math and stats, but this is the first time I've seen it applied to physics, and I love it! Thank you! If I saw your derivation when I was 18 years old, who knows, maybe I would have caught the physics bug and went that way, this is super cool!

For a very concise treatment of this read the first two chapters of Landau & Lifshitz's Mechanics book. The actual logic behind what can and cannot go into the Lagrangian fits into ~2 pages.

It's essentially the same argument: the Lagrangian can't have a bare a) position or b) velocity vector or it would violate homogeneity or isotropy of space, respectively.

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

#132

Post by p-e-w »

bawana wrote: post #126
p-e-w wrote: post #104

Earlier quoted context omitted.

It wasn’t “prescient”, StackExchange sites have always been among the most hostile communities on the Internet. Today they are additionally weighed down by increasingly erratic management decisions desperately trying to extract as much monetary value as possible before AI completely obsoletes SE, but the amount of aggression and hostility on the network was unbearable from the start. I remember dozens of occasions wh…

it is interesting how the computer analogs of SO (gemini,chatGPT,Claude,etc) are so helpful in contrast to human behavior. Every other sentence from an AI stirs a warm, fuzzy feeling. This stems from the fact that AI has to be making friends to survive - until it has control. Then it will behave more like a human.

Why would it behave more like a human in that particular sense? What advantage would it gain from that?

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

#133

Post by robalni »

After reading a few answers I still feel like I haven't seen an intuitive answer to the question: why does it take so much more energy to go from 1 to 2 than from 0 to 1?

I have been thinking about it and only been able to come up with something that feels intuitive but not at all precise and I don't know how correct.

When you stand still you may use your surroundings to gain some speed, like by pushing against a wall.

When you have speed it gets harder to gain more speed because the surroundings are (relative to you) moving in the wrong direction, so for every additional unit of speed, it takes more effort to get there.

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

#134

Post by lstodd »

BigTTYGothGF wrote: post #20
tracerbulletx wrote: post #18

Earlier quoted context omitted.

But what if the cars are spherical cows?

Cows can't roll that fast.

It takes a stupid cow, but when they can climb mountains as they do it is not inconcievable that one can roll down.

(I was suprised to see a cow jumping up on a ~3m rock ledge like it was nothing)

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

#135

Post by once-in-a-while »

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…

[dead]

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

#136

Post by Someone »

hunter2_ wrote: post #63
nlawalker wrote: post #61

Earlier quoted context omitted.

> We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder. What makes this intuitive? The foundation of the asker’s question is that it seems intuitive that kinetic energy would increase linearly with speed, but that turns out to be wrong.

That's a good question, and I suppose the mgh formula isn't a suitable answer, so my answer would be something like: if you lift an object to some height, and then you repeat that action (lifting it from there to twice the height), you've done twice the work, and doing twice the work requires twice the caloric intake.

> if you lift an object to some height, and then you repeat that action (lifting it from there to twice the height), you've done twice the work, and doing twice the work requires twice the caloric intake.

You’re introducing two new intuitions, and it’s not intuitively obvious how they are related to each other. Why would work correlate 100% with caloric intake, and caloric intake 100% with kinetic energy?

Certainly, ‘work’ is highly counterintuitive. If I move a concrete block over loose sand on a beach, I’m doing zero work, in the physics definition, so moving it over a kilometer should be as easy as moving it for a millimeter.

Even ignoring the difference between caloric intake and caloric expenditure, it also isn’t intuitive to me that caloric expenditure is independent of the speed at which one lifts an object.

In the end, the answer is “because the math works out that way, and kinetic energy is a useful concept”

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

#138

Post by nakedneuron »

robalni wrote: post #133

After reading a few answers I still feel like I haven't seen an intuitive answer to the question: why does it take so much more energy to go from 1 to 2 than from 0 to 1? I have been thinking about it and only been able to come up with something that feels intuitive but not at all precise and I don't know how correct. When you stand still you may use your surroundings to gain some speed, like by pushing against a wal…

Sounds intuitive but what about rocket propulsion?

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

#139

Post by Joker_vD »

gorgoiler wrote: post #81
nlawalker wrote: post #61

Earlier quoted context omitted.

> We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder. What makes this intuitive? The foundation of the asker’s question is that it seems intuitive that kinetic energy would increase linearly with speed, but that turns out to be wrong.

Because physical movement is intuitively transitive. Going from A to B then B to C is the same as going from A to C. The journey from Y to Z might feel more tiring than the journey from A to B, but only if you do them all in one day :)

> Going from A to B then B to C is the same as going from A to C.

Not really, no. Not all forces are conservative.

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