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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)

#61

Post by nlawalker »

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.

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.

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

#63

Post by hunter2_ »

nlawalker wrote: post #61
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. 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.

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

#64

Post by cubic_earth »

card_zero wrote: post #60
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…

Nice. Nitpick: in the middle paragraph you put "speed 10" instead of 100.

Fixed. Thanks.

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

#65

Post by Tazerenix »

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 formulas:

1. They are not boost invariant: Galilean relativity is violated. That means there is necessarily a privileged reference frame (i.e. an aether) in which the universe is at rest, and all dynamics must be understood relative to this reference frame.

2. Newton's first law has a pathological interpretation in regards to the above reference frame: If ma = |v|F and |v| = 0 (i.e. you are at rest relative to the aether), then a = 0 no matter what F is. That is, for objects which are stationary with respect to the aether, no motion is possible regardless of what force is applied.

It is still true that objects in motion (relative to the aether) remain in motion unless acted upon by an outside force, and Newton's third law is still true, but such a universe basically makes no sense.

You could essentially argue from the anthropic principle that such a universe would have such pathological dynamics that it could not permit life, and therefore we cannot observe it.

This is the contrapositive of the argument presented on stackexchange. There they say "given Galilean relativity, you get the quadratic scaling law". This argument says "if you don't have the quadratic scaling law, you don't have relativity".

The point of the counterfactual is a bit like Richard Feynman's "why" argument [1]. There is no fundamental reason why this kind of dynamics couldn't exist. We can only ever reduce our explanation to a more fundamental intuition we have about the same universe we live in (i.e. from kinetic energy scaling laws to Galilean relativity). But without a mathematical proof of the incoherence even in principle of the alternative, its perfectly valid to imagine an alternative universe with different dynamics. It's just not our universe.

[1] https://www.youtube.com/watch?v=36GT2zI8lVA

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

#67

Post by jmalicki »

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!

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

#68

Post by itemize123 »

c1ccccc1 wrote: post #53
drivebyhooting wrote: post #44

I don’t find the answer convincing. It assumes one can measure heat at a distance and it is a conserved quantity between reference frames. Energy is actually not a conserved quantity in Galilean relativity.

Energy is conserved in Galilean relativity. The thing you're trying to say is that it's not invariant across reference frames. The answer linked above actually takes advantage of the fact that energy is not the same in different reference frames in order to make the argument work. I think you are overthinking the heat thing. If you have a train car full of hot water and you slow the train down (extracting kinetic ene…

thinking aloud here - so it seems like 2 things are taken as intuitive here:

a) energy is conserved in any frame of reference. b) energy can vary in 2 frame of references.

but then what it feels like is that when you reference the energy as mE(v), the v is actually not the only variable, and it will be more like mE(v, v_moving_reference)?

so we also must take intuitive that c) E(v, v_moving_reference) == E(v - v_moving_reference)

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

#69

Post by throwawaytea »

nlawalker wrote: post #61
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. 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 things like energy are relative. So if you label the ground 0, and go up 10 feet, you get x energy. Going up another exact same x from your 10 foot ladder spot you could now call 0 again, would mean you gain x energy again. Since they're both the same height, and you gained the same energy, you could infer double the height has double energy.
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